Formal Verification of Software Propositional and Predicate Logic Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU B. Beckert: Formal Verification of Software – p.1 Signature Σ Propositional variables = fp0; p1; : : :g Formulas The propositional variables p 2 Σ are formulas If A; B are formulas, then :A (A ^ B) (A _ B) (A ! B) (A $ B) are formulas Propositional Logic: Syntax Special symbols ( ) : ^ _ ! $ B. Beckert: Formal Verification of Software – p.2 Formulas The propositional variables p 2 Σ are formulas If A; B are formulas, then :A (A ^ B) (A _ B) (A ! B) (A $ B) are formulas Propositional Logic: Syntax Special symbols ( ) : ^ _ ! $ Signature Σ Propositional variables = fp0; p1; : : :g B. Beckert: Formal Verification of Software – p.2 Propositional Logic: Syntax Special symbols ( ) : ^ _ ! $ Signature Σ Propositional variables = fp0; p1; : : :g Formulas The propositional variables p 2 Σ are formulas If A; B are formulas, then :A (A ^ B) (A _ B) (A ! B) (A $ B) are formulas B. Beckert: Formal Verification of Software – p.2 Conjunctive formulas Type α ::A (A ^ B) :(A _ B) :(A ! B) Disjunctive formulas Type β :(A ^ B) (A _ B) (A ! B) Propositional Logic: Unified Notation Introduced by Smullyan, 1968 B. Beckert: Formal Verification of Software – p.3 Disjunctive formulas Type β :(A ^ B) (A _ B) (A ! B) Propositional Logic: Unified Notation Introduced by Smullyan, 1968 Conjunctive formulas Type α ::A (A ^ B) :(A _ B) :(A ! B) B. Beckert: Formal Verification of Software – p.3 Propositional Logic: Unified Notation Introduced by Smullyan, 1968 Conjunctive formulas Type α ::A (A ^ B) :(A _ B) :(A ! B) Disjunctive formulas Type β :(A ^ B) (A _ B) (A ! B) B. Beckert: Formal Verification of Software – p.3 β β1 β2 :(A^B) :A :B A_B A B A!B :A B Propositional Logic: Unified Notation Non-literal formulas and their corresponding “logical” sub-formulas α α1 α2 A^B A B :(A_B) :A :B :(A!B) A :B ::A A A B. Beckert: Formal Verification of Software – p.4 Propositional Logic: Unified Notation Non-literal formulas and their corresponding “logical” sub-formulas α α1 α2 β β1 β2 A^B A B :(A^B) :A :B :(A_B) :A :B A_B A B :(A!B) A :B A!B :A B ::A A A B. Beckert: Formal Verification of Software – p.4 Valuation Extension of interpretation to formulas as follows: valI(p) = I(p) 8 true if I(p) = false = > valI(:p) <> false if I(p) = true > :> Propositional Logic: Semantics Interpretation Function I : Σ ! ftrue; falseg B. Beckert: Formal Verification of Software – p.5 Propositional Logic: Semantics Interpretation Function I : Σ ! ftrue; falseg Valuation Extension of interpretation to formulas as follows: valI(p) = I(p) 8 true if I(p) = false = > valI(:p) <> false if I(p) = true > :> B. Beckert: Formal Verification of Software – p.5 8 true if valI(β1) = true > > = > or valI(β2) true = > valI(β) <> false if valI(β1) = false > > > and val (β2) = false > I :> Propositional Logic: Semantics 8 true if valI(α1) = true > > = > and valI(α2) true = > valI(α) <> false if valI(α1) = false > > > or val (α2) = false > I :> B. Beckert: Formal Verification of Software – p.6 Propositional Logic: Semantics 8 true if valI(α1) = true > > = > and valI(α2) true = > valI(α) <> false if valI(α1) = false > > > or val (α2) = false > I :> 8 true if valI(β1) = true > > = > or valI(β2) true = > valI(β) <> false if valI(β1) = false > > > and val (β2) = false > I :> B. Beckert: Formal Verification of Software – p.6 Propositional Logic: Semantics 8 true if valI(A) = valI(B) = > valI(A$B) <> false if val (A) 6= val (B) > I I :> B. Beckert: Formal Verification of Software – p.7 Object variables Var = fx0; x1; : : :g Signature Triple Σ = hFΣ; PΣ; αΣi consisting of set FΣ of functions symbols set PΣ of predicate symbols function αΣ : FΣ [ PΣ ! N assigning aritys to function and predicate symbols Predicate Logic: Syntax Additional special symbols “,” “8” “9” B. Beckert: Formal Verification of Software – p.8 Signature Triple Σ = hFΣ; PΣ; αΣi consisting of set FΣ of functions symbols set PΣ of predicate symbols function αΣ : FΣ [ PΣ ! N assigning aritys to function and predicate symbols Predicate Logic: Syntax Additional special symbols “,” “8” “9” Object variables Var = fx0; x1; : : :g B. Beckert: Formal Verification of Software – p.8 Predicate Logic: Syntax Additional special symbols “,” “8” “9” Object variables Var = fx0; x1; : : :g Signature Triple Σ = hFΣ; PΣ; αΣi consisting of set FΣ of functions symbols set PΣ of predicate symbols function αΣ : FΣ [ PΣ ! N assigning aritys to function and predicate symbols B. Beckert: Formal Verification of Software – p.8 Atoms If p 2 PΣ, αΣ(p) = n, and t1; : : : ; tn terms, then p(t1; : : : ; tn) is an atom Predicate Logic: Syntax Terms variables x 2 Var are terms if f 2 FΣ, αΣ( f ) = n, and t1; : : : ; tn terms, then f (t1; : : : ; tn) is a term B. Beckert: Formal Verification of Software – p.9 Predicate Logic: Syntax Terms variables x 2 Var are terms if f 2 FΣ, αΣ( f ) = n, and t1; : : : ; tn terms, then f (t1; : : : ; tn) is a term Atoms If p 2 PΣ, αΣ(p) = n, and t1; : : : ; tn terms, then p(t1; : : : ; tn) is an atom B. Beckert: Formal Verification of Software – p.9 Literals If A is an atom, then A and :A are literals Predicate Logic: Syntax Formulas Atoms are formulas If A; B are formulas, x 2 Var, then :A; (A ^ B); (A _ B); (A ! B); (A $ B); 8xA; 9xA are formulas B. Beckert: Formal Verification of Software – p.10 Predicate Logic: Syntax Formulas Atoms are formulas If A; B are formulas, x 2 Var, then :A; (A ^ B); (A _ B); (A ! B); (A $ B); 8xA; 9xA are formulas Literals If A is an atom, then A and :A are literals B. Beckert: Formal Verification of Software – p.10 Formula φ = : leq(y; x) ! 9z (:leq(z; x) ^ :leq(y; z)) Atom Scope of | {z } | {z 9z } Example Signature Σ≤ = hf0; a; b; f g; fin_iv; leqg; αi with α(0) = α(a) = α(b) = 0 α( f ) = 1 α(leq) = 2 α(in_iv) = 3 (in interval) B. Beckert: Formal Verification of Software – p.11 Example Signature Σ≤ = hf0; a; b; f g; fin_iv; leqg; αi with α(0) = α(a) = α(b) = 0 α( f ) = 1 α(leq) = 2 α(in_iv) = 3 (in interval) Formula φ = : leq(y; x) ! 9z (:leq(z; x) ^ :leq(y; z)) Atom Scope of | {z } | {z 9z } B. Beckert: Formal Verification of Software – p.11 Existential formulas Type δ :8xA 9xA Predicate Logic: Unified Notation Extension of unified notation for propositional logic Universal formulas Type γ 8xA :9xA B. Beckert: Formal Verification of Software – p.12 Predicate Logic: Unified Notation Extension of unified notation for propositional logic Universal formulas Type γ 8xA :9xA Existential formulas Type δ :8xA 9xA B. Beckert: Formal Verification of Software – p.12 Predicate Logic: Unified Notation γ- and δ-formulas and their corresponding “logical” sub-formulas γ γ1(x) δ δ1(x) 8xA(x) A(x) :8xA(x) :A(x) :9xA(x) :A(x) 9xA(x) A(x) B. Beckert: Formal Verification of Software – p.13 Variable assignment A function λ : Var ! D Predicate Logic: Semantics Interpretation A pair D = hD; Ii where D an arbitrary non-empty set, the universe I an interpretation function α for f 2 FΣ: I( f ) : D ( f ) ! D α for p 2 PΣ: I(p) : D (p) ! ftrue; falseg B. Beckert: Formal Verification of Software – p.14 Predicate Logic: Semantics Interpretation A pair D = hD; Ii where D an arbitrary non-empty set, the universe I an interpretation function α for f 2 FΣ: I( f ) : D ( f ) ! D α for p 2 PΣ: I(p) : D (p) ! ftrue; falseg Variable assignment A function λ : Var ! D B. Beckert: Formal Verification of Software – p.14 valD;λ(p(t1; : : : ; tn)) = I(p)(valD;λ(t1); : : : ; valD;λ(tn)) 8 true if val ;λd (A) = true for all d 2 D = D x valD;λ(8xA) <> false otherwise :> 8 true if val ;λd (A) = true for some d 2 D = D x valD;λ(9xA) <> false otherwise :> valD;λ defined for propositional operators in the same way as valI. Predicate Logic: Semantics Valuation Extension of interpretation and variable assignment to formulas valD;λ(x) = λ(x) for x 2 Var valD;λ( f (t1; : : : ; tn)) = I( f )(valD;λ(t1); : : : ; valD;λ(tn)) B. Beckert: Formal Verification of Software – p.15 Predicate Logic: Semantics Valuation Extension of interpretation and variable assignment to formulas valD;λ(x) = λ(x) for x 2 Var valD;λ( f (t1; : : : ; tn)) = I( f )(valD;λ(t1); : : : ; valD;λ(tn)) valD;λ(p(t1; : : : ; tn)) = I(p)(valD;λ(t1); : : : ; valD;λ(tn)) 8 true if val ;λd (A) = true for all d 2 D = D x valD;λ(8xA) <> false otherwise :> 8 true if val ;λd (A) = true for some d 2 D = D x valD;λ(9xA) <> false otherwise :> valD;λ defined for propositional operators in the same way as valI. B. Beckert: Formal Verification of Software – p.15 Predicate Logic: Semantics Example D = R I(0) = 0 I(a) = −1 I(b) = 1 8 R ! R = I( f ) <> x 7! x2 :> I(leq) = true iff x ≤R y I(in_iv) = true iff x 2 [a; b] B. Beckert: Formal Verification of Software – p.16 Satisfiable Φ ist satisfiable iff there are an interpretation D and a variable assignment λ s.t. Φ valD;λ(A) = true for all A 2 Validity A is valid iff all interpretations are a model of A Predicate Logic: Semantics Model An interpretation D is model of a set Φ of formulas iff Φ valD;λ(A) = true for all λ and all A 2 .