Dafny is widely used in teaching[citation needed] because it provides a simple, integrated introduction to formal specification and verification; it is regularly featured in software verification competitions (e.g. VSTTE'08,[3] VSCOMP'10,[4] COST'11,[5] and VerifyThis'12[6]).
Dafny was designed as a verification-aware programming language, requiring verification along with code development. It thus fits the "Correct by Construction" software development paradigm. Verification proofs are supported by a mathematical toolbox that includes mathematical integers and reals, bit-vectors, sequences, sets, multisets, infinite sequences and sets, induction, co-induction, and calculational proofs. Verification obligations are discharged automatically, given sufficient specification. Dafny uses some program analysis to infer many specification assertions, reducing the burden on the user of writing specifications. The general proof framework is that of Hoare logic.
Dafny provides methods for implementation which may have side-effects and functions for use in specification which are pure.[9] Methods consist of sequences of statements following a familiar imperative style whilst, in contrast, the body of a function is simply an expression. Any side-effecting statements in a method (e.g. assigning an element of an array parameter) must be accounted for by noting which parameters can be mutated, using the modifies clause. Dafny also provides a range of immutable collection types including: sequences (e.g. seq<int>), sets (e.g. set<int>), maps (map<int,int>), tuples, inductive datatypes and mutable arrays (e.g. array<int>).
The following illustrates many of the features in Dafny, including the use of preconditions, postconditions, loop invariants and loop variants.
methodmax(arr:array<int>)returns(max:int)// Array must have at least one elementrequiresarr.Length>0// Return cannot be smaller than any element in arrayensuresforallj:int::j>=0&&j<arr.Length==>max>=arr[j]// Return must match some element in arrayensuresexistsj:int::j>=0&&j<arr.Length&&max==arr[j]{max:=arr[0];vari:int:=1;//while(i<arr.Length)// Index at most arr.Length (needed to show i==arr.Length after loop)invarianti<=arr.Length// No element seen so far larger than maxinvariantforallj:int::j>=0&&j<i==>max>=arr[j]// Some element seen so far matches maxinvariantexistsj:int::j>=0&&j<i&&max==arr[j]// arr.Length - i decreases at every step and is lower-bounded by 0decreasesarr.Length-i{// Update max if larger element encounteredif(arr[i]>max){max:=arr[i];}// Continue through arrayi:=i+1;}}
This example computes the maximum element of an array. The method's precondition and postcondition are given with the requires and ensures clauses (respectively). Likewise, the loop invariant and loop variant are given through the invariant and decreases clauses (respectively).
Loop invariants
The treatment of loop invariants in Dafny differs from traditional Hoare logic. Variables mutated in a loop are treated such that (most) information known about them prior to the loop is discarded. Information required to prove properties of such variables must be expressed explicitly in the loop invariant. In contrast, variables not mutated in the loop retain all information known about them beforehand. The following example illustrates using loops:
This fails verification because Dafny cannot establish that (sum + arr[i]) >= 0 holds at the assignment. From the precondition, intuitively, forall i :: 0 <= i < arr.Length ==> arr[i] >= 0 holds in the loop since arr[i] := arr[i]; is a NOP. However, this assignment causes Dafny to treat arr as a mutable variable and drop information known about it from before the loop. To verify this program in Dafny we can either (a) remove the redundant assignment arr[i] := arr[i];; or (b) add the loop invariant invariant forall i :: 0 <= i < arr.Length ==> arr[i] >= 0
Dafny additionally employs limited static analysis to infer simple loop invariants where possible. In the example above, it would seem that the loop invariant invariant i >= 0 is also required as variable i is mutated within the loop. Whilst the underlying logic does require such an invariant, Dafny automatically infers this and, hence, it can be omitted at the source level.
Proof features
Dafny includes features which further support its use as a proof assistant. Although proofs of simple properties within a function or method (as shown above) are not unusual for tools of this nature, Dafny also allows the proof of properties between one function and another. As is common for a proof assistant, such proofs are often inductive in nature. Dafny is perhaps unusual in employing method invocation as a mechanism for applying the inductive hypothesis. The following illustrates:
datatypeList=Nil|Link(data:int,next:List)functionsum(l:List):int{matchlcaseNil=>0caseLink(d,n)=>d+sum(n)}predicateisNatList(l:List){matchlcaseNil=>truecaseLink(d,n)=>d>=0&&isNatList(n)}lemmaNatSumLemma(l:List,n:int)requiresisNatList(l)&&n==sum(l)ensuresn>=0{matchlcaseNil=>// Discharged AutomaticallycaseLink(data,next)=>{// Apply Inductive HypothesisNatSumLemma(next,sum(next));// Check what known by Dafnyassertdata>=0;}}
Here, NatSumLemma proves a useful property betweensum() and isNatList() (i.e. that isNatList(l) ==> (sum(l) >= 0)). The use of a ghost method for encoding lemmas and theorems is standard in Dafny with recursion employed for induction (typically, structural induction). Case analysis is performed using match statements and non-inductive cases are often discharged automatically. The verifier must also have complete access to the contents of a function or predicate in order to unroll them as necessary. This has implications when used in conjunction with access modifiers. Specifically, hiding the contents of a function using the protected modifier can limit what properties can be established about it.
^Leino, Rustan (2010). Dafny: An Automatic Program Verifier for Functional Correctness. Proceedings of the Conference on Logic for Programming, Artificial Intelligence, and Reasoning. pp. 348–370. doi:10.1007/978-3-642-17511-4_20.
^Bormer, Thorsten; et al. (2011). The COST IC0701 Verification Competition 2011. Proceedings of the Conference on Formal Verification of Object-Oriented Software. pp. 3–21. CiteSeerX10.1.1.396.6170. doi:10.1007/978-3-642-31762-0_2.
^de Moura, Leonardo; Bjørner, Nikolaj (2008). Z3: An Efficient SMT Solver. Proceedings of the Conference on Tools and Algorithms for the Construction and Analysis. pp. 337–340. doi:10.1007/978-3-540-78800-3_24.
