Dafny
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Paradigm | Imperative,functional |
---|---|
Designed by | K. Rustan M. Leino |
Developer | Microsoft Research,AWS[citation needed] |
First appeared | 2009 |
Stable release | 3.7.2
/ July 14, 2022 |
Typing discipline | Static, strong, safe |
License | MIT License |
Filename extensions | .dfy |
Website | dafny |
Dafnyis animperativeandfunctionalcompiled languagethat compiles to otherprogramming languages,such asC#,Java,JavaScript,GoandPython.It supportsformal specificationthroughpreconditions,postconditions,loop invariants,loop variants,termination specifications and read/write framing specifications. The language combines ideas from thefunctionalandimperativeparadigms; it includes support forobject-oriented programming.Features includegeneric classes,dynamic allocation,inductive datatypesand a variation ofseparation logicknown asimplicit dynamic frames[1]for reasoning about side effects.[2]Dafny was created by Rustan Leino atMicrosoft Researchafter his previous work on developing ESC/Modula-3,ESC/Java,and Spec#.
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 ofHoare logic.
Dafny builds on the Boogieintermediate languagewhich uses theZ3 automated theorem proverfor discharging proof obligations.[7][8]
Data types
[edit]Dafny provides methods for implementation which may haveside-effectsand functions for use in specification which arepure.[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 themodifies
clause. Dafny also provides a range ofimmutablecollection types including: sequences (e.g.seq<int>
), sets (e.g.set<int>
), maps (map<int,int>
), tuples, inductive datatypes andmutablearrays (e.g.array<int>
).
Imperative features
[edit]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 element
requiresarr.Length>0
// Return cannot be smaller than any element in array
ensuresforallj:int::j>=0&&j<arr.Length==>max>=arr[j]
// Return must match some element in array
ensuresexistsj: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 max
invariantforallj:int::j>=0&&j<i==>max>=arr[j]
// Some element seen so far matches max
invariantexistsj:int::j>=0&&j<i&&max==arr[j]
// arr.Length - i decreases at every step and is lower-bounded by 0
decreasesarr.Length-i
{
// Update max if larger element encountered
if(arr[i]>max){
max:=arr[i];
}
// Continue through array
i:=i+1;
}
}
This example computes the maximum element of an array. The method's precondition and postcondition are given with therequires
andensures
clauses (respectively). Likewise, the loop invariant and loop variant are given through theinvariant
anddecreases
clauses (respectively).
Loop invariants
[edit]The treatment of loop invariants in Dafny differs from traditionalHoare 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:
methodsumAndZero(arr:array<int>)returns(sum:nat)
requiresforalli::0<=i<arr.Length==>arr[i]>=0
modifiesarr
{
vari:int:=0;
sum:=0;
//
while(i<arr.Length){
sum:=sum+arr[i];
arr[i]:=arr[i];
i:=i+1;
}
}
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 sincearr[i]:= arr[i];
is aNOP.However, this assignment causes Dafny to treatarr
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 assignmentarr[i]:= arr[i];
;or (b) add the loop invariantinvariant forall i:: 0 <= i < arr.Length ==> arr[i] >= 0
Dafny additionally employs limitedstatic analysisto infer simple loop invariants where possible. In the example above, it would seem that the loop invariantinvariant i >= 0
is also required as variablei
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
[edit]Dafny includes features which further support its use as aproof assistant.Although proofs of simple properties within afunction
ormethod
(as shown above) are not unusual for tools of this nature, Dafny also allows the proof of properties between onefunction
and another. As is common for aproof assistant,such proofs are ofteninductivein 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{
matchl
caseNil=>0
caseLink(d,n)=>d+sum(n)
}
predicateisNatList(l:List){
matchl
caseNil=>true
caseLink(d,n)=>d>=0&&isNatList(n)
}
lemmaNatSumLemma(l:List,n:int)
requiresisNatList(l)&&n==sum(l)
ensuresn>=0
{
matchl
caseNil=>
// Discharged Automatically
caseLink(data,next)=>{
// Apply Inductive Hypothesis
NatSumLemma(next,sum(next));
// Check what known by Dafny
assertdata>=0;
}
}
Here,NatSumLemma
proves a useful propertybetweensum()
andisNatList()
(i.e. thatisNatList(l) ==> (sum(l) >= 0)
). The use of aghost method
for encodinglemmas and theoremsis standard in Dafny with recursion employed for induction (typically,structural induction).Case analysisis performed usingmatch
statements and non-inductive cases are often discharged automatically. The verifier must also have complete access to the contents of afunction
orpredicate
in order to unroll them as necessary. This has implications when used in conjunction withaccess modifiers.Specifically, hiding the contents of afunction
using theprotected
modifier can limit what properties can be established about it.
See also
[edit]References
[edit]- ^Smans, Jan; Jacobs, Bart; Piessens, Frank (2009).Implicit Dynamic Frames: Combining Dynamic Frames and Separation Logic(PDF).Proceedings of the Conference on European Conference on Object Oriented Programming. pp. 148–172.doi:10.1007/978-3-642-03013-0_8.
- ^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.
- ^Leino, Rustan; Monahan, Rosemary (2010).Dafny Meets the Verification Benchmarks Challenge(PDF).International Conference on Verified Software: Theories, Tools, and Experiments. pp. 112–116.doi:10.1007/978-3-642-15057-9_8.
- ^Klebanov, Vladimir; et al. (2011).The 1st Verified Software Competition: Experience Report.Proceedings of the Conference on Formal Methods. pp. 154–168.CiteSeerX10.1.1.221.6890.doi:10.1007/978-3-642-21437-0_14.
- ^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.
- ^Huisman, Marieke; Klebanov, Vladimir; Monahan, Rosemary (2015)."VerifyThis 2012"(PDF).International Journal on Software Tools for Technology Transfer.17(6): 647–657.doi:10.1007/s10009-015-0396-8.S2CID14301377.
- ^"Z3 Homepage".GitHub.2019-02-07.
- ^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.
- ^"Dafny Programming Language".2022-07-14.
Further reading
[edit]- Meyer, Bertrand; Nordio, Martin, eds. (2012).Tools for Practical Software Verification: International Summer School, LASER 2011, Elba Island, Italy, Revised Tutorial Lectures.Springer.ISBN978-3642357459.
- Sitnikovski, Boro (2022).Introducing Software Verification with Dafny Language: Proving Program Correctness.Apress.ISBN978-1484279779.