Propositional and Predicate Logic

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1 Formal Verification of Software Propositional and Predicate Logic Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU B. Beckert: Formal Verification of Software p.1

2 Propositional Logic: Syntax Special symbols ( ) B. Beckert: Formal Verification of Software p.2

3 Propositional Logic: Syntax Special symbols ( ) Signature Propositional variables Σ = {p 0, p 1,...} B. Beckert: Formal Verification of Software p.2

4 Propositional Logic: Syntax Special symbols ( ) Signature Propositional variables Σ = {p 0, p 1,...} Formulas The propositional variables p 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

5 Propositional Logic: Unified Notation Introduced by Smullyan, 1968 B. Beckert: Formal Verification of Software p.3

6 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

7 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

8 Propositional Logic: Unified Notation Non-literal formulas and their corresponding logical sub-formulas α α 1 α 2 A B A B (A B) A B (A B) A B A A A B. Beckert: Formal Verification of Software p.4

9 Propositional Logic: Unified Notation Non-literal formulas and their corresponding logical sub-formulas α α 1 α 2 A B A B (A B) A B (A B) A B β β 1 β 2 (A B) A B A B A B A B A B A A A B. Beckert: Formal Verification of Software p.4

10 Propositional Logic: Semantics Interpretation Function I : Σ {true, false} B. Beckert: Formal Verification of Software p.5

11 Propositional Logic: Semantics Interpretation Function I : Σ {true, false} Valuation Extension of interpretation to formulas as follows: val I (p) = I(p) val I ( p) = true false if I(p) = false if I(p) = true B. Beckert: Formal Verification of Software p.5

12 Propositional Logic: Semantics val I (α) = true false if val I (α 1 ) = true and val I (α 2 ) = true if val I (α 1 ) = false or val I (α 2 ) = false B. Beckert: Formal Verification of Software p.6

13 Propositional Logic: Semantics val I (α) = val I (β) = true false true false if val I (α 1 ) = true and val I (α 2 ) = true if val I (α 1 ) = false or val I (α 2 ) = false if val I (β 1 ) = true or val I (β 2 ) = true if val I (β 1 ) = false and val I (β 2 ) = false B. Beckert: Formal Verification of Software p.6

14 Propositional Logic: Semantics val I (A B) = true false if val I (A) = val I (B) if val I (A) val I (B) B. Beckert: Formal Verification of Software p.7

15 Predicate Logic: Syntax Additional special symbols, B. Beckert: Formal Verification of Software p.8

16 Predicate Logic: Syntax Additional special symbols, Object variables Var = {x 0, x 1,...} B. Beckert: Formal Verification of Software p.8

17 Predicate Logic: Syntax Additional special symbols, Object variables Var = {x 0, x 1,...} Signature Triple Σ = F Σ, P Σ, α Σ consisting of set F Σ of functions symbols set P Σ of predicate symbols function α Σ : F Σ P Σ N assigning aritys to function and predicate symbols B. Beckert: Formal Verification of Software p.8

18 Predicate Logic: Syntax Terms variables x Var are terms if f F Σ, α Σ ( f ) = n, and t 1,..., t n terms, then f (t 1,..., t n ) is a term B. Beckert: Formal Verification of Software p.9

19 Predicate Logic: Syntax Terms variables x Var are terms if f F Σ, α Σ ( f ) = n, and t 1,..., t n terms, then f (t 1,..., t n ) is a term Atoms If p P Σ, α Σ (p) = n, and t 1,..., t n terms, then p(t 1,..., t n ) is an atom B. Beckert: Formal Verification of Software p.9

20 Predicate Logic: Syntax Formulas Atoms are formulas If A, B are formulas, x Var, then A, (A B), (A B), (A B), (A B), xa, xa are formulas B. Beckert: Formal Verification of Software p.10

21 Predicate Logic: Syntax Formulas Atoms are formulas If A, B are formulas, x Var, then A, (A B), (A B), (A B), (A B), xa, xa are formulas Literals If A is an atom, then A and A are literals B. Beckert: Formal Verification of Software p.10

22 Example Signature Σ = {0, a, b, f }, {in_iv, leq}, α 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

23 Example Signature Σ = {0, a, b, f }, {in_iv, leq}, α 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) }{{} Atom z ( leq(z, x) leq(y, z)) }{{} Scope of z B. Beckert: Formal Verification of Software p.11

24 Predicate Logic: Unified Notation Extension of unified notation for propositional logic Universal formulas Type γ xa xa B. Beckert: Formal Verification of Software p.12

25 Predicate Logic: Unified Notation Extension of unified notation for propositional logic Universal formulas Type γ xa xa Existential formulas Type δ xa xa B. Beckert: Formal Verification of Software p.12

26 Predicate Logic: Unified Notation γ- and δ-formulas and their corresponding logical sub-formulas γ γ 1 (x) δ δ 1 (x) xa(x) A(x) xa(x) A(x) xa(x) A(x) xa(x) A(x) B. Beckert: Formal Verification of Software p.13

27 Predicate Logic: Semantics Interpretation A pair D = D, I where D an arbitrary non-empty set, the universe I an interpretation function for f F Σ : I( f ) : D α( f ) D for p P Σ : I(p) : D α(p) {true, false} B. Beckert: Formal Verification of Software p.14

28 Predicate Logic: Semantics Interpretation A pair D = D, I where D an arbitrary non-empty set, the universe I an interpretation function for f F Σ : I( f ) : D α( f ) D for p P Σ : I(p) : D α(p) {true, false} Variable assignment A function λ : Var D B. Beckert: Formal Verification of Software p.14

29 Predicate Logic: Semantics Valuation Extension of interpretation and variable assignment to formulas val D,λ (x) = λ(x) for x Var val D,λ ( f (t 1,..., t n )) = I( f )(val D,λ (t 1 ),..., val D,λ (t n )) B. Beckert: Formal Verification of Software p.15

30 Predicate Logic: Semantics Valuation Extension of interpretation and variable assignment to formulas val D,λ (x) = λ(x) for x Var val D,λ ( f (t 1,..., t n )) = I( f )(val D,λ (t 1 ),..., val D,λ (t n )) val D,λ (p(t 1,..., t n )) = I(p)(val D,λ (t 1 ),..., val D,λ (t n )) true if val D,λ d val D,λ ( xa) = x (A) = true for all d D false otherwise true if val D,λ d val D,λ ( xa) = x (A) = true for some d D false otherwise val D,λ defined for propositional operators in the same way as val I. B. Beckert: Formal Verification of Software p.15

31 Predicate Logic: Semantics Example D = R I(0) = 0 I(a) = 1 I(b) = 1 I( f ) = R R x x 2 I(leq) = true iff x R y I(in_iv) = true iff x [a, b] B. Beckert: Formal Verification of Software p.16

32 Predicate Logic: Semantics Model An interpretation D is model of a set Φ of formulas iff val D,λ (A) = true for all λ and all A Φ. Notation: D Φ B. Beckert: Formal Verification of Software p.17

33 Predicate Logic: Semantics Model An interpretation D is model of a set Φ of formulas iff val D,λ (A) = true for all λ and all A Φ. Notation: D Φ Satisfiable Φ ist satisfiable iff there are an interpretation D and a variable assignment λ s.t. val D,λ (A) = true for all A Φ B. Beckert: Formal Verification of Software p.17

34 Predicate Logic: Semantics Model An interpretation D is model of a set Φ of formulas iff val D,λ (A) = true for all λ and all A Φ. Notation: D Φ Satisfiable Φ ist satisfiable iff there are an interpretation D and a variable assignment λ s.t. val D,λ (A) = true for all A Φ Validity A is valid iff all interpretations are a model of A B. Beckert: Formal Verification of Software p.17

35 Predicate Logic: Semantics Consequence A formula A is a consequence of Φ iff all models of Φ are models of A as well Notation: Φ A B. Beckert: Formal Verification of Software p.18

36 Predicate Logic: Semantics Consequence A formula A is a consequence of Φ iff all models of Φ are models of A as well Notation: Φ A Equivalent formulas Two formulas are equivalent iff they are consequences of each other B. Beckert: Formal Verification of Software p.18

37 Predicate Logic: Semantics Consequence A formula A is a consequence of Φ iff all models of Φ are models of A as well Notation: Φ A Equivalent formulas Two formulas are equivalent iff they are consequences of each other Satisfiability equivalent formulas Two formulas are satisfiability equivalent iff they are either both satisfiable or both unsatisfiable B. Beckert: Formal Verification of Software p.18

38 Substitutions Substitution Function σ : Var Term Written as: {x 1 t 1,..., x n t n } where σ(x) = t i x if x = x i for 1 i n otherwise B. Beckert: Formal Verification of Software p.19

39 Substitutions Substitution Function σ : Var Term Written as: {x 1 t 1,..., x n t n } where σ(x) = t i x if x = x i for 1 i n otherwise Extension to terms and formulas By replacing all free occurrences of variables x by σ(x) B. Beckert: Formal Verification of Software p.19

40 Substitutions Example: φ = leq(y, x) z( leq(z, x) leq(y, z)) σ = {x a, y w, z c} φσ = leq(w, a) z( leq(z, a) leq(w, z)) B. Beckert: Formal Verification of Software p.20

41 Substitutions Example: φ = leq(y, x) z( leq(z, x) leq(y, z)) σ = {x a, y w, z c} φσ = leq(w, a) z( leq(z, a) leq(w, z)) Note Substitution forbidden in cases such as: φ as above and σ = {y f (z)} B. Beckert: Formal Verification of Software p.20

42 Typed Signatures Definition A typed Signature is a tuple Σ = (S,, F, P, α), where S is a finite set of types (or sorts) is a partial ordering on S F, P are sets of function and predicate symbols (as before) α : F P S assigns argument and domain types to function and predicate symbols B. Beckert: Formal Verification of Software p.21

43 Typed Signatures The function α α( f ) = Z 1... Z n Z means: f is a symbol for functions assigning to n-tuples of elements of type Z 1... Z n an element of type Z α(p) = Z 1,..., Z n means: p is a symbol for relations on n-tuples of elements of types Z 1,..., Z n B. Beckert: Formal Verification of Software p.22

44 Typed Signatures The function α α( f ) = Z 1... Z n Z means: f is a symbol for functions assigning to n-tuples of elements of type Z 1... Z n an element of type Z α(p) = Z 1,..., Z n means: p is a symbol for relations on n-tuples of elements of types Z 1,..., Z n Variables are typed as well For each type Z S there is an infinite set of variables of type Z B. Beckert: Formal Verification of Software p.22

45 Typed Signatures: Terms If x is a variable of type Z, then x is a term of type Z If t 1,..., t n are terms of types Y 1,..., Y n f is a functions symbol with α( f ) = Z 1 Z n Z Y i Z i for all 1 i n then f (t 1,..., t n ) is a term of type Z. B. Beckert: Formal Verification of Software p.23

46 Typed Signatures: Formulas If t 1,..., t n are terms with types Y 1,..., Y n p is a predicate symbol α(p) = Z 1 Z n Y i Z i for all 1 i n then p(t 1,..., t n ) is a typed (or well-sorted) formula B. Beckert: Formal Verification of Software p.24

47 Typed Signatures: Formulas If t 1,..., t n are terms with types Y 1,..., Y n p is a predicate symbol α(p) = Z 1 Z n Y i Z i for all 1 i n then p(t 1,..., t n ) is a typed (or well-sorted) formula If t, s are terms of sorts X and Y with X Y or Y X, then t. = s is a typed formula B. Beckert: Formal Verification of Software p.24

48 Typed Signatures: Formulas If t 1,..., t n are terms with types Y 1,..., Y n p is a predicate symbol α(p) = Z 1 Z n Y i Z i for all 1 i n then p(t 1,..., t n ) is a typed (or well-sorted) formula If t, s are terms of sorts X and Y with X Y or Y X, then t. = s is a typed formula If A, B are typed formulas, then so are A (A B) (A B) (A B) B. Beckert: Formal Verification of Software p.24

49 Typed Signatures: Formulas If t 1,..., t n are terms with types Y 1,..., Y n p is a predicate symbol α(p) = Z 1 Z n Y i Z i for all 1 i n then p(t 1,..., t n ) is a typed (or well-sorted) formula If t, s are terms of sorts X and Y with X Y or Y X, then t. = s is a typed formula If A, B are typed formulas, then so are A (A B) (A B) (A B) If A is a typed formula and x is a typed variable, then are typed formulas xa xa B. Beckert: Formal Verification of Software p.24

50 Typed Interpretations Given a signature Σ = (S,, F, P, α) Interpretation A pair (D, I) such that {D Z Z S} is a family of non-empty sets with D = S {D Z Z S} D Z1 D Z2 if Z 1 Z 2 I( f ) : D Z1 D Zn D Z if α( f ) = Z 1 Z n Z I(p) D Z1 D Zn if α(p) = Z 1 Z n B. Beckert: Formal Verification of Software p.25

51 Typed Substitutions Typed substitution A substitution is well-sorted if for each variable x, the type of the term σ(x) is a sub-type of the type of x B. Beckert: Formal Verification of Software p.26

52 Special Type Structures A type structure (S, ) is discrete, in case Z 1 Z 2 only if Z 1 = Z 2 B. Beckert: Formal Verification of Software p.27

53 Special Type Structures A type structure (S, ) is discrete, in case Z 1 Z 2 only if Z 1 = Z 2 a tree structure, in case U Z 1 and U Z 2 implies Z 2 Z 1 oder Z 1 Z 2 B. Beckert: Formal Verification of Software p.27

54 Special Type Structures A type structure (S, ) is discrete, in case Z 1 Z 2 only if Z 1 = Z 2 a tree structure, in case U Z 1 and U Z 2 implies Z 2 Z 1 oder Z 1 Z 2 a lattice, in case that for any two sorts Z 1, Z 2 there is an infimum U, i.e. U Z 1 and U Z 2 W U for every sort W S with W Z 1 and W Z 2 B. Beckert: Formal Verification of Software p.27

Predicate Logic: Sematics Part 1

Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with