Introduction

Viper is a verification infrastructure that simplifies the development of program verifiers and facilitates rapid prototyping of verification techniques and tools. In contrast to similar infrastructures such as Boogie and Why3, Viper has strong support for permission logics such as separation logic and implicit dynamic frames. It supports permissions natively and uses them to express ownership of heap locations, which is useful to reason about heap-manipulating programs and thread interactions in concurrent software.

The Viper infrastructure, shown above, provides an intermediate language as well as two verification back-ends, one based on symbolic execution and one based on verification condition generation. Both back-ends ultimately use the SMT solver Z3 to discharge proof obligations. Front-end tools translate different source languages and verification approaches into the Viper language and thereby benefit from its tool support and automation.

The Viper intermediate language is a simple sequential, object-based, imperative programming language. Even though it has been designed as an intermediate language, the Viper language supports both high-level features that are convenient to express verification problems manually as well as powerful low-level features that are useful to encode source languages and verification techniques automatically.

The following simple example shows a method that computes the sum of the first n positive natural numbers. The method declaration includes a precondition (the assertion following requires) to restrict the argument to non-negative values. This expresses the fact that method sum can only be assumed to behave correctly (in particular, do not crash) for non-negative values. The method declaration also includes a postcondition (the assertion following ensures) to guarantee properties of the result variable res. The postcondition is the only information that a caller of the method sum can use to correlate the call result and argument. In particular, in absence of post-condition a caller of method sum cannot make use of the fact that the result is non-negative. This would be required, for example, in order to be able to call sum on its own result. Method preconditions and postconditions together make up a method’s specification.

Viper verifies partial correctness of program statements; that is, verification guarantees that if a program state is reached, then the properties specified at that program state are guaranteed to hold. For example, the postcondition of sum is guaranteed to hold whenever a call to sum terminates. Verification of loops also requires specification: the loop in sum‘s body needs a loop invariant (if omitted, the default loop invariant is true, which is typically not strong enough to prove interesting properties of the program). The loop invariant in sum could also be written in one line with the boolean operator && placed between the two assertions.

method sum(n: Int) returns (res: Int)
  requires 0 <= n
  ensures  res == n * (n + 1) / 2
{
  res := 0
  var i: Int := 0;
  while(i <= n)
    invariant i <= (n + 1)
    invariant res == (i - 1) * i / 2
  {
    res := res + i
    i := i + 1
  }
}

This tutorial gives an overview of the features of the Viper language and explains their syntax and semantics. We provide examples and exercises throughout, to illustrate the basic usage of these features. We encourage readers to experiment with the examples and often suggest variations of the presented examples to try out. The tutorial does not aim to explain the workings of the Viper verifiers in general, nor the advanced usage of Viper’s language features for building custom verification tools: for these topics, we refer interested readers to our Viper-related research papers.

If you have comments, questions or feedback about Viper, including this tutorial, we would be happy to receive them! Please send your emails to viper@inf.ethz.ch

Structure of Viper Programs

For a type safe Viper program to be correct, all methods and functions in it must be successfully verified against their specifications. The implementation of a Viper method consists of certain imperative building blocks (such as branch conditions, loops, etc.) whereas the specification consists of assertions (contracts, loop invariants, etc.). Statements (or operations) may change the program state, whereas assertions cannot. In contrast, assertions can talk about properties of a particular state — so they can be used to specify the behavior of operations. What the implementation and the specification have in common is that they both make use of expressions. For all of these building blocks, Viper supports different means of abstraction.

Methods can be viewed as a means of abstraction over a sequence of operations (i.e. the execution of a potentially-unbounded number of statements). The caller of a method observes its behavior exclusively through the method’s signature and its specification (its preconditions and postconditions). Viper method calls are treated modularly: for the caller of a method, the method’s implementation can be thought of as invisible. Calling a method may result in modifications to the program state, therefore method calls cannot be used in specifications. On the one hand, the caller of a method must first satisfy the assertions in its precondition in order to obtain the assertions from its postcondition. On the other hand, in order to verify a method itself, Viper must prove that the method’s implementation adheres to the method’s specification.

Functions can be viewed as a means of abstraction over (potentially state-dependent) expressions. Generally, the caller of a function observes three things. First, the precondition of the function is checked in the state in which the function is called. The precondition may include assertions denoting resources, such as permissions to field locations that the the function may read from. Second, the function application’s result value is equated with the expression in the function’s body (if provided); note that this usage of the function’s implementation is a difference from the handling of methods. Third, the function’s postconditions are assumed. The postcondition of a function may not contain resource assertions (e.g. denoting field permissions): all resources from the precondition are automatically returned after the function application. Refer to the section on functions for more details.

Predicates can be viewed as a means of abstraction over assertions (including resources such as field permissions). The body of a predicate is an assertion. Unlike functions, predicates are not automatically inlined: to replace a predicate with its body, Viper provides an explicit unfold statement. An unfold is an operation that changes the program state, replacing the predicate resource with the assertions specified by its body. The dual operation is called a fold: folding a predicate replaces the resources specified by its body with an instance of the predicate itself. Refer to the section on predicates for more details.

Below you can find a brief description and examples of the language constructs that can be used to write a Viper program. Note that the order in which top-level declarations are written is not important, as names are resolved against all declarations of the program (including later declarations).

Language overview

Top-level declarations

Fields

field val: Int
field next: Ref

• Declared by keyword field
• Every object has all fields (there are no classes in Viper)
• See the permission section for examples

Methods

method QSort(xs: Seq[Ref]) returns (ys: Seq[Ref])
  requires ... // precondition
  ensures ... // postcondition
{
  // body (optional)
}

• Declared by keyword method
• Have input and output parameters (e.g., xs and ys)
• Method calls can modify the program state; see the section on statements for details
  • Hence calls cannot be used in specifications
• Modular verification of methods and method calls
  • The body may include some number of statements (including recursive calls)
  • The body is invisible at call site (changing the body does not affect client code)
  • The precondition is checked before the method call (more precisely, it is exhaled)
  • The postcondition is assumed after the method call (more precisely, it is inhaled)
• See the permission section for more details and examples

Functions

function gte(x: Ref, a: Int): Int
  requires ... // precondition
  ensures ... // postcondition
{
  ...  // body (optional)
}

• Declared by keyword function
• Have input parameters (e.g., x and a) and one output return value
• The keyword result may be used in a function’s postcondition to refer to the return value
• Function applications may read but not modify the program state
  • Function applications can be used in specifications
  • Permissions (and resource assertions in general) may be mentioned in a function’s precondition, but not in its postcondition
• If a function has a body, the body is a single expression (possibly including recursive calls to functions)
• Unlike methods, function applications are not handled modularly (for functions with bodies): changing the body of a function affects client code
• See the section on functions for details

Predicates

predicate list(this: Ref) {
  ... // body (optional)
}

• Declared by keyword predicate
• Have input parameters (e.g., head)
• Typically used to abstract over assertions and to specify the shape of recursive data structures
• Predicate instances (e.g. list(x)) are resource assertions in Viper
• See the section on predicates for details

Domains

domain Pair[A, B] {
  function getFirst(p: Pair[A, B]): A
  // other functions

  axiom ax_1 {
    ... // axiom body
  }
  // other axioms
}

• Declared by keyword domain
• Have a name, which is introduced as an additional type in the Viper program
• Domains may have type parameters (e.g. A and B above)
• A domain’s body (delimited by braces) consists of a number function declarations, followed by a number of axioms
  • Domain functions (functions declared inside a domain) may have neither a body nor a specification; these are uninterpreted total mathematical functions
  • Domain axioms consist of name (following the axiom keyword), and a definition enclosed within braces (which is a boolean expression which may not read the program state in any way)
• Useful for specifying custom mathematical theories
• See the section on domains for details

Macros

define plus(a, b) (a+b)
define link(x, y) {
  assert x.next == null 
  x.next := y
}

• Declared by keyword define
• C-style, syntactically expanded macros
  • Macros are not type-checked until after expansion
  • However, macro bodies must be well-formed assertions / statements
• May have any number of (untyped) parameter names (e.g. a and b above)
• The are two kinds of macros
  • Macros defining assertions (or expressions) include the macro definition whitespace-separated afterwards (e.g. plus above)
  • Macros defining statements include their definitions within braces (e.g. link above)
• See the array domain encoding for an example

Built-in types

• Ref for references (to objects, except for the built-in Ref constant null)
• Bool for Boolean values
• Int for mathematical (unbounded) integers
• Rational for mathematical (unbounded) rationals (this type is expected to be deprecated in the summer 2023 release)
• Perm for permission amounts (see the section on permissions for details)
• Seq[T] for immutable sequences with element type T
• Set[T] for immutable sets with element type T
• Multiset[T] for immutable multisets with element type T
• Map[T, V] for immutable maps with key type T and value type V
• Additional types can be defined via domains

Imports

Local Import:

import "path/to/local.vpr"

Standard Import:

import <path/to/provided.vpr>

Imports provide a simple mechanism for splitting a Viper program across several source files using the local import. Further, it also makes it possible to make use of programs provided by Viper using the standard import.

• A relative or absolute path to a Viper file may be used (according to the Java/Unix style)
• import adds the imported program as a preamble to the current one
• Transitive imports are supported and resolved via depth-first traversal of the import graph
• The depth-first traversal mechanism enforces that each Viper file is imported at most once, including in the cases of multiple (indirect) imports of the same file or of recursive imports.

Permissions

Introduction

Reasoning about the heap of a Viper program is governed by field permissions, which specify the heap locations that a statement, an expression or an assertion may access (read and/or modify). Heap locations can be accessed only if the corresponding permission is held by the currently verified method. The simplest form of field permission is the exclusive permission to a heap location x.f; it expresses that the current method may read and write to the location, whereas other methods or threads are not allowed to access it in any way.

Every Viper operation, i.e., every statement, expression, or assertion, has an implicit or explicit specification expressing which field permissions the operation requires, i.e., which locations it accesses. The part of the heap denoted this way is called the footprint of an operation. Permissions enable preserving information about heap values in Viper: as long as the footprint of an expression is disjoint from the footprint of a method call, it can be concluded that the call does not change the expression’s value, even if the concrete method implementation is unknown. Preserving properties this way is called framing: e.g. we might say that the value of an expression is framed across a method call, or in general, across a statement. For example: the footprints of the expression x.f == 0 and the field assignment statement x.f := 1 are not disjoint, and the property x.f == 0 can therefore not be framed across the assignment. In contrast, the property could be framed across y.f := 0 if y and x were known to not be aliases. Permissions can also be used to guarantee non-aliasing, as will be discussed in more detail later.

In pre- and postconditions, and generally in assertions, permission to a field is denoted by an accessibility predicate: an assertion which is written acc(x.f). An accessibility predicate in a method’s precondition can be understood as an obligation for a caller to transfer the permission to the callee, thereby giving it up. Consequently, an accessibility predicate in a postcondition can be understood as a permission transfer in the opposite direction: from callee to caller.

A simple example is given next: a method inc that increments the field x.f by an arbitrary value i.

field f : Int

method inc(x: Ref, i: Int)
  requires acc(x.f)
  ensures true
{
  x.f := x.f + i
}

The program above declares a single integer (Int) field f, and the aforementioned increment method. The reference type (Ref) is built in; values of this type (other than the special value null) represent objects in Viper, which are the possible receivers for field accesses. Method inc‘s specification (sometimes called its contract, in other languages) is expressed via its precondition (requires) and postcondition (ensures).

In the remainder of this section we proceed as follows: After a brief description of Viper’s program state, we introduce the basic permission-related features employed by Viper. In particular, we discuss permission-related statements in the Viper language, as well as permission-related assertions, when such assertions are well-defined, and how they can be used to specify properties such as (non-)aliasing. Finally, we present a refined notion of permission supported by Viper: fractional permissions, which enable simultaneous access to the same heap location by multiple methods or (when modelling these in Viper) concurrent threads.

Viper’s Program State

Permissions are a part of a Viper program’s state, alongside the values of variables and heap locations. Fields are only the first of several kinds of resource that will be explained in this tutorial; access to each resource is governed by appropriate permissions. Different permissions can be held at different points in a Viper program: e.g., after allocating new memory on the heap, we would typically also add the permission to access those locations. In the next subsection, we will see the primitive operations Viper provides for manipulating the permissions currently held.

A program state in Viper consists of:

• The values of all variables in scope: these include local variables, method input parameters (which cannot be assigned to in Viper), and method return parameters (which can) of the current method execution. Verification in Viper is method-modular: each method implementation is verified in isolation and, thus, the program state does not contain an explicit call stack.
• The permissions to field locations held by the current method execution.
• The values of those field locations to which permissions are currently held. Other field locations cannot be accessed.

The initial state of each method execution contains arbitrary values (of the appropriate types) for all variables in scope, and no permissions to heap locations. Permissions can be obtained through a suitable precondition (as in the inc example above); preconditions can also constrain the values of parameters and heap locations. Field locations to which permission is newly obtained will also contain arbitrary values.

Inhaling and Exhaling

As previously mentioned, accessibility predicates in a method’s precondition, such as acc(x.f) in the precondition of inc, can be understood as specifying that permission to a field (here x.f) must be transferred from caller to callee. The process of gaining permission (which happens in the callee), is called inhaling permissions; the opposite process of losing permission (in the caller) is called exhaling. Both operations thus update the amount of currently held permissions: from a caller’s perspective, permissions required by a precondition are removed before the call and permissions guaranteed by a postcondition are gained after the call returns. From a callee’s perspective, the opposite happens.

Similar permission transfers also happen at other points in a Viper program; most notably, when verifying loops: a loop invariant specifies the permissions transferred (1) from the enclosing context to the first loop iteration, (2) from one loop iteration to the next, and (3) from the last loop iteration back to the enclosing context. Inside a loop body, heap locations may only be accessed if the required permissions have been explicitly transferred from the surrounding context to the loop body via the loop invariant.

In addition to specifying which permissions to transfer, Viper assertions may also specify constraints on values, just like in traditional specification languages. For example, a precondition acc(x.f) && x.f > 0 requires permission to location x.f and that its value is positive. Note that the occurrence of x.f inside acc(x.f) denotes the location (in compiler parlance, x.f as an lvalue); the meaning of an accessibility predicate is independent of the value of x.f as an expression (as used, e.g., in x.f > 0).

Consider now the call in the first line of method client in the example below: set‘s precondition specifies that the permission to a.f is transferred from the caller to the callee, and that i must be greater than a.f. Thus, method client has to exhale the permission to a.f (which is inhaled by set) and the caller has to prove that a.f < i (which it currently cannot). Conversely, the postcondition causes the permission to be transferred back to the caller when set terminates, i.e., it is inhaled by method client, and the caller gains the knowledge that a.f == 5. When verifying method set itself, the opposite happens: permission to x.f is inhaled before the method body is verified, alongside the fact that x.f < i. After the body has been verified, permission to x.f are exhaled and it is checked that x.f‘s value is indeed i.

field f: Int

method client(a: Ref)
  requires acc(a.f)
{
  set(a, 5)
  a.f := 6
}

method set(x: Ref, i: Int)
  requires acc(x.f) && x.f < i
  ensures  acc(x.f) && x.f == i
{
  x.f := i
}

Note that when encoding, e.g., a garbage-collected source language into Viper, the design choice that any excess permissions get leaked is convenient; it allows heap-based data to simply go out of scope and become unreachable. However, in the case of set in the example above, such leaking is presumably not the intention. Viper can also be used to check that certain permissions are not leaked; see the perm expression in the section on expressions for more details.

Inhale and Exhale Statements

To enable the encoding of programming language features that are not directly supported by Viper, such as forking and joining threads or acquiring and releasing locks, Viper allows one to explicitly exhale or inhale permissions via the statements exhale A and inhale A, where A is a Viper assertion such as method set‘s precondition acc(x.f) && i < x.f. From a caller’s perspective, set‘s pre- and postcondition can be seen as syntactic sugar for appropriate exhale and inhale statements before and after a call to the method.

The informal semantics of exhale A is as follows:

1. Assert that all value constraints in A hold; if not, verification fails
2. Assert that all permissions denoted (via accessibility predicates) by A are currently held; if not, verification fails
3. Remove the permissions denoted by A

The informal semantics of inhale A is as follows:

1. Add the permissions denoted by A to the program state
2. Assume that all value constraints in A hold

As an example, consider the following Viper program (ignoring, for the moment, the commented-out lines):

field f: Int

method set_inex(x: Ref, i: Int) {
  // x.f := i
  inhale acc(x.f)
  x.f := i
  // exhale acc(x.f)
  // x.f := i
}

Unlike the previous example, this method has no pre- and postcondition (no requires/ensures). This means that we start verification of the method body in a state with no permissions. The statement inhale acc(x.f) causes the corresponding permission to be added to the state, allowing the assignment on the following line to verify.

Self-Framing Assertions

Some Viper expressions and assertions come with conditions under which they are well-defined: e.g., partial operations (such as division) must not be applied outside of their domains (such as 1/n if n is potentially zero). Well-definedness conditions in Viper guarantee not only that assertions have a meaningful semantics, but that this semantics will be consistent across multiple contexts in which specifications are evaluated. Examples are Viper method specifications and loop invariants: preconditions (postconditions) are evaluated both at the beginning (end) of verifying the method body and before (after) each call to the method; loop invariants are evaluated before and after a loop, as well as at the beginning and end of the loop body. Such assertions are therefore checked to be guaranteed well-defined in all states they can possibly be evaluated in.

As an example, the assertion n < i/j is not well-defined in general; it cannot be used in e.g. a method precondition unless that precondition also guarantees that the value of j cannot be 0. The assertion j > 0 && n < i/j is well-defined, since the first conjunct is well-defined by itself, and ensures the well-definedness of the second conjunct. In general, the (short-circuiting) order of evaluation of logical connectives is taken into account for well-definedness conditions. For example, j != 0 ==> n < i/j is also well-defined (the right hand side’s value is only used when the left hand side is true, which guarantees its well-definedness condition).

Well-definedness in Viper also requires that all heap locations read by the assertion are accessible, i.e., that the corresponding permissions are held. Again, this must be the case for all possible states in which the assertion could be evaluated. To ensure this property, Viper requires specification assertions to be self-framing: i.e., each such assertion must include permission to at least the locations it reads. As an example, acc(x.f) and acc(x.f) && x.f > 0 are self-framing, whereas x.f > 0 and acc(x.f.g) are not: in the latter two cases, the meanings of the assertions depend on the value of the field x.f, to which permission is not included.

Viper checks well-definedness, and thus also self-framedness, according to a left-to-right evaluation order. The assertion acc(x.f) && 0 < x.f is therefore accepted as self-framing, but 0 < x.f && acc(x.f) is not. This restriction is typically easy to work around in practice.

The assertions in explicit inhale and exhale statements need not be self-framing because they are evaluated in only one program state; Viper will simply check that the well-definedness conditions for their assertions (e.g., that sufficient permissions are held) are true in that program state.

Exclusive Permissions

Permissions to field locations as described so far are exclusive; it is not possible to hold permission to a location more than once. This built-in principle can indirectly guarantee non-aliasing between references: inhaling the assertions acc(x.f) and acc(y.f) implies x != y because otherwise, the exclusive permission to acc(x.f) would be held twice. This is demonstrated by the following program:

field f: Int

method exclusivity(x: Ref, y: Ref) {
  inhale acc(x.f)
  inhale acc(y.f)
  exhale x != y
}

In Viper, accessibility predicates can be conjoined via &&; the resulting assertion requires the sum of the permissions required by its two conjuncts. Therefore, the two statements inhale acc(x.f); inhale acc(y.f) (semicolons are required in Viper only if statements are on the same line) are equivalent to the single statement inhale acc(x.f) && acc(y.f). In both cases, the obtained permissions imply that x and y cannot be aliases. Intuitively, the statement inhale acc(x.f) && acc(y.f) can be understood as inhaling permission to acc(x.f), and in addition to that, inhaling the permission to acc(y.f). Technically, this conjunction between resource assertions is strongly related to the separating conjunction from separation logic; formal details of the connection (and how to encode standard separation logic into Viper) can be found in this paper.

We can now see how exclusive permissions enable framing and modular verification, as illustrated by the next example below. Here, method client is able to frame the property b.f == 3 across the call to inc(a, 3) because holding permission to both a.f and b.f implies that a and b cannot be aliases, and because method inc‘s specification states that inc only requires the permission to a.f. Since permission to b.f is retained, the value of b.f can be framed across the method call. Informally, and thinking more operationally, the method would not be able to modify this field location, since it lacks the necessary permission to do so.

field f: Int

method inc(x: Ref, i: Int)
  requires acc(x.f)
  ensures  acc(x.f)
  ensures  x.f == old(x.f) + i
{
  x.f := x.f + i
}

method client(a: Ref, b: Ref) {
  inhale acc(a.f) && acc(b.f)

  a.f := 1
  b.f := 3

  inc(a, 3)

  assert b.f == 3
}

Note:

• The expression old(x.f) in method inc‘s postcondition denotes the value that x.f had at the beginning of the method call. In general, an old expression old(e) causes all heap-dependent subexpressions of e (in particular, field accesses and calls to heap-dependent functions) to be evaluated in the initial state of the corresponding method call. Note that variables are not heap-dependent; their values are unaffected by old.
• Method specifications can contain multiple requires and ensures clauses; this has the same meaning as if all requires assertions were conjoined, and likewise for ensures.

Fractional Permissions

Exclusive permissions are too restrictive for some applications. For instance, it is typically safe for multiple threads of a source program to concurrently access the same heap location as long as all accesses are reads. That is, read access can safely be shared. However, if any thread potentially writes to a heap location, no other should typically be allowed to concurrently read it (otherwise, the program has a data race). To support encoding such scenarios, Viper also supports fractional permissions with a permission amount between 0 and 1. Any non-zero permission amount allows read access to the corresponding heap location, but only the exclusive permission (1) allows modifications.

The general form of an accessibility predicate for field permissions is acc(e.f, p), where e is a Ref-typed expression, f is a field name, and p denotes a permission amount. Permission amounts are denoted by write for exclusive permissions, none for zero permission, quotients of two Int-typed expressions i1/i2 to denote a fractional permission; any Perm-typed expression may be used here. Perm is the type of permission amounts, which is a built-in type that can be used like any other type. The permission amount parameter p is optional and defaults to write. For example, acc(e.f), acc(e.f, write) and acc(e.f, 1/1) all have the same meaning.

The next example illustrates the usage of fractional permissions to distinguish between read and write access: there, method copyAndInc requires write permission to x.f, but only read permission (we arbitrarily chose 1/2, but any non-zero fraction would suffice) to y.f.

field f: Int

method copyAndInc(x: Ref, y: Ref)
  requires acc(x.f) && acc(y.f, 1/2)
  ensures  acc(x.f) && acc(y.f, 1/2)
{
  x.f := y.f + 1
}

Fractional permissions to the same location are summed up: inhaling acc(x.f, p1) && acc(x.f, p2) is equivalent to inhaling acc(x.f, p1 + p2), and analogously for exhaling. As before, inhaling permissions maintains the invariant that write permission to a location are exclusive. With fractional permission in mind, this can be rephrased as maintaining the invariant that the permission amount to a location never exceeds 1.

To illustrate this, consider the next example (and its exercises): there, the assert statement fails because holding one third permission to each x.f and y.f does not imply that x and y cannot be aliases, since the sum of the individual permission amounts does not exceed 1.

field f: Int

method test(x: Ref, y: Ref) {
  inhale acc(x.f, 1/2) && acc(y.f, 1/2)
  assert x != y
}

While fractional permissions no longer always guarantee non-aliasing between references, as demonstrated by the previous example, they still enable framing, e.g., across method calls: from a caller’s perspective, holding on to a non-zero permission amount to a location x.f across a method call guarantees that the value of x.f cannot be affected by the call. That is, because the callee would need to obtain write permission, i.e., permission amount 1, which cannot happen as long as the caller retains its fraction.

The next example illustrates the use of fractional permissions for framing: there, method client can frame the property b.f == 3 across the call to copyAndInc because client retains half the permission to b.f across the call. Note that the postcondition of copyAndInc does not explicitly state that the value of y.f remains unchanged.

field f: Int

method copyAndInc(x: Ref, y: Ref)
  requires acc(x.f) && acc(y.f, 1/2)
  ensures  acc(x.f) && acc(y.f, 1/2)
  ensures  x.f == y.f + 1
{
  x.f := y.f + 1
}

method client(a: Ref, b: Ref) {
  inhale acc(a.f) && acc(b.f)

  a.f := 1
  b.f := 3

  copyAndInc(a, b)

  assert b.f == 3 && a.f == 4
}

Predicates

So far, we have only discussed the specification of programs with a statically-finite number of field locations; our specifications must enumerate all relevant locations in order to express the necessary permissions. However, realistic programs manipulate data structures of unbounded size. Viper supports two main features for specifying unbounded data structures on the heap: predicates and quantified permissions.

Viper predicates are top level declarations, which give a name to a parameterised assertion; a predicate can have any number of parameters, and its body can be any self-framing Viper assertion using only these parameters as variable names. Predicate definitions can be recursive, allowing them to denote permission to and properties of recursive heap structures such as linked lists and trees. Viper predicates may also be declared with no body (in this case, the predicate is abstract), which can be useful for representing assertions whose definitions should be hidden for information hiding reasons, or in more advanced applications of Viper, for adding new kinds of resource assertion to the verification problem at hand.

Predicates are introduced in top-level declarations of the form

predicate P(...) { A }

or

predicate P(...)

where P is a globally-unique predicate name, followed by a possibly-empty list of parameters, and A is the predicate body, that is, an assertion that may include usages of P as well as P‘s parameters. Viper checks that predicate bodies are self-framing. Declarations of the second form (that is, without body) introduce abstract predicates.

A predicate instance is written P(e1,...,en), and is a second kind of resource assertion in Viper: as for field permissions, predicate instances can be inhaled and exhaled (added to and removed from the program state), and the Viper program state includes how many instances of which predicates are currently held.

In Viper, a predicate instance is not directly equivalent to the corresponding instantiation of the predicate’s body, but these two assertions can be explicitly exchanged for one another. The statement unfold P(...) exchanges a predicate instance for its body; fold P(...) performs the inverse operation. Abstract predicates cannot be folded or unfolded.

In the following example, the predicate tuple represents permission to the left and right fields of some tuple (note that this is not a keyword in Viper). The method requires permission to the fields this.left and this.right. Intuitively speaking, the tuple predicate required by the precondition contains permissions to the fields this.left and this.right. Holding the predicate is not enough to be allowed to access these fields; the corresponding permissions, however, can be obtained by unfolding the predicate. On the last line of the method’s body, these permission are folded back into the tuple predicate that is given back to the caller.

field left: Int
field right: Int

predicate tuple(this: Ref) {
  acc(this.left) && acc(this.right)
}

method setTuple(this: Ref, l: Int, r: Int)
  requires tuple(this)
  ensures tuple(this)
{
  unfold tuple(this)
  this.left := l
  this.right := r
  fold tuple(this)
}

Viper supports assertions of the form unfolding P(...) in A, which temporarily unfolds the predicate instance P(...) for (only) the evaluation of the assertion A. It is useful in contexts where statements are not allowed such as within method specifications and other assertions. For instance, in the example above we could add a postcondition unfolding tuple(this) in this.left == l && this.right == r to express that the entries of the tuple are set to l and r, respectively.

An unfold operation exchanges a predicate instance for its body; roughly speaking, the predicate instance is exhaled, and its body inhaled. Such an operation causes a verification failure if the predicate instance is not held. A fold operation exchanges a predicate body for a predicate instance; roughly speaking, the body is exhaled, and the instance inhaled. In both cases, however, in contrast to a standard exhale operation, these exhales do not remove information about the locations whose permissions have been exhaled because these permissions are still held, but perhaps folded (or no longer folded) into a predicate.

Formally, recursive predicate definitions are interpreted with respect to their least fixpoint interpretations; informally, this implies the built-in assumption that any given predicate instance has a finite (but potentially unbounded) number of predicate instances folded within it. Note that a predicate instance may represent a statically-unknown set of permissions. Holding a predicate instance in a Viper state can be thought of as indirectly holding all of these permissions (though unfolding the predicate will be necessary to make direct use of them).

Analogously to field permissions, it is possible for a program state to hold fractions of predicate instances (unlike for field permissions, these can also be permission amounts larger than 1): this is denoted by accessibility predicates of the shape acc(P(...), p). The simple syntax P(...) has the same meaning as acc(P(...)), which in turn has the same meaning as acc(P(...), write). Folding or unfolding a fraction of a predicate effectively multiplies all permission amounts used for resources in the predicate body by the corresponding fraction. In the example below, one half of a tuple predicate is given to the method. Unfolding this half of the predicate yields half a permission for each of the fields this.left and this.right; which are sufficient permissions to read the fields.

method addTuple(this: Ref) returns (sum: Int)
  requires acc(tuple(this), 1/2)
  ensures acc(tuple(this), 1/2)
{
  unfold acc(tuple(this), 1/2)
  sum := this.left + this.right
  fold acc(tuple(this), 1/2)
}

The next example is an extract from an encoding of a singly-linked list implementation. The predicate list represents permission to the elem and next fields of all nodes in a null-terminated list. The append method requires an instance of the list predicate for its this parameter and returns this predicate instance to its caller. The body unfolds the predicate in order to get access to the fields of this and folds it back before terminating.

The statement n := new(elem, next) models object creation: it assigns a fresh reference to n and inhales write permission to n.elem and n.next. Notice that unfold list(this) will exchange the predicate instance for its body, which includes the predicate instance list(this.next) if this.next != null. This is important to understand why (when the first branch of the if-condition is taken) two fold statements are needed: one for list(n) and another for list(this): since this.next (or n) is no longer null, folding list(this) depends on first folding list(n).

field elem: Int
field next: Ref

predicate list(this: Ref) {
  acc(this.elem) && acc(this.next) &&
  (this.next != null ==> list(this.next))
}

method append(this: Ref, e: Int)
  requires list(this)
  ensures  list(this)
{
  unfold list(this)

  if (this.next == null) {
    var n: Ref

    n := new(elem, next)
    n.elem := e
    n.next := null
    this.next := n

    fold list(n)
  } else {
    append(this.next, e)
  }
  fold list(this)
}

It is often useful to declare predicates with several arguments, such as the following list segment predicate, which is commonly used in separation logic. The predicate’s first argument denotes the start of the list segment, the second argument its end (i.e., the node directly after the segment) and the third argument, a value-typed mathematical sequence, represents the values stored in the segment.

field elem : Int
field next : Ref

predicate lseg(first: Ref, last: Ref, values: Seq[Int])
{
  first != last ==>
    acc(first.elem) && acc(first.next) &&
    0 < |values| &&
    first.elem == values[0] &&
    lseg(first.next, last, values[1..])
}

method removeFirst(first: Ref, last: Ref, values: Seq[Int])
  returns  (value: Int, second: Ref)
  requires lseg(first, last, values)
  requires first != last
  ensures  lseg(second, last, values[1..])
{
  unfold lseg(first, last, values)

  value := first.elem
  second := first.next
}

Functions

Just as predicates can be used to define parameterised (and potentially-recursive) assertions, Viper functions define parameterised and potentially-recursive expressions. A function can have any number of parameters, and always returns a single value; evaluation of a Viper function (just as any other Viper expression) is side-effect free. Function applications may occur both in code and in specifications: anywhere that Viper expressions may occur.

Functions are introduced in top-level declarations of the form:

function f(...): T
  requires A
  ensures  E1
{ E2 }

or

function f(...): T
  requires A
  ensures  E1

where f is a globally-unique function name, followed by a possibly-empty list of parameters, and a result type T. Functions may have an arbitrary number of pre- and postconditions; each precondition (indicated by the requires keyword) consists of an assertion A, whereas each postcondition (indicated by the ensures keyword) consists of an expression E1. That is, preconditions may contain resource assertions such as accessibility predicates, but postconditions must not (this difference from methods reflects the fact that function applications are side-effect-free, and so the pre- and post-states of a function application are the same; one can think of function preconditions as also being implicit additional postconditions). The result of the function is referred to using the keyword result in postconditions.

The following example defines a function listLength that takes a null-terminated simply-linked list and computes its length. As shown in the body of listLength, function applications are written simply as the function name followed by appropriately-typed arguments in parentheses. The precondition of listLength expresses the fact that the function application can only be evaluated when the corresponding list predicate instance is held, while the post-condition expresses the fact that the length of a list is always a non-negative integer.

field elem: Int
field next: Ref

predicate list(this: Ref) {
  acc(this.elem) && acc(this.next) &&
  (this.next != null ==> list(this.next))
}

function listLength(l:Ref) : Int
  requires list(l)
  ensures  result > 0
{ 
  unfolding list(l) in l.next == null ? 1 : 1 + listLength(l.next) 
}

The function body declaration { E2 } (if provided) must contain an expression E2 of (return) type T; it may contain function invocations, including recursive invocations of f itself. Function declarations of the latter form (that is, without a function body) introduce abstract functions, which may be useful for information hiding reasons, or to model functions which need not or cannot be directly implemented, e.g., because they model externally-justified information about the encoded program (such as the behaviour of library code). The following example illustrates this by adding a capacity function, intended to model a capacity suitable for storing the elements of the list in an array-like container.

function capacity(l:Ref): Int
  requires list(l)
  ensures  listLength(l) <= result && result <= 2 * listLength(l)

Note that functions declared in Viper domains (i.e. domain functions) are considered by Viper to be abstract state-independent total functions. As such, they can neither have a body nor be equipped with any pre-/postconditions; see the domains section for more details. In contrast, top-level functions can be state-dependent; the ability for function preconditions to include permissions allows them to depend on not only the values of their parameters, but also on heap locations to which their preconditions require permissions.

Viper checks that the function body and any postconditions are framed by the preconditions; that is, the preconditions must require all permissions that are needed to evaluate the function body and the postconditions. Moreover, Viper verifies that the postconditions can be proven to hold for the result of the function. In order to enable function termination checks, which are not performed by default, users can specify termination measures, as discussed in the chapter on termination. As the checking of a recursive function definition is essentially a proof by induction on the unrolling of the definition, not checking termination can lead to unsound behaviour. The following example yields such an inconsistency by means of a non-terminating function:

function bad() : Int
  ensures 0 == 1
{ bad() }

Due to the least fixpoint interpretation of predicates, any recursive function whose recursive calls occur inside an unfolding expression are guaranteed to be terminating, as in the case of the listLength function above. Consequently, predicates, and other common well-founded orders, are standard termination measures provided by Viper.

For non-abstract functions, Viper reasons about function applications in terms of the function bodies. That is, in contrast to methods, it is not always necessary to provide a postcondition in order to convey information to the caller of a function. Nevertheless, postconditions are useful for abstract functions and in situations where the property expressed in the postcondition does not directly follow from unfolding the function body once but, for instance, requires induction. In the case of the listLength function, the non-negativity of the result is indeed an inductive property, and is not exploitable by Viper unless stated in the postcondition.

For every function application, Viper checks that the function precondition is true in the current state, and then assumes the value of the function application to be equal to the function body (if provided), as well as assuming any postconditions. Fully expanding function bodies cannot work for recursive functions. Instead, functions are by-default expanded only once; additional expansions are triggered when unfolding or folding a predicate that is mentioned in the function’s preconditions. This feature allows one to traverse recursive structures and simultaneously reason about the permissions and values. For example, since predicate list was mentioned in the precondition of function listLength earlier, the body of any function call listLength(l) is unfolded whenever a predicate list(l) is. This is why the following implementation of the capacity function can be successfully verified:

function capacity(l:Ref): Int
  requires list(l)
  ensures  listLength(l) <= result && result <= 2 * listLength(l)
{ unfolding list(l) in l.next == null
  ? 1
  : unfolding list(l.next) in l.next.next == null
  ? 2
  : 3 + capacity(l.next.next) }

The example below is an alternative version of the previously shown list segment example from the predicates section: instead of using a predicate parameter for the abstract representation of the list segment (as a mathematical sequence), a function is introduced that computes the abstraction. This usage of functions to eliminate predicate parameters which are redundant (in the sense that their values can instead be computed given any predicate instance and its other parameters) is common in Viper.

field elem: Int
field next: Ref

predicate lseg(this: Ref, last: Ref) {
  this != last ==>
    acc(this.elem) && acc(this.next) &&
    this.next != null &&
    lseg(this.next, last)
}

function values(this: Ref, last: Ref): Seq[Int]
  requires lseg(this, last)
{
  unfolding lseg(this, last) in 
    this == last
      ? Seq[Int]()
      : Seq(this.elem) ++ values(this.next, last)
}

method removeFirst(this: Ref, last: Ref) returns (first: Int, rest: Ref)
  requires lseg(this, last)
  requires this != last
  ensures  lseg(rest, last)
  ensures  values(rest, last) == old(values(this, last)[1..])
{
  unfold lseg(this, last)

  first := this.elem
  rest := this.next
}

The values function requires an lseg predicate instance in its precondition to obtain the permissions to traverse the list. Its body uses an unfolding expression to obtain the predicate instance required for the recursive application.

Quantifiers

Viper’s assertions can include forall and exists quantifiers, with the following syntax:

forall [vars] :: [triggers] A

exists [vars] :: e

Here [vars] is a list of comma-separated declarations of variables which are being quantified over, [triggers] consists of a number of so-called trigger expressions in curly braces (explained next), and A (and e) is a Viper assertion (respectively, boolean expression) potentially including the quantified variables. Unlike existential quantifiers, forall quantifiers in Viper may contain resource assertions; this possibility is explained in the section on quantified permissions.

Trigger expressions take a crucial role in guiding the SMT solver towards a quick solution, by restricting the instantiations of a quantified assertion. In particular, when a forall-quantified assertion is a hypothesis for a proof goal, the triggers inform the SMT solver to instantiate the quantifier only when it encounters expressions (which are not themselves under a quantifier) of forms matching the trigger. Let’s first illustrate with an example:

assume forall i: Int, j: Int :: {f(i), g(j)} f(i) && i < j ==> g(j)
assert ...f(h(5))...g(k(7))... // some proof goal

Here, assuming that the SMT solver encounters both the expressions f(h(5)) and g(k(7)), the body of the quantification will be instantiated with i == h(5) and j == k(7), obtaining f(h(5)) && h(5) < k(7) ==> g(k(7)). If no other pairs of expressions matching the triggers are encountered, no other instantiations of the quantifier will be made.

In general, a forall quantifier can have any number of sets of trigger expressions; these are written one after the other, each enclosed within braces. Multiple such sets prescribe alternative triggering conditions; multiple expressions within a single trigger set prescribe that expressions matching each of the trigger expressions must be encountered before an instantiation may be made.

You can check how triggers affect the verification in the following examples:

• in restrictive_triggers the triggers are too restrictive and do not allow the right instantiation of the quantifier;
• in dangerous_triggers the bad choice of the triggers leads to an infinite loop of instantiations (in this case, each instantiation results in a new expression which matches the trigger): a problem known as matching loop. In this case, the SMT solver times out without providing an answer.
• in good_triggers the choice of the triggers allows the SMT solver to quickly provide the right answer, preventing the problematic matching loop of the previous example.

function magic(i:Int) : Int

method restrictive_triggers()
{
  // Our definition of `magic`, with a very restrictive trigger
  assume forall i: Int :: { magic(magic(magic(i))) }
    magic(magic(i)) == magic(2 * i) + i

  // The following should verify. However, the verification fails
  // because the triggers are too restrictive and the quantifier
  // can not be intantiated.
  assert magic(magic(10)) == magic(20) + 10
}

method dangerous_triggers()
{
  // Our definition of `magic`, with a matching loop
  assume forall i: Int :: { magic(i) }
    magic(magic(i)) == magic(2 * i) + i

  // The following should fail, because our definition says nothing
  // about this equality. However, if you uncomment the assertion
  // the verification will time out and give no answer because of the
  // matching loop caused by instantiating the quantifier.
  // assert magic(magic(10)) == magic(87987978) + 10
}

method good_triggers()
{
  // Our definition of `magic`
  assume forall i: Int :: { magic(magic(i)) }
    magic(magic(i)) == magic(2 * i) + i
  
  // The following verifies, as expected
  assert magic(magic(10)) == magic(20) + 10

  // The following fails, as expected, because our definition says
  // nothing about this equality. The verification terminates
  // quickly because we don't have matching loops; the SMT solver
  // quickly exhausts the available quantifier instantiations. 
  assert magic(magic(10)) == magic(87987978) + 10
}

There are a number of restrictions on what can be used as a set of trigger expressions:

1. Each quantified variable must occur at least once in a trigger set.
2. Each trigger expression must include at least one quantified variable.
3. Each trigger expression must have some additional structure (typically a function application); a quantified variable alone cannot be used as a trigger expression.
4. Arithmetic and boolean operators may not occur in trigger expressions.
5. Accessibility predicates (the acc keyword) may not be used in trigger expressions.

Applications of both domain and top-level Viper functions can be used in trigger expressions, as can field dereference expressions (e.g. x.f) and Viper’s built-in sequence and set operators. Note that the types of trigger expressions are not restricted; in particular, there is no requirement that trigger expressions are boolean-typed.

If no triggers are specified, Viper will infer them automatically with a heuristics based on the body of the quantifier. In some unfortunate cases this automatic choice will not be good enough and can lead to either incompletenesses (necessary instantiations which are not made) or matching loops; it is recommended to always specify triggers on Viper quantifiers.

The underlying tools currently have limited support for existential quantifications: the syntax for exists does not allow the specification of triggers (which play a dual role for existential quantifiers, in controlling the potential witnesses/instantiations considered when proving an existentially-quantified formula), so existential quantifications should be used sparingly due to the risk of matching loops. This limitation is planned to be lifted in the near future.

For more details on triggers and the e-matching approach to quantifier instantiation, we recommend the Programming with Triggers paper.

Quantified Permissions

Viper provides two main mechanisms for specifying permission to a (potentially unbounded) number of heap locations: recursive predicates and quantified permissions. While predicates can be a natural choice for modelling entire data structures which are traversed in an orderly top-down fashion, quantified permissions enable point-wise specifications, suitable for modelling heap structures which can be traversed in multiple directions, random-access data structures such as arrays, and unordered data structures such as general graphs.

The basic idea is to allow resource assertions such as acc(e.f) to occur within the body of a forall quantifier. In particular, the e receiver expression can be parameterised by the quantified variable, specifying permission to a set of different heap locations: one for each instantiation of the quantifier.

As a simple example, we can model a “binary graph” (in which each node has at most two outgoing edges) in the heap, in terms of a set of nodes, using the following quantified permission assertion: forall n:Ref :: { n.first }{ n.second } n in nodes ==> acc(n.first) && acc(n.second). Such an assertion provides permission to access the first and second fields of all nodes n (as explained in the previous section on quantifiers, the { n.first }{ n.second } syntax denotes triggers). To usefully model a graph, one would typically also require the nodes set to be closed under the graph edges, so that a traversal is known to stay within these permissions; this is illustrated in the following example:

field first : Ref
field second : Ref

method inc(nodes: Set[Ref], x: Ref)
  requires forall n:Ref :: { n.first } n in nodes ==> 
    acc(n.first) && 
    (n.first != null ==> n.first in nodes)
  requires forall n:Ref :: { n.second } n in nodes ==> 
    acc(n.second) && 
    (n.second != null ==> n.second in nodes)
  requires x in nodes
{
  var y : Ref
  if(x.second != null) {
    y := x.second.first // permissions covered by preconditions
  }
}

Receiver Expressions and Injectivity

In the above examples, the receiver expressions used to specify permissions (the e expression in acc(e.f)) were always the quantified variable itself. This is not a requirement; for example, in the following code, quantified permissions are used along with a function address in the exhale statement, to exhale permission to multiple field locations:

field val : Int

function address(i:Int) : Ref
// ensures forall j:Int, k:Int :: j != k ==> address(j)!=address(k)

method test()
{
  inhale acc(address(3).val, 1/2)
  inhale acc(address(2).val, 1/2)
  inhale acc(address(1).val, 1/2)
  exhale forall i:Int :: 1<=i && i<=3 ==> acc(address(i).val, 1/2)
}

The expression address(i) implicitly defines a mapping between instances i of the quantifier and receiver references address(i). Such receiver expressions cannot be fully-general: Viper imposes the restriction that this mapping must be provably injective: for any two instances of such a quantifier, the verifier must be able to prove that the corresponding receiver expressions are different. As usual, this condition can be proven using any information available at the particular program point. In addition, injectivity is only required for instances of the quantifier for which permission is actually required; in the above example, the restriction amounts to showing that the references address(1), address(2) and address(3) are pairwise unequal. In the following exercises, this is illustrated more thoroughly.

The injectivity restriction imposed by Viper has the consequence that, when considering permissions required via quantified permissions, one can equivalently think about these per instantiation of the quantified variable, or per corresponding instance of the receiver expression.

Magic Wands

Note: this section introduces a somewhat advanced feature of the Viper language, which users who are just starting-out with Viper may wish to skip over for the moment.

When reasoning with unbounded data structures, it can often be useful to specify properties of partial versions of these data structures. For example, during an iterative traversal of a linked list, one typically needs a specification relating the prefix of the list already visited, to a view of the overall data structure. Directly specifying such prefixes (or more generally, instances of data structures with a “hole”), tends to lead to auxiliary predicate definitions (e.g. an lseg predicate for list segments), which in turn necessitate additional lemmas or ghost methods for converting between multiple views of the same structure.

Viper includes a powerful alternative mechanism, which is often useful for evading these problems: the magic wand connective. A magic wand assertion is written A --* B, where A and B are Viper assertions. Like permissions to field locations and instances of predicates, magic wands are a type of resource, instances of which can be held in a Viper program state; these can be added and removed from the state via inhale and exhale operations, just as for other resources.

A magic wand instance A --* B abstractly represents a resource which, if combined with the resources represented by A, can be exchanged for the resources represented by B. For example, A could represent the permissions to the postfix of a linked-list (where we are now), and B could represent the permissions to the entire list; the magic wand then abstractly represents the leftover permissions to the prefix of the list. In this case, both the postfix A and a magic wand A --* B could be given up, to reobtain B, describing the entire list. This “giving up” step, is called applying the magic wand, and is directed in Viper with an apply statement:

inhale A
inhale A --* B
apply A --* B
assert B // succeeds; would fail before the apply statement

To understand a typical use-case for magic wands more concretely, consider the following iterative code for appending two linked lists:

field next : Ref
field val : Int

method append_iterative(l1 : Ref, l2: Ref)
  {
    if(l1.next == null) { // easy case
      l1.next := l2
    } else {
      var tmp : Ref := l1.next

      while(tmp.next != null)
      {
        tmp := tmp.next
      }
      tmp.next := l2
    }
  }

Using a standard linked-list predicate list, and a function elems to fetch the sequence of elements stored in a linked-list, we can specify the intended behaviour of our method, and add a first-attempt at a loop invariant, as follows:

field next : Ref
field val : Int

predicate list(start : Ref)
{
  acc(start.val) && acc(start.next) &&
      (start.next != null ==> list(start.next))
}

function elems(start: Ref) : Seq[Int]
  requires list(start)
{
  unfolding list(start) in (
    (start.next == null ? Seq(start.val) :
     Seq(start.val) ++ elems(start.next) ))  
}

method append_iterative(l1 : Ref, l2: Ref)
  requires list(l1) && list(l2) && l2 != null
  ensures list(l1)
  ensures elems(l1) == old(elems(l1) ++ elems(l2))
  {
    unfold list(l1)
    if(l1.next == null) { // easy case
      l1.next := l2; fold list(l1)
    } else {
      var tmp : Ref := l1.next
      var index : Int := 1 // extra variable: useful for specification

      while(unfolding list(tmp) in tmp.next != null)
        invariant index >= 0
        invariant list(tmp)
        // what about the prefix of the list?
      {
        unfold list(tmp)
        var prev : Ref := tmp // extra variable: useful for specification
        tmp := tmp.next
        index := index + 1
      }
      unfold list(tmp)
      tmp.next := l2
      fold list(tmp)
      // how do we get back to list(l1) ?
    }
  }

In this version of the code, we’ve added the extra variables index (representing how far through the linked-list the tmp reference is), and prev; both will be convenient for writing a specification later in this section. As commented in the file, the specification is not sufficient to verify the code. The loop invariant tracks permission to the postfix linked-list (referenced by tmp). However, it includes neither permissions nor value information about the prefix of the list between l1 and the tmp reference. Since these permissions are not tracked by the loop invariant, they are effectively leaked during the loop execution; with the loop invariant given there is no way after the loop to obtain permission to the entire list (the predicate instance list(l1)), as required in the postcondition.

We can now employ magic wands to retain the lost permissions. In order to retain (during execution of the loop) the permissions to the previously-visited nodes in the list, we use a magic wand list(tmp) --* list(l1). This magic wand represents sufficient resources to guarantee that if we give up a list(tmp) predicate along with this wand, we can obtain a list(l1) predicate; conceptually, it represents the permissions to the earlier segment of the list between l1 and tmp.

field next : Ref
field val : Int

predicate list(start : Ref)
{
  acc(start.val) && acc(start.next) &&
    (start.next != null ==> list(start.next))
}

function elems(start: Ref) : Seq[Int]
  requires list(start)
{
  unfolding list(start) in (
    (start.next == null ? Seq(start.val) :
     Seq(start.val) ++ elems(start.next) ))  
}

method appendit_wand(l1 : Ref, l2: Ref)
  requires list(l1) && list(l2) && l2 != null
  ensures list(l1) // && elems(l1) == old(elems(l1) ++ elems(l2))
  {
    unfold list(l1)
    if(l1.next == null) { // easy case
      l1.next := l2; fold list(l1)
    } else {
      var tmp : Ref := l1.next
      var index : Int := 1

      // package the magic wand required in the loop invariant below
      package list(tmp) --* list(l1)
      { // show how to get from list(tmp) to list(l1):
        fold list(l1) // also requires acc(l1.val) && acc(l1.next)
      }

      while(unfolding list(tmp) in tmp.next != null)
        invariant index >= 0
        invariant list(tmp)// && elems(tmp) == old(elems(l1))[index..]
        invariant list(tmp) --* list(l1) // magic wand instance
      {
        unfold list(tmp)
        var prev : Ref := tmp
        tmp := tmp.next
        index := index + 1

        package list(tmp) --* list(l1) // package new magic wand
        { // we get from list(tmp) to list(l1) by ...
          fold list(prev)
          apply list(prev) --* list(l1)
        }  
      }
      unfold list(tmp)
      tmp.next := l2
      fold list(tmp)
      apply list(tmp) --* list(l1) // regain predicate for whole list
    }
  }

The additional loop invariant includes an instance of a magic wand: list(tmp) --* list(l1). Such a magic wand instance denotes a new kind of resource in Viper (in addition to field permissions and predicate instances); as such, it can be inhaled and exhaled just as other resource assertions. This particular magic wand instance can (when applied), be used up to exchange any list(tmp) predicate instance for a list(l1) predicate instance. In this example, the magic wand notionally represents the permissions to the prefix of the list between l1 and tmp. These magic wand instances are created via package operations, which are explained below.

Package operations

There are two ways in which a magic wand instance can be added to the resources held in the program state: they can be inhaled (just as any other Viper resource), or we can instruct Viper to construct a new magic wand instance with a package statement. As an example to explain the feature, we will consider the package before the loop in the code example above. A package statement consists of the keyword followed by the desired magic wand instance (in this case, list(tmp) --* list(l1)), along with an optional block of code delimited by braces. The role of a package statement is to create (and justify the creation of) a new magic wand instance in the following way:

• A subset of the resources held in the current state must be identified as necessary for justifying the new magic wand instance. These must be sufficient resources to ensure that, given the additional resources described in the wand left-hand-side, those on the wand’s right-hand-side can be produced. This set of resources is taken from the current state, and is called the footprint of the newly-created magic wand instance. In our example, these are the field permissions acc(l1.val) && acc(l1.next) (since, along with the wand’s left-hand-side list(tmp), these are sufficient to guarantee the wand’s right-hand-side list(l1)).
• The package operation must check that, given the selected footprint of resources from the current state, in any heap in which the wand’s left-hand-side assertion is held, the combination of these resources can be exchanged for the wand right-hand-side. Any field locations framed by permissions in the wand’s footprint will be assumed to be unchanged for this check (e.g. l1.next == tmp is known in our example, since permission to l1.next is in the wand’s footprint). The check is made during the package operation by successively attempting to match the resources required on the right-hand-side with resources provided on the left; if not found on the left-hand-side, the resources must instead be found in the current state (or else the package operation fails with a verification error), and are taken for the wand’s footprint. See our ECOOP’15 paper for more details of how wand footprints are chosen.
• It is often the case that the combination of the wand’s left-hand-side and footprint do not directly yield the wand’s right-hand-side, but instead can do so after a number of additional operations are performed. These operations can be specified in the block attached to the package statement. In our example, the wand left-hand-side list(tmp) combined with the wand footprint acc(l1.val) && acc(l1.next) are sufficient to guarantee list(l1) after performing the operation fold list(l1), which is given in the block.

Heap-dependent expressions

So far, we have used magic wands with only resource assertions on their left- and right-hand-sides. However, in order to prove functional properties about our programs, we will as usual need to employ heap-dependent expressions (e.g. field dereferences and function calls) in our specifications. For example, to prove the postcondition elems(l1) == old(elems(l1) ++ elems(l2)) we need a magic wand which guarantees something about the elements of the lists in our program.

Magic wands can include any Viper assertions and expressions, provided that the left- and right-hand-sides are self-framing assertions. Heap dependent expressions used inside magic wands do not refer to the values in the current heap. Instead, a heap dependent expression on the left of a magic wand refers to the heap when the wand is applied; those on the right-hand-side refer to the heap resulting from the application of the wand.

Recall that the magic wand used so far is list(tmp) --* list(l1). One way to finish our example is to express that the elements that list(l1) will have after applying the magic wand are guaranteed to be a prefix of the elements from the original list, plus whichever elements list(tmp) has at the point of applying the magic wand. Informally, we might consider writing something like elems(l1) == old(elems(l1)[..index]) ++ elems(tmp). However, this expression is not quite well-defined; we never hold both the predicate list(l1) and the predicate list(tmp) at the same time; list(tmp) is given up as part of applying the magic wand itself. In order to evaluate heap-dependent expressions in the state corresponding to the wand’s left-hand-side, Viper supports the construct: old[lhs](e), which may only be used in the right-hand-side of a magic wand, and causes the expression e to be evaluated in the corresponding left-hand state. Thus, we can express our desired magic wand as: list(tmp) --* list(l1) && elems(l1) == old(elems(l1)[..index]) ++ old[lhs](elems(tmp)). The corresponding full version of the example above is presented below:

field next : Ref
field val : Int

predicate list(start : Ref)
{
  acc(start.val) && acc(start.next) &&
    (start.next != null ==> list(start.next))
}

function elems(start: Ref) : Seq[Int]
  requires list(start)
{
  unfolding list(start) in (
    (start.next == null ? Seq(start.val) :
     Seq(start.val) ++ elems(start.next) ))  
}

method appendit_wand(l1 : Ref, l2: Ref)
  requires list(l1) && list(l2) && l2 != null
  ensures list(l1) && elems(l1) == old(elems(l1) ++ elems(l2))
  {
    unfold list(l1)
    if(l1.next == null) {
      l1.next := l2
      fold list(l1)
    } else {
      var tmp : Ref := l1.next
      var index : Int := 1

      package list(tmp) --* list(l1) && elems(l1) == old(elems(l1)[..index]) ++ old[lhs](elems(tmp))
      {
        fold list(l1)
      }

      while(unfolding list(tmp) in tmp.next != null)
        invariant index >= 0
        invariant list(tmp) && elems(tmp) == old(elems(l1))[index..]
        invariant list(tmp) --* list(l1) && elems(l1) == old(elems(l1)[..index]) ++ old[lhs](elems(tmp))
      {
        unfold list(tmp)
        var prev : Ref := tmp
        tmp := tmp.next
        index := index + 1

        package list(tmp) --* list(l1) && elems(l1) == old(elems(l1)[..index]) ++ old[lhs](elems(tmp))
        {
          fold list(prev)
          apply list(prev) --* list(l1) && elems(l1) == old(elems(l1)[..index-1]) ++ old[lhs](elems(prev))
        }  
      }
      unfold list(tmp)
      tmp.next := l2
      fold list(tmp)
      apply list(tmp) --* list(l1) && elems(l1) == old(elems(l1)[..index]) ++ old[lhs](elems(tmp))
    }
  }

Statements

This section gives a complete overview of Viper statements. Most of them have been used throughout this tutorial, but we describe them in more detail here.

Assignments

Viper supports the following forms of assignments:

Assignments to local variables and result parameters of methods work as expected. It is not possible to assign to input method parameters. Assignment to heap locations requires the full permission to the heap location (here, e1.f). Methods may have any number of result parameters; method call statements use the appropriate number (and types) of variables on the left-hand side (using the same variable twice on the left-hand side is disallowed). A method call can be understood as an exhale of the method precondition, a reassignment of the variables used to store result parameters, and inhale of the method postcondition. Finally, a new statement creates a new object and inhales exclusive permission to all (possibly none) fields listed comma-separated within the parentheses. As a special case, x := new(*) inhales permission to all fields declared in the Viper program. Note that neither method calls nor object creation are expressions. Hence, they must not occur as receivers, method arguments, etc.; instead of nesting these constructs, one typically assigns their results first to local variables, and then uses these.

Control Structures

Viper supports two main control structures: conditional statements and loops. Conditional statements have the following form:

if(E) {
  S1
} else {
  S2
}

where the else branch is optional. The semantics is as expected.

Loops have the following form:

while(b)
  invariant A
{
  S
}

Loops are verified as follows: with respect to the scope surrounding the loop, verification amounts to replacing the loop with the following operations:

1. Exhale the loop invariant;

2. Havoc (assign arbitrary values) to all locals and result parameters that get assigned to by the loop body S (the so-called loop targets);

3. Inhale the loop invariant and the negation of the loop condition b; the verification of any code following the loop proceeds from this state.

Note that this approach provides two forms of framing: first, locals and result parameters that are not loop targets are always known to be unchanged by the loop. Second, the value of those heap locations for which the context of the loop holds some permission that is not transferred into the loop via the loop invariant are also known to be unchanged. Consequently, it is not necessary to write explicit loop invariants to preserve information about these variables and locations, as is illustrated by the following example:

field f: Int
field g: Int

method Foo(this: Ref, n: Int)
  requires acc(this.f) && acc(this.g)
{
  var i: Int := n
  var x: Int := 3
  this.f := 5

  while(0 <= i)
    invariant acc(this.g)
  {
    this.g := this.g + 1
    i := i - 1
  }
  
  assert this.f == 5
  assert x == 3
}

The loop body is verified starting from a state that contains no permissions and in which all loop targets have arbitrary values (all other variables in scope will be known to have the same values as they did immediately before the loop). Verification of the loop body then proceeds as follows:

1. Inhale the loop invariant and the loop condition b;

2. Verify the loop body S;

3. Exhale the loop invariant.

Analogous to functions, Viper does also not check loop (or method) termination per default, see the chapter on termination for more details. Alternatively, custom termination checks can be encoded through suitable assertions (see the labelled old expressions described in the section on expressions).

Assertion Checking

• exhale A and inhale A are explained in the section on permissions.
• assert A is similar to exhale A, but it does not remove any permissions.
• assume A is similar to inhale A, but it does not add any permissions.
• refute A tries to show that A holds for all executions that reach the statement, and causes a verification error if this is the case. In other words, if A is provable in some, but not all, execution paths, then the statement still passes.

Note that inhale acc(...) adds the given permission to the current state, whereas assume acc(...) only keeps executing if the given permission was already in the current state.

Verifier Directives

• unfold and fold are explained in the section on predicates
• package and apply are explained in the section on magic wands
• Note that unfolding is an expression, not a statement

Expressions and Assertions

Expressions

Viper supports a number of different kinds of expressions, which can be evaluated to a value of one of the types supported in Viper.

The primitive types supported in Viper are booleans (Bool), integers (Int), permission amounts (Perm), denoting real numbers, and references to heap objects (Ref). In addition, there are built-in parameterised set (Set[T]), multiset (Multiset[T]), sequence (Seq[T]), and map (Map[T, U]) types, and users can define custom types using domains.

Evaluating an expression never changes the state of the program, i.e., expression evaluation has no side effects. However, expression evaluation comes with well-definedness conditions for some expressions: evaluating an expression can cause a verification failure if the expression is not well-defined in the current program state; this leads to a verification error. As an example, the expression x % y is not well-defined if y is equal to zero, and the expression o.f is only well-defined if the current method has the permission to read o.f (which also implies that o is not null).

In the following, we list the different kinds of expressions, remark on their well-definedness conditions (if any) and the value that they evaluate to.

Expressions of multiple types

• field access e.f: to be well-defined, this requires at least some permission to read the field location; returns a value of the type of the field.

• function application f(...): the function can either be a domain function or a top-level, (potentially heap-dependent) function. In the latter case, for a function application to be well-defined the function’s precondition must be fulfilled, and in both cases, the argument expressions must be well-defined and have the expected types. Evaluates to a value of the return type of the function. See the respective sections for more information on top-level functions and domains.

• typed function application (f(...) : Type): a variant of the above that additionally enforces that the return type of the function application to be the one given in the expression. This is particularly useful with domains with type parameters, for example (Nil() : List[Bool]). The parentheses are mandatory.

• local variable and parameter evaluation x: read the current value of the named variable or parameter. Note that it is possible to read local variables which have not been assigned to; in this case, the expression will evaluate to an arbitrary value of its type.

• conditional expressions e1 ? e2 : e3, where e1 has type Bool and e2 and e3 must have the same type; evaluates to e2 if e1 is true, and otherwise to e3. Short-circuiting evaluation is taken into account when checking well-definedness conditions; e.g. e2 need only be well-defined when e1 evaluates to true.

• unfolding expressions unfolding acc(P(...), p) in e: Requires that the current method has at least the permission amount p of the predicate instance P(...). Evaluates e in a context where (this amount of) the predicate instance has been temporarily unfolded (i.e., P is no longer available, but its contents are).

• old and labelled-old expressions old(e) and old[l](e): Evaluates expression e in a different (older) version of the heap. In the first case, this is the heap at the start of the current method call; in the second, it is the heap at the position of the label l. For the second expression to be well-defined, the label l must be in the same method as the old-expression, and must have been encountered by the time the old-expression is evaluated; as a result, old-expressions cannot be used in functions, predicates or preconditions (they also cannot be used in postconditions but for a different reason: they would be meaningless for the caller). It is not supported to refer back to a label inside a loop body from outside of the loop body. Note that old expressions does not affect the value of variable references; old(x.f), for example, evaluates to the value that the field f of the object that the variable x currently points to had at the beginning of the method.

• function results result: Can only be used in postconditions of top-level Viper functions, and refers to the result of the function application; it therefore has the type of the return value of the function the postcondition belongs to.

• let expressions let v == (e1) in e2: Evaluates e1, and subsequently evaluates e2 in a context where the variable v is bound to the value of e1 (currently, the parentheses are necessary). Let expressions are often convenient when one needs to write an expression with many repetitions, or one that concerns several different heaps. For example, if one wishes to evaluate the argument of a function call f(x.f) in the current state and the function application itself in the current method’s old state, this can be expressed by using a let expression as follows: let y == (x.f) in old(f(y)).

Integer expressions

• constants, e.g. -2, 1023123909. Integers in Viper are signed and unbounded.

• unary minus -e: Negates the value of e if e is itself an integer.

• addition, subtraction, multiplication, division, modulo (e1 + e2, e1 - e2, e1 * e2, e1 / e2, e1 % e2). For all of these, both operands are expected to be integers, and the result will be an integer. These operators are overloaded; similar operators exist returning Perm-typed values; Viper will choose the appropriate operator based on the type information available. The only observable ambiguity is for division, since integer division truncates and Perm-typed division does not. In ambiguous cases, the alternative syntax e1 \ e2 can be used to force Int-typed division (Perm-typed is otherwise the default).

Boolean expressions

• constants true and false

• conjunction e1 && e2, disjunction e1 || e2, implication e1 ==> e2. For all cases, both operands are expected to be booleans, and the result will be a boolean. Viper has short-circuiting semantics for these operators, meaning that, for example, in e1 && e2, e2 will only be evaluated if e1 evaluates to true. In particular, this means that e2 only has to be well-defined if e1 is true.

• equality e1 == e2 and inequality e1 != e2 on all types. Both operands have to be of the same type, and the result is a boolean.

• quantifiers forall and exists. See the section on quantifiers for more details.

• For-perm quantifiers forperm x: Ref [x.f] :: e. This expression serves as a quantifier over all objects for which a permission to the specified field is held by the current method. Inside the expression e in the body, the variable x points to an object for which a positive amount of permission to x.f is held. The entire expression is true if e is true for every such object, and false otherwise. As an example, forperm x: Ref [x.g] :: x.g > 0 is true if and only if, for all objects to whose g-fields the current method holds a permission, the value of the g-field is positive.

forperm expressions are useful for implementing leak checks. For example, by asserting forperm x: Ref [x.f] :: false we can check that in the current context we do not hold any permission to the field f. Note that forperm expressions are evaluated in the current heap, including side-effects caused during exhale operations, as illustrated in the following example:

inhale acc(x.f)
exhale forperm x: Ref [x.f] :: false // Would fail because we have acc(x.f)
acc(x.f) && forperm x: Ref [x.f] :: false // Would succeed because we do not have acc(x.f) anymore.

This is useful, for example, for checking that after the method postcondition is exhaled, the method body does not have any permission left which would be leaked.

Forperm expressions may not be used in functions or predicates.

• equivalence or bi-implication e1 <==> e2, where both expressions are of type Bool, is equivalent to writing e1 == e2.

• negation !e: Negates the boolean expression e.

• integer or perm comparisons e1 < e2, e1 <= e2, e1 > e2, e1 >= e2 require that the operands are either both of type Int or both of type Perm, and return a Bool.

Set and multiset expressions

Viper has a built-in parametric finite set type Set[T], where T is a type parameter that describes the type of the elements of the set. Sets are immutable (i.e. represent mathematical sets).

• empty set: Set[T]() returns an empty set of type Set[T]. The type argument T is optional if the type is clear from the context; then simply Set() can be written.

• set literals: Set(e1, e2, ..., en) evaluates to a set containing only the n enumerated values and has the type Set[T], where T is the common type of the enumerated expressions.

• e1 union e2, e1 intersection e2, e1 setminus e2 perform the respective set operations. In all cases, e1 and e2 must be sets of the same type, and the entire expression has the same type as its operands. These operators each define a set; the operand sets will not be modified (Viper sets are immutable).

• e1 subset e2 has type Bool and evaluates to true if and only if e1 is a subset of e2. Again, both operands have to be sets of the same type.

• Similarly, e1 in e2, where e2 has the type Set[T] for some T and e1 has type T, evaluates to true if and only e1 is a member of e2.

• set cardinality |s| evaluates to a non-negative integer that represents the cardinality of s.

Similar to sets, Viper supports multisets:

• Literals can either be empty multisets (Multiset[T]()) or non-empty ones (Multiset(e, ...)) and work like their set counterparts.

• As for sets, the operations union, intersection, setminus, subset and cardinality |s| are supported.

• e1 in e2, where e1 has type T and e2 has type Multiset[T], denotes the multiplicity of e1 in e2.

Sequence expressions

Viper’s built-in sequence type Seq[T] represents immutable finite sequences of elements of type T.

• empty sequence: Seq[T]() evaluates to an empty sequence of type Seq[T]. As with empty set literals, the type argument only has to be stated explicitly if it is not clear from the surrounding context.

• sequence literal: Seq(e1, e2, ..., en), where e1-en are expressions that all have a common type T, represents a sequence of type Seq[T] of n elements, whose elements are the argument expressions (i.e., the first element is e1, the second e2 etc., and the last is en.

• integer sequence literals: [e1..e2], where e1 and e2 are Int-typed, represent the sequence of integers ranging from e1 until, but excluding, e2 (or the empty sequence, if e2 is less than or equal to e1).

• sequence lookup: s[e], where s is an expression of type Seq[T] for some T and e is an integer, returns the element at index e in the sequence. As a well-definedness condition, e must be known to be non-negative and less than the length of s, otherwise this expression will result in a verification error.

• sequence concatenation: e1 ++ e2 results in a new sequence whose elements are the elements of e1, followed by those of e2.

• sequence update: s[e1 := e2], where e1 has type Int, s has type Seq[T] for some T and e2 is of type T, denotes the sequence that is identical to s, except that the element at index e1 is e2 (the operation does not change the original sequence’s value, but rather defines a new sequence).

• sub-sequence operations: s[e1..e2], where s is a sequence and e1 and e2 are integers, returns a new sequence that contains the elements of s starting from index e1 until (but not including) index e2. The values of e1 and e2 need not be valid indices for the sequence; for negative e1 the operator behaves as if e1 were equal to 0, for e1 larger than |s| the empty sequence will result (and vice versa for e2). Optionally, either the first or the second index can be left out (leading to expressions of the form s[..e] and s[e..]), in which case only elements at the end or at the beginning of s are dropped, respectively.

• sequence length: |s| returns the length of a sequence as an integer.

• sequence member check: e in s evaluates to true if e is in the sequence s.

Map expressions

Viper has a built-in parametric partial map type Map[K, V], where K is a type parameter that describes the type of the keys in the map, and V similarly describes the type of the values in the map. Like sets, maps are immutable (i.e., they represent mathematical maps), and they map a finite amount of keys to values.

• empty map: Map[K, V]() returns an empty map of type Map[K, V], i.e., a map whose domain is empty. As with empty set literals, the type arguments are optional if the type is clear from the context; then simply Map() can be written.

• map literals: Map(e1 := e2, e3 := e4, ...) evaluates to a map containing only the explicitly enumerated mappings (i.e, mapping e1 to e2, e3 to e4 etc.), where all keys must have a common type K and all values must have a common type V s.t. the resulting expression has the type Map[K, V]. If multiple key-value pairs have the same key, the map contains the value of the latest such pair. I.e., Map(1 := 3, 1 := 5) will contain value 5 for key 1.

• map domain and range: If m is a map of type Map[K, V], then domain(m) is the finite domain (of type Set[K]) of m, i.e., the set of keys it contains. Similarly, range(m) denotes the range (or image) of m, which has type Set[V].

• map cardinality: |m| denotes the cardinality of the map m, i.e., the cardinality of its domain.

• map lookup: If m is a map of type Map[K, V] and e has a type K, then m[e] looks up the value of key e in m. As a well-definedness condition, e must be known to be in the domain of m, otherwise this expression will result in a verification error.

• map membership: e in m, where m has type Map[K, V] and e has type K, is true iff e is in the domain of m.

• map update: If m is a map of type Map[K, V], e1 has type K and e2 has type V, then m[e1 := e2] is a map that is identical to m except the key-value-pair e1 -> e2 has been added (or its value updated).

Perm expressions

Expressions of type Perm are real numbers and are usually used to represent permission amounts (though they can be used for other purposes).

• Fractional permission expressions e1/e2, where both e1 and e2 are integers, evaluate to a Perm value whose numerator is e1 and whose denominator is e2. A well-definedness condition is that e2 must not equal 0. e1/e2 can also denote a permission amount divided by an integer if e1 is an expression of type Perm and e2 is an expression of type Int.

• The Perm-typed literals none and write denote no permission and a full permission, corresponding to 0/1 and 1/1, respectively.

• The special literal wildcard denotes an unspecified positive amount. The specific value represented by wildcard is not fixed but is chosen anew every time a wildcard expression is encountered. In particular, if a wildcard amount of permission to a field or predicate is to be exhaled, a value less than the currently held amount is chosen, so that it is always possible to exhale (or assert having) a wildcard amount of any permission, unless no permission at all is currently held. However, exhaling and subsequently inhaling a wildcard permission to a location will not restore the initial permission amount, since it will not be known that the inhaled wildcard amount is equal to the exhaled one. The wildcard permission amount provides a convenient way to implement duplicable read-only resources, which, for example, can be used to model immutable data.

• perm(e.f): Evaluates to the amount of permission the current method has to the specified location. Similarly, perm(P(e1, ..., en)) returns the amount of permission held for the specified predicate instance. Similarly to forperm, the values of perm expressions take into account the side-effects of e.g. ongoing exhale operations. For example exhale acc(x.f) && perm(x.f) > none should always fail, while exhale perm(x.f) > none && acc(x.f) will succeed (if the full permission to x.f was held before the exhale).

• Permissions can be added (e1 + e2), subtracted (e1 - e2), multiplied (e1 * e2) or divided (e1 / e2). In the first two cases, e1 and e2 must both be Perm-typed. In the case of multiplication, either two Perm-typed expressions or one Perm-typed and one Int-typed are possible. For divisions, the first expression must be Perm-typed and the second of type Int (though of course ordinary fractions, where both expressions are Int-typed, also exist). The results in all cases will be Perm-typed expressions.

Reference expressions

Reference-typed expressions denote either objects or the special value null (denoted by the null literal). The most important fact about null is the built-in assertion that permissions to fields of null cannot exist; Viper will deduce from holding a field permission that the receiver of the field location is non-null. The analogous assumption does not hold e.g. for predicate instances; it is possible for null to be the value of a predicate parameter.

Assertions

In addition to expressions, Viper supports three kinds of resource assertions: accessibility predicates for fields, predicate instances, and magic wands.

• boolean expressions: any boolean expression can also be used as an assertion. This includes disjunctions; note, however, that Viper does not support disjunctions whose disjuncts contain resource assertions (such as accessibility predicates). If such a disjunction is desired where at least one of the operands is a pure expression, this can be rewritten as a conditional expression.

• Field permissions are denoted by accessibility predicates acc(e.f) and acc(e.f, p), where e is an expression of type Ref, f is a field in the program, and p is an expression of type Perm, denotes a permission amount of p to the field e.v. The former is equivalent to writing acc(e.f, write).

• Predicate instances can be denoted by acc(P(...)) and acc(P(...), p) (the latter denotes a permission amount p of a predicate instance P(...)). Where the amount is write, the simpler syntax P(...) can also be used.

• Magic wands A1 --* A2 represent the permission to get assertion A2 in exchange for giving up assertion A1.

• Conjunctions A1 && A2 (similar to the separating conjunction * in separation logic); see the section on fractional permissions.

• Implication b ==> A, denotes that assertion A holds if the boolean expression b is true. Note that only pure expressions may occur on the left of implications, not general assertions.

• Quantified assertions forall x: T :: A, where A is a Viper assertion potentially containing resource assertions, such as accessibility predicates acc(e.f, p).

• Conditional assertions e ? A1 : A2 denote that A1 holds if the (pure) boolean expression e is true, and otherwise A2 holds.

• Inhale-exhale assertions [A1, A2] behave like A1 when they are inhaled, but like A2 when they are exhaled. As an example, inhale [acc(x.f), acc(x.g)] is equivalent to inhale acc(x.f). On the other hand, the assertion [acc(x.f), acc(x.g)] in a method precondition means that when calling the method, the caller has to give up (exhale) a permission to x.g, but when verifying the method body, the verifier inhales permission to x.f at the beginning.

Domains

Domains enable the definition of additional types, mathematical functions, and axioms that provide their properties. Syntactically, domains consist of a name (for the newly-introduced type), and a block in which a number of function declarations and axioms can be introduced. The following example shows a simple domain declaration:

domain MyDomain {
  function foo(): Int
  function bar(x: Bool): Bool

  axiom axFoo { foo() > 0 }
  axiom axBar { bar(true) }
  axiom axFoobar { bar(false) ==> foo() == 3 }
}

The functions declared in a domain are global: one can apply them in any other scope of the Viper program. Functions declared inside a domain are called domain functions; these are more restricted than the standard Viper functions described previously. In particular, domain functions cannot have preconditions; they can be applied in any state. They are also always abstract, i.e., cannot have a defined body. The typical way to attach any meaning to these otherwise-uninterpreted functions, is via domain axioms.

Domain axioms are also global: they define properties of the program which are assumed to hold in all states. Since there is no restriction on the states to which an axiom applies, it must be well-defined in all states; for this reason, domain axioms cannot refer to the values of heap locations, nor to permission amounts (e.g., via perm expressions). In practice, domain axioms are standard (often quantified) first-order logic assertions. Axiom names are used only for readability of the code, but are currently not optional.

domain Neg {
  function not(cond: Bool): Bool
}
method TestNeg() {
  
}

In the remainder of this section, we will illustrate some use-cases of domains for defining additional types and concepts within a Viper program.

Declaring New Types

The following domain defines a new type MyType with two type parameters:

domain MyType[A,B] {
  function constructor_a(x: A): MyType[A,B]
  function constructor_b(y: B): MyType[A,B]
  function bin_oper(a: MyType[A,B], b: MyType[A,B]): MyType[A,B]
}

In this demo we create a new type called MyType. It’s defined with two type parameters A and B. The encoding provides two constructors and a binary operation.

The following client code demonstrates how the type MyType can be used in a Viper program:

method client(a: MyType[Bool, Ref])
  requires a == bin_oper(a,a)
{
  var b: MyType[Int, Bool] := constructor_a(11)
  var c: MyType[Int, Bool] := constructor_b(true)
  var d: MyType[Int, Bool] := bin_oper(b,c)
  
  // The following code will be rejected by the typechecker,
  //  because a and d are created with different type parameters.
  var e: MyType[Int, Bool] := bin_oper(a,d)
}

Note that Viper does not support type parameters as part of function and method declarations; the only type parameters available are those of the enclosing domain. This has the practical implication that one must instantiate all type parameters of a domain type before they can be used outside of the domain definition. This instantiation is usually implicit; so long as the type-checker can figure out the necessary instantiations, then they need not be provided.

By default, Viper provides only two built-in operations for domain types: == and !=. Other operations must be encoded via domain functions and given a meaning via appropriate domain axioms.

Encoding a list ADT

For a more realistic demo, we can encode an algebraic list datatype. We start with the function declarations.

domain List[T] {
  /* Constructors */

  function Nil(): List[T]
  function Cons(head: T, tail: List[T]): List[T]

  /* Destructors */

  function head_Cons(xs: List[T]): T  // requires is_Cons(xs)
  function tail_Cons(xs: List[T]): List[T] // requires is_Cons(xs)

  /* Constructor types */

  function type(xs: List[T]): Int
  unique function type_Nil(): Int
  unique function type_Cons(): Int

  /* Axioms omitted */
}

A mathematical list needs two constructors: Nil, which takes no parameters, and Cons, which takes a scalar value of type T and the tail list (of type List[T]) and returns the new list.

An algebraic datatype requires a deconstructor for each parameter of each of its constructors (alternatively, if one implements tuple types, one can write a single destructor for each constructor). In the case of a list ADT, we will need only two deconstructors (for the parameters of Cons). The deconstructors are called head_Cons and tail_Cons, respectively.

Finally, we encode the integer function type which denotes whether the type of the list constructor is Nil or Cons. In this case, we could use a Boolean function for this purpose. However, there could be more than two constructors of an ADT in the general case.

Notice that we use the keyword unique to declare domain functions whose values (while not concretely known) should be guaranteed different from those returned by any other unique domain functions. Sometimes the value of a function is irrelevant by itself, as long as it is guaranteed to be unique, as in the case of identification functions type_Nil and type_Cons.

The full axiomatisation of these functions and some use cases are available in the Encoding ADTs example.

Axiomatising Custom Theories

Custom axioms can specify the semantics of mathematical functions. Functions typically have parameters, hence the corresponding axioms usually have a universal quantification over a range of possible values as arguments.

domain MyInteger {
  function create_int(x: Int): MyInteger
  function get_value(a: MyInteger): Int
  function sum(a: MyInteger, b: MyInteger): MyInteger

  axiom axSum {
    forall a: MyInteger, b: MyInteger ::
      sum(a,b) == create_int( get_value(a) + get_value(b) )
  }
}

Triggers

The axiom axSum uses a universal quantifier (forall) to express the value of sum(a,b) for arbitrary values of the arguments a and b, respectively. In order to apply this axiom, the SMT-solver must instantiate the quantified variables with concrete values in a given context. So, as explained in the section on quantifiers, it is highly important to choose good triggers for the quantifier to avoid performance issues during the verification. Poorly selected triggers may as well disallow the instantiation of the quantifier in cases in which the instantiation is needed.

In the example above, one would intuitively expect the axiom axSum to be applied whenever the function sum is applied to some pair of arguments. Therefore, it makes sense to specify { sum(a,b) } as the trigger, as in the example below:

domain MyInteger {
  function create_int(x: Int): MyInteger
  function get_value(a: MyInteger): Int
  function sum(a: MyInteger, b: MyInteger): MyInteger

  axiom axSum {
    forall a: MyInteger, b: MyInteger :: { sum(a,b) }
      sum(a,b) == create_int( get_value(a) + get_value(b) )
  }
}

Recall that a proper trigger must mention all quantified variables.

If the trigger is omitted, the Viper tools attempt to infer possible triggers based on the body of the quantifier. However, this can result in unreliable behavior in general; it is highly recommended to write triggers for all quantifiers in a Viper program. For example, if we remove the explicit trigger from the case above, any of the following triggering terms could be inferred:

• { sum(a,b) }
• { get_value(a), get_value(b) }
• { sum(a,b) }{ get_value(a), get_value(b) }
• { get_value(a), get_value(b) }{ sum(a,b) }

Recall that, for example, create_int( get_value(a) + get_value(b) ) is not a valid trigger, because it contains the arithmetic operator +.

Sometimes it is beneficial to choose expressions as triggers which are not present in the body of the quantifier. For instance, this can help avoiding matching loops.

Axiom-Caused Unsoundness

Just as with any other kind of assumption in a Viper program, one can introduce unsoundness by adding domain axioms. For the simplest case, consider the following example:

domain Foo {
  axiom Unsound {
    false
  }
}

method smokeTest()
{
  assert false // verifies
}

Viper is able to verify this program, because the axiom Unsound is assumed to be true in all states of the program, including those in the body of method test. While unsoundnesses in user-defined axioms can be much more subtle than in this example, the simple strategy of checking that one cannot assert false at various program points is often effective for catching basic mistakes, especially if some other surprising verification result has already been observed.

domain MyInteger {
  // copied from above without modification

  function create_int(x: Int): MyInteger
  function get_value(a: MyInteger): Int
  function sum(a: MyInteger, b: MyInteger): MyInteger

  axiom axSum {
    forall a: MyInteger, b: MyInteger :: { sum(a,b) }
      sum(a,b) == create_int( get_value(a) + get_value(b) )
  }
}

method test1(a: MyInteger, b: MyInteger)
{
    assert get_value(a) + get_value(b) == get_value(sum(a, b))
}

method test2(i: Int, j:Int) 
{
    assert sum(create_int(i), create_int(j)) == create_int(i + j)
}

Encoding an Array Datatype

We proceed next with an example encoding of integer arrays. Note that arrays are not built-in in Viper, so we have to model an array type. Since array locations should be on the heap, we do this with a combination of a domain, a field, and quantified permissions. Let’s first consider a suitable domain:

domain IArray {
  function slot(a: IArray, i: Int): Ref
  function len(a: IArray): Int
  function first(r: Ref): IArray
  function second(r: Ref): Int

  axiom all_diff {
    forall a: IArray, i: Int :: { slot(a,i) }
      first(slot(a,i)) == a && second(slot(a,i)) == i
  }

  axiom len_nonneg {
    forall a: IArray :: { len(a) }
      len(a) >= 0
  }
}

The idea behind this encoding is to use a Viper object with a single field to model each element of an array. The slot function plays the role of mapping an array and integer index to the corresponding object. We express that the slot function is an injective mapping via the axiom all_diff. The two functions first and second together play the role of an inverse to the slot(a,i) function. Intuitively, this axiom expresses that for any pair (a, i), slot(a,i) defines a unique object (which will provide the memory location for the array slot).

In addition, we encode the function len for specifying the length of an array, and the axiom len_nonneg which expresses that this function never returns a negative value.

Because the elements of the array will be located on the heap, we need a way of specifying permissions to a range of array elements. We can use quantified permissions for this, and create the following macro definition for specifying permissions to the whole array:

field val: Int
define access(a)
  (forall j: Int :: 0 <= j && j < len(a) ==> acc(slot(a,j).val))

Note that, in Viper, we cannot refer to a specific program state from within domains. In particular, we cannot mention slot(a,j).val inside the axioms of IArray, since its meaning depends on a particular heap, and its well-definedness condition depends on whether one holds the appropriate permissions. The state-dependent part of the array encoding is taken care of outside of this domain. In particular, we can now use access in the following client code:

method client()
{
  // Create an integer array with three elements
  var a: IArray
  inhale len(a) == 3
  inhale access(a) // access to all array slots

  // Initialize the elements of an array
  var i: Int := 0
  while ( i < len(a) )
    invariant 0 <= i && i <= len(a)
  {
    slot(a,i).val := -i // models a[i] := -i
    i := i + 1
  }
}

This method will not verify without adding a suitable loop invariant. If we still try, Viper will report the following message:

Assignment might fail. There might be insufficient permission to access slot(a, i).val.

Therefore, we need to add the permissions to the array to the loop invariant:

    invariant access(a)

For a complete example, see the Viper encoding of Array Max, by elimination.

ADT plugin

adt List[T] {
    Nil()
    Cons(value: T, tail: List[T])
}

The ADT plugin is an extension to Viper that enables declarations of algebraic datatypes. These consist of one or more constructors, each of which has zero or more arguments. Any instance of such a datatype then corresponds to exactly one of these constructors. There is syntax to identify which constructor an instance corresponds to, as well as syntax to extract the arguments given to that constructor. As in the example above, ADTs can have type parameters and can be recursive.

Internally, the ADT syntax is translated into domains and domain function applications.

The plugin is enabled by default, and can be disabled with the command-line option --disableAdtPlugin.

Discriminators

For every constructor of the ADT, a discriminator is generated. Syntactically, this is a field with the name is<Constructor> with a Bool type.

assert Cons(1, Cons(2, Nil())).isCons
assert Nil().isNil

Destructors

For every argument of every constructor of the ADT, a destructor is generated. This is a field that extracts the given argument out of the constructor, and is only well-defined if the ADT instance is known to correspond to the given constructor.

assert Cons(1, Nil()).value == 1
assert Cons(1, Nil()).tail == Nil()

// this expression is not well-defined:
// assert Nil().value == 0

Derived methods

Similarly to derivable methods in Haskell or Rust, the ADT plugin provides a syntax to derive certain operations for ADTs.

import <adt/derives.vpr>

adt Name[...] {
    ...
} derives {
    ...
}

The block after the derives keyword is a list of functions that should be derived for the declared ADT. At the moment, the only supported operation is contains, but this may change in future Viper versions.

Note that the import <adt/derives.vpr> is required for derived methods to work.

contains

When the contains operation is derived, the function contains becomes available for the given ADT. Given a value and an ADT instance, it evaluates to true if the former value is found inside the ADT instance, as one of its constructor arguments. This evaluation works recursively.

import <adt/derives.vpr>

adt List[T] {
    Nil()
    Cons(value: T, tail: List[T])
} derives {
    contains
}

method client() {
    var x: List[Int] := Cons(42, Cons(33, Nil()))

    // test for the "value" argument of type Int in x
    assert contains(42, x)
    assert contains(33, x)

    // test for the "tail" argument of type List[Int] in x
    assert contains((Nil() : List[Int]), x)
    assert contains(Cons(33, Nil()), x)
}

By default, all arguments of every constructor of an ADT are considered for the contains operation. An optional comma-separated blocklist of arguments can be declared, which will cause the contains operation to ignore the named arguments:

import <adt/derives.vpr>

adt Tree[T] {
    Leaf()
    Node(left: Tree[T], val: T, right: Tree[T])
} derives {
    contains without left, right
}

method client() {
    var x: Tree[Int] := Node(Leaf(), 42, Node(Leaf(), 33, Leaf()))
    assert contains(42, x)

    // will fail to prove, because "right" is in the blocklist
    assert contains(Node(Leaf(), 33, Leaf()), x)
}

Termination

In the context of Viper, reasoning about termination serves (at least) two purposes: the first is to prove that methods and loops terminate, the second is to ensure that Viper functions are well-defined. The second aspect is crucial even for front-ends that are not concerned with total correctness, because specifications, often expressed via Viper’s pure functions, would be meaningless if the involved functions were not well-defined.

Front-ends can encode their own termination proofs, but since proving termination is a crucial verification task with subtle pitfalls, Viper has support for termination proofs built in, based on the well-known approach of termination measures (ranking functions).

In this section, we will introduce Viper’s support for termination proofs: first, we describe how to specify termination measures, then we explain termination proofs for (mutually) recursive functions. Lastly, we discuss our experimental support for termination of methods and loops.

Termination Measures and Decreases Clauses

To prove termination of a recursive function, users must specify a termination measure: a Viper expression whose value is checked to decrease at each recursive call, with respect to a well-founded order. Termination measures are provided via decreases clauses:

decreases <tuple>

Here, <tuple> denotes the termination measure: a tuple of comma-separated expressions. In this tutorial, we interchangeably refer to <tuple> as termination measure and decreases tuple. For functions and methods, a decreases tuple can be defined at the position of a precondition, for a loop, at the position of an invariant.

A common example for termination is the standard factorial function, which terminates because its argument decreases with respect to the usual well-founded order over non-negative numbers.

import <decreases/int.vpr>

function factorial(n:Int) : Int
  requires 0 <= n
  decreases n
{ n == 0 ? 1 : n * factorial(n-1) }

Viper successfully verifies that factorial terminates: at each recursive invocation, it checks that 1. the termination measure n decreases and 2. remains non-negative, i.e. cannot decrease infinitely often. The corresponding well-founded order over non-negative numbers is defined in the file decreases/int.vpr, which is part of Viper’s standard library.

Hint: If you run the Viper verifier Silicon (symbolic-execution-based) with --printTranslatedProgram then you can inspect the verification conditions generated for termination checks.

Predefined Well-Founded Orders

Viper’s standard library provides definitions of well-founded orders for most types built into Viper, all of which can be imported from the decreases folder. The following table lists all provided orders; we write s1 <_ s2 if s1 is less than s2 with respect to the order.

All definitions are straightforward, except the last one, which is concerned with using predicate instances as termination measures. Due to the least fixed-point interpretation of predicates, any predicate instance has a finite depth, i.e., can be unfolded only a finite number of times. This implies that a predicate instance p1, which is nested inside a predicate instance p2, has a smaller (and non-negative) depth than p2.

Viper uses this nesting depth to enable termination checks based on predicate instances, as illustrated by the next example, the recursive computation of the length of a linked list: intuitively, the remainder of the linked list, represented by predicate instance list(this), is used as the termination measure. This works because the recursive call is nested under the unfolding of list(this), and takes the smaller predicate instance list(this.next).

import <decreases/predicate_instance.vpr>

field next: Ref

predicate list(this: Ref) {
  acc(this.next) &&
  (this.next != null ==> list(this.next))
}

function length(this: Ref): Int
  decreases list(this)
  requires list(this)
{
  unfolding list(this) in this.next == null ? 1 : 1 + length(this.next)
}

Final remarks:

• Note that PredicateInstance is an internal Viper type, and currently supported only in decreases tuples. The nested function is also internal and cannot be used by users.
• For simplicity, all standard well-founded orders can be imported via decreases/all.vpr.
• Users can also define custom well-founded orders.

General Syntax of Decreases Tuples

In the previous examples, the termination measures were always single expressions. However, it is not always possible to find a single expression whose value decreases at each call, and Viper therefore also supports tuples of expressions as termination measures. The well-founded order for tuples is the lexicographical order over the tuple elements.

More precisely, let (a1, a2, ...) and (b1, b2, ...) be two non-empty tuples of finite (and for now, equal) length. The well-founded order <_ that is used to compare the two tuples is defined as follows:

(a1, a2, ...) <_ (b1, b2, ...)
    <==>
a1 <_ b1  ||  a1 == b1 && (a2, ...) <_ (b2, ...) 

Special cases, such as empty tuples, tuples of different length, and tuples of different types will be discussed later.

A typical example of a function for which a tuple as termination measure is used, is the Ackermann function:

import <decreases/int.vpr>

function ack(m:Int, n:Int):Int
  decreases m, n
  requires m >= 0
  requires n >= 0
  ensures result >= 0
{
  m == 0 ? n + 1 :
  n == 0 ? ack(m - 1, 1) :
           ack(m - 1, ack(m, n - 1))
}

For the first recursive call ack(m - 1, 1), and the second (outer) recursive call ack(m - 1, ack(m, n - 1)), the first expression of the tuple (i.e. m) decreases. Hence, the other part of the tuple is not required to decrease in this situation. In the third (inner) recursive call ack(m, n - 1) the first expression of the tuple is unchanged, while the second tuple expression (i.e., n) decreases.

The well-founded order over tuples need not be imported (and cannot be customised). However, the well-founded orders of the types appearing in the tuple must be.

Special Decreases Tuples

Viper supports three decreases tuples whose semantics require further explanation.

Empty

Consider the following pair of functions, where compute performs the actual computation and facade merely provides a default argument to compute:

function facade(i: Int): Int { compute(i, 0) }

function compute(i: Int, j: Int): Int { i + j }

Both functions have no decreases clause, and Viper thus won’t generate termination checks. This may be fine now, since the functions obviously terminate, but if compute were changed to be recursive, potential non-termination might go unnoticed. To account for future code changes, users could use “artificial” constants as termination measures, but Viper offers a better solution: to ensure termination checks, users can specify empty tuples, and Viper’s call-graph analysis (performed to detect mutual recursion) effectively infers suitable constants.

function facade(i: Int): Int
  decreases
{ compute(i, 0) }

function compute(i: Int, j: Int): Int
  decreases
{ i + j }

Wildcard

Specifying a termination measure might not always be an option: it could be too cumbersome to express; it could be considered a problem to deal with later; or it could be that the function does not terminate in an operational sense, but is nevertheless well-defined. Simply omitting the decreases clause, however, might not be an option, e.g. because the function is called from another function, for which termination checks are generated.

For such cases, Viper offers the wildcard measure _, which expresses that a function is to be considered terminating, although no termination checks are generated. I.e., a wildcard measure is effectively an unchecked assumption.

function collatz(n: Int): Int
  requires 1 <= n
  decreases _ // I can't be bothered right now
{
  n == 1     ? n :
  n % 2 == 0 ? collatz(n / 2) :
               collatz(3 * n + 1)
}

Star

To explicate that a function is not checked for termination, and may thus not terminate, Viper supports the star measure *. This special measure is equivalent to providing no decreases clause at all, but may be useful for documentation purposes.

function nonterm(): Int
  decreases * // Whole clause could as well have been omitted
{ 1 + nonterm() }

Custom Well-Founded Orders

As previously mentioned, Viper offers predefined orders for its built-in types, plus predicates. However, via domains, Viper also enables users to define their own types. In order to use values of custom types as termination measures, users can either define a mapping from custom to some built-in type, or they can directly define a well-founded order over the custom type.

In the remainder of this subsection, both approaches will be illustrated using a combination of the MyInt example (from the earlier subsection on domains) and a factorial function operating on MyInt. In the example below, the destructor get is used to map a MyInt to a regular Int, which indirectly allows using MyInt in the function’s decreases clause.

import <decreases/int.vpr>

domain MyInt {
  function put(i: Int): MyInt   // Constructor
  function get(m: MyInt): Int   // Destructor
  function dec(m: MyInt): MyInt // Decrement by one

  axiom { forall i: Int   :: {put(i)} get(put(i)) == i }
  axiom { forall m: MyInt :: {dec(m)} dec(m) == put(get(m) - 1) }
}

function factorial(m: MyInt): Int
  requires 0 <= get(m)
  decreases get(m)
{ get(m) == 0 ? 1 : get(m) * factorial(dec(m)) }

Alternatively, a well-founded order for MyInt itself can be defined, by instantiating the two special functions decreasing and bounded for MyInt. The necessary function declarations are provided by the standard library file decreases/declaration.vpr, and look as follows:

domain WellFoundedOrder[T] {
  // v1 is less than v2
  function decreasing(v1: T, v2: T): Bool

  // v is bounded
  function bounded(v: T): Bool
}

Function decreasing is used to define an order between two values v1 and v2 of a custom type T, whereas function bounded is used to define a lower bound on such values. Suitable definitions for both suffices to establish a well-founded order (where, as before, <_ denotes well-founded less-than over type T):

v1 <_ v2 <==> decreasing(v1, v2) && bounded(v2)

The necessary properties of decreasing and bounded for values of type T can be defined via axioms. For the MyInt type from before, suitable axioms would be:

domain MyIntWellFoundedOrder {
  axiom {
    forall i1: MyInt, i2: MyInt :: {decreasing(i1, i2)}
      get(i1) < get(i2) <==> decreasing(i1, i2)
  }
  axiom {
    forall m: MyInt :: {bounded(m)}
      0 <= get(m) <==> bounded(m)
  }
}

Mutual Recursion

For mutually recursive functions, Viper implements the following approach (as, e.g., Dafny does as well): given a mutually recursive function fun, Viper verifies that fun function’s termination measure decreases at each indirectly recursive call. By transitivity, this implies that the measure decreased by the time a recursive invocation of fun takes place.

A simple case of mutual recursion is illustrated next, by functions is_even and is_odd:

import <decreases/int.vpr>

function is_even(x: Int): Bool
  requires x >= 0
  decreases x
{
  x == 0 ? true : is_odd(x-1)
}

function is_odd(y: Int): Bool
  requires y >= 0
  decreases y
{
  y == 0 ? false : is_even(y-1)
}

Consider function is_even: its termination measure x decreases at the indirectly recursive call is_odd(x-1), where is_odd‘s termination measure y is instantiated with x-1. Analogously, is_odd‘s termination measure decreases at the call is_even(y-1).

In the example above, the two termination measures are tuples of equal length and type. However, this is not required of mutually recursive functions in order to prove their termination. Consider the next example (which verifies successfully):

import <decreases/int.vpr>
import <decreases/bool.vpr>

function fun1(y: Int, b: Bool): Int
  decreases y, b
{
    (b     ? fun1(y, false) : 1)
  + (y > 0 ? fun2(y-1)      : 2)
}

function fun2(x: Int): Int
  decreases x
{
  x > 0 ? fun1(x-1, true) : 3
}

At indirectly recursive calls, two decreases tuples are compared by lexicographical order of their longest commonly typed prefix (as does, e.g., Dafny). E.g., for the indirectly recursive call fun2(y-1) in function fun1, Viper verifies that y-1 <_ y, while for the recursive call fun1(y, false), it verifies that y <_ y || (y == y && false <_ b).

Conditional Termination

We previously presented three different kinds of termination measures: a tuple e1, e2, ..., wildcard (_) for “just assume that something is decreased”, and star (*) for “no termination guarantees”. It may sometimes be desirable to combine different measures, e.g., in order to reduce proof efforts by providing a concrete decreases tuple for certain invocations only, complemented with wildcard for the remaining invocations.

To account for this, Viper supports conditional termination clauses for the first two kinds of measures (tuple, wildcard):

decreases <tuple> ... if <condition>
decreases _ if <condition>

Here, <condition> is a boolean expression under which the decreases clause is considered. Omitting the condition, as in all previous examples, is equivalent to if true. Consequently, the condition false is equivalent to not providing the decreases clause at all.

The following example illustrates combined conditional termination clauses: function sign promises to decrease x if positive, and something (wildcard) if x is negative. In case x is zero, function sign does not (promise to) terminate.

import <decreases/int.vpr>

function sign(x: Int): Int
  decreases x if 1 <= x
  decreases _ if x <= -1
  decreases * // can also simply be omitted
{
  x == 0 ? sign(x) :
  1 < x  ? sign(x - 1) :
  x < -1 ? sign(x + 1) :
           x
}

function caller(x: Int): Int
  decreases // must terminate
{
  sign(x)
}

We refer to the condition 1 <= x of the clause decreases x if 1 <= x as the tuple condition, and to the condition x <= -1 of the clause decreases _ if x <= -1 as the wildcard condition. The disjunction of the two, i.e. 1 <= x || x <= -1, is the termination condition: the effective condition under which the function (promises to) terminate.

When verifying termination of function sign, the following happens for the recursive invocations:

• sign(x): the termination check is vacuous because the invocation happens under the condition x == 0, for which sign does not promise to terminate.
• sign(x - 1): termination measure x is checked to decrease, since the call happens under the condition 1 < x (which implies that the tuple condition held in the prestate).
• sign(x + 1): the termination check is again vacuous because the call happens under the condition x < -1, which is the wildcard condition and termination is therefore assumed.

When verifying termination of function caller, the termination condition (1 <= x || x <= -1) of function sign is checked. Since it might not hold, a verification error is reported.

Abstract Functions

Termination specifications may also be provided for abstract functions, i.e., those without a body. For such functions, Viper still checks that the specifications are well-defined, which includes that pre- and postconditions terminate. Moreover, when an abstract function is invoked, the previously described call-site checks are made.

When Viper performs a call-graph analysis to determine (mutually) recursive functions, abstract functions are assumed to not (mutually) recurse through their omitted bodies.

Statement Termination (Experimental)

Function Calls inside Methods

The development of the experimental termination checks for statements was (partly) guided by Viper’s intended use as an intermediate verification language. This shaped, among other things, how function calls are currently handled.

Front-ends may use Viper functions to encode (pure) program procedures, but front-ends may also use functions to encode proof-relevant properties. Similarly, Viper methods may be used to encode (impure) program procedures, or to encode proof lemmas. Moreover, encoded program procedures could contain Viper statements corresponding to actual statements from the source program, interspersed with Viper statements that encode additional proof steps.

Consider, e.g., a Viper function elems(ll) that denotes the set of all elements in a potentially cyclic linked-list ll. A statement such as inhale es == elems(ll) could be part of a sequence of program statements – in which case elems(ll) should probably terminate – or it could be an additional proof step, in which case it would suffice if elems(ll) were well-defined.

Function calls inside Viper methods (as opposed to calls inside functions) are therefore (currently) not checked for termination, and front-ends would have to generate appropriate checks where necessary. Potential alternatives include (1) to always check termination – probably too restrictive; (2) to enable users to specify termination requirements at call-site, e.g. via attributes/annotations – but the latter are currently not available in Viper.

Checking Loop Termination

Viper while loops accept decreases clauses at the position of invariants, as illustrated by the following code snippet:

while (0 < i)
  decreases i
{
  i := i - 1
}

Given such a specification, Viper will check that the termination measure decreases if the loop body is executed again. A successful verification then implies that the loop is always bounded, i.e. that a finite unrolling of the loop exists. Consequently, Viper does (currently) not check that each statement, including nested loops, in the loop body terminates, as illustrated by the next code snippet:

method m() // might not terminate

while (0 < i)
  decreases i // verifies successfully (finite unrolling exists)
{
  i := i - 1
  m()
  while (true) {}
}

We have chosen this approach for the following reasons (but are happy to receive feedback and discuss alternatives):

• Minimality: Interpreting loop termination as “a finite unrolling exists” can be seen as the weakest reasonable requirement, compared to the stronger (additional) requirement of all body statements having to terminate (note: the same argument does not apply to functions).

• Uniformity: When proving method termination (discussed next), the current approach directly exhibits the same behaviour for both syntactic while loops and loops encoded via gotos and labels (but see the note below). If loop bodies had to terminate as well, it would pose the question of which statements belonged to a goto-loop, which in general cannot be answered unambiguously (e.g. when inferring a while-loop from a goto-loop). This uniformity also extends to manually unrolled loops, as some front-ends might do.

• Expressiveness: If loop body termination checks are required, they can be encoded easily (potentially via a synthetic method that only hosts the loop body).

• Familiarity: The same approach is used in Dafny (but then again, Dafny was not designed as an intermediate verification language)

Checking Method Termination

The currently implemented approach to checking termination of methods is similar to the previously described approach for functions: decreases clauses can be specified at the position of preconditions; if provided, Viper checks that the termination measure decreases in each directly or indirectly recursive method invocation, and that other method calls, as well as loops, terminate. As previous described, function applications inside methods are not checked for termination.

A straightforward example is method sum, shown next:

import <decreases/int.vpr>

method sum(n: Int) returns (res: Int)
  requires 0 <= n
  decreases
  ensures res == n * (n + 1) / 2
{
  res := 0
  var i: Int := 0;

  while (i <= n)
    invariant i <= (n + 1)
    invariant res == (i - 1) * i / 2
    decreases n-i
  {
    res := res + i
    i := i + 1
  }
}

Final remark: Viper (currently) performs a call-graph analysis for methods to detect mutually recursive methods. Analogous to the case of functions, this is done for convenience, to unburden users from having to write “artificial” constant termination measures for non-recursive methods. This could be considered a deviation from Viper’s otherwise method-modular verification approach – please let us know if this poses a problem in your use case of Viper.

More Information

For further information about the Viper project in general, please check out the project page. This includes links to papers about the language, and about encoding a variety of verification problems into Viper. More examples to try online can be found here. However, for more serious usage of Viper, we strongly recommend downloading a version to use locally, for example via our VSCode plugin. See the project page for more details.

If you have comments, questions or feedback about Viper, including this tutorial, we would be happy to receive them! Please send your emails to viper@inf.ethz.ch