Add B3 Verifier: SMT-based verification for B3 programs #307
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| /- | ||
| Copyright Strata Contributors | ||
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| SPDX-License-Identifier: Apache-2.0 OR MIT | ||
| -/ | ||
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| import Strata.Languages.B3.DDMTransform.DefinitionAST | ||
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| /-! | ||
| # Function-to-Axiom Transformation | ||
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| Transforms B3 programs by splitting function definitions into declarations and axioms. | ||
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| While SMT-LIB 2.6 provides `define-fun-rec` for mutually recursive definitions, | ||
| we use quantified axioms for broader solver compatibility and to maintain consistency | ||
| with our verification approach. By converting function bodies to axioms with quantifiers, | ||
| we enable verification of programs with mutually recursive functions across different | ||
| SMT solvers. | ||
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| TODO: Add config option to use `define-fun` for non-recursive functions instead of | ||
| quantified axioms. This could improve solver performance for simple function definitions. | ||
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| ## Example: Simple Function | ||
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| Input: | ||
| ``` | ||
| function abs(x : int) : int { | ||
| if x >= 0 then x else -x | ||
| } | ||
| ``` | ||
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| Output: | ||
| ``` | ||
| function abs(x : int) : int | ||
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| axiom forall x : int pattern abs(x) abs(x) == (if x >= 0 then x else -x) | ||
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Contributor
There was a problem hiding this comment. As we chatted offline, one day we ought to have a config option that allows the definition of at least non-recursive functions using There was a problem hiding this comment. Note that B3 already supports this. |
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| ``` | ||
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| ## Example: Mutually Recursive Functions | ||
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| Input: | ||
| ``` | ||
| function isEven(n : int) : int { | ||
| if n == 0 then 1 else isOdd(n - 1) | ||
| } | ||
| function isOdd(n : int) : int { | ||
| if n == 0 then 0 else isEven(n - 1) | ||
| } | ||
| ``` | ||
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| Output: | ||
| ``` | ||
| function isEven(n : int) : int | ||
| function isOdd(n : int) : int | ||
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| axiom forall n : int pattern isEven(n) isEven(n) == (if n == 0 then 1 else isOdd(n - 1)) | ||
| axiom forall n : int pattern isOdd(n) isOdd(n) == (if n == 0 then 0 else isEven(n - 1)) | ||
| ``` | ||
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| ## Example: Function with Precondition | ||
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| Input: | ||
| ``` | ||
| function sqrt(x : int) : int | ||
| when x >= 0 | ||
| { | ||
| ... | ||
| } | ||
| ``` | ||
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| Output: | ||
| ``` | ||
| function sqrt(x : int) : int | ||
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| axiom forall x : int pattern sqrt(x) (x >= 0) ==> (sqrt(x) == ...) | ||
| ``` | ||
| -/ | ||
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| namespace Strata.B3.Transform | ||
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| open Strata.B3AST | ||
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| def transformFunctionDecl (decl : B3AST.Decl α) : Option (B3AST.Decl α × B3AST.Decl α) := | ||
| match decl with | ||
| | .function _m name params resultType tag body => | ||
| match body.val with | ||
| | some bodyAnn => | ||
| match bodyAnn with | ||
| | FunctionBody.functionBody m whens bodyExpr => | ||
| let funcDecl := B3AST.Decl.function m name params resultType tag (Ann.mk body.ann none) | ||
| let paramList := params.val.toList | ||
| let funcCallArgs : Array (Expression α) := | ||
| params.val.mapIdx (fun i _param => Expression.id m i) | ||
| let funcCall := Expression.functionCall m name ⟨m, funcCallArgs⟩ | ||
| let equality := Expression.binaryOp m (BinaryOp.eq m) funcCall bodyExpr | ||
| let axiomBody := | ||
| if whens.val.isEmpty then | ||
| equality | ||
| else | ||
| let whenExprs := whens.val.toList.filterMap (fun w => | ||
| match w with | When.when _m cond => some cond) | ||
| let whenConj := match whenExprs with | ||
| | [] => Expression.literal whens.ann (Literal.boolLit whens.ann true) | ||
| | first :: rest => rest.foldl (fun acc e => | ||
| Expression.binaryOp whens.ann (BinaryOp.and whens.ann) acc e | ||
| ) first | ||
| Expression.binaryOp whens.ann (BinaryOp.implies whens.ann) whenConj equality | ||
| -- Create pattern with function call | ||
| let pattern := Pattern.pattern m ⟨m, #[funcCall]⟩ | ||
| let axiomExpr := paramList.foldr (fun param body => | ||
| match param with | ||
| | FParameter.fParameter _m _inj paramName paramTy => | ||
| let varDecl := VarDecl.quantVarDecl m paramName paramTy | ||
| Expression.quantifierExpr m | ||
| (QuantifierKind.forall m) | ||
| ⟨m, #[varDecl]⟩ ⟨m, #[pattern]⟩ body | ||
| ) axiomBody | ||
| let axiomDecl := Decl.axiom m ⟨m, #[]⟩ axiomExpr | ||
| some (funcDecl, axiomDecl) | ||
| | none => none | ||
| | _ => none | ||
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| def functionToAxiom (prog : B3AST.Program α) : B3AST.Program α := | ||
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| match prog with | ||
| | Program.program m decls => | ||
| Id.run do | ||
| let mut typeDeclsRev : List (B3AST.Decl α) := [] | ||
| let mut funcDeclsRev : List (B3AST.Decl α) := [] | ||
| let mut funcAxiomsRev : List (B3AST.Decl α) := [] | ||
| let mut otherDeclsRev : List (B3AST.Decl α) := [] | ||
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| for decl in decls.val.toList do | ||
| match decl with | ||
| | .typeDecl _ _ | .tagger _ _ _ => | ||
| typeDeclsRev := decl :: typeDeclsRev | ||
| | .function _ _ _ _ _ body => | ||
| match body.val with | ||
| | some bodyAnn => | ||
| match bodyAnn with | ||
| | FunctionBody.functionBody _ _ _ => | ||
| match transformFunctionDecl decl with | ||
| | some (funcDecl, axiomDecl) => | ||
| funcDeclsRev := funcDecl :: funcDeclsRev | ||
| funcAxiomsRev := axiomDecl :: funcAxiomsRev | ||
| | none => otherDeclsRev := decl :: otherDeclsRev | ||
| | none => | ||
| funcDeclsRev := decl :: funcDeclsRev | ||
| | .axiom _ _ _ => | ||
| funcAxiomsRev := decl :: funcAxiomsRev | ||
| | _ => | ||
| otherDeclsRev := decl :: otherDeclsRev | ||
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| let finalDecls := typeDeclsRev.reverse ++ funcDeclsRev.reverse ++ funcAxiomsRev.reverse ++ otherDeclsRev.reverse | ||
| return Program.program m ⟨decls.ann, finalDecls.toArray⟩ | ||
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| end Strata.B3.Transform | ||
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