First-order logic Syntax and semantics

Transcription

1 1 / 43 First-order logic Syntax and semantics Mario Alviano University of Calabria, Italy A.Y. 2017/2018

2 Outline 2 / 43 1 Motivation Why more than propositional logic? Intuition 2 Syntax Terms Formulas 3 Semantics Structures Valuation

3 Propositional logic: is it enough? (1) We want to represent: Socrates is a human Humans are mortal From this, we want to draw the conclusion: Socrates is mortal In propositional logic Variable SH for Socrates is a human Variable HM for Humans are mortal Variable SM for Socrates is mortal We define Γ := {SH, SH SM} From this, we can draw the conclusion: Γ = SM Where is HM?!? This is not what we wanted to express! 5 / 43

4 6 / 43 Propositional logic: is it enough? (2) Humans are mortal is not an atom! Mario is a human Humans are mortal Mario is mortal We talk about objects, not about propositions! But... There is no concept of object in propositional logic

5 8 / 43 Towards first-order logic Start from propositional logic Atoms are no longer indivisible Compose atoms from: terms, representing objects, and predicates, i.e., statements about terms Example Socrates is a term Mortal is a predicate

6 9 / 43 Functions Socrates father Represents exactly one object But we refer to it from another object Function symbols map some objects to other objects

7 10 / 43 Variables Humans are mortal If some object is human then it is mortal We are not referring to any specific object We need a variable ranging over objects Note: Such variables are completely different from propositional variables! Think about them as object variables

8 11 / 43 Quantifiers Humans are mortal Actually wants to express all humans are mortal And not some humans are mortal And certainly not no humans are mortal Quantifiers express for all objects, or some object exists

9 14 / 43 Function symbols and constants Let F be a countable set of function symbols An arity (nonnegative integer) is associated with each function symbol Function symbols with arity 0 are constants (i.e., objects) Examples socrates (arity 0) father (arity 1) firstsonof (arity 2) supercalifragilistichespiralidoso287 (arity 6)

10 15 / 43 Variables Let V be a countable set of (object) variables Note: V and F are disjoint! Examples x, y, z _human _xiknve

11 16 / 43 Terms (1) Build terms from function symbols and variables, respecting arities Terms: Inductive definition If v is a variable symbol then v is a term If f is a function symbol of arity 0 then f is a term If f is a function symbol of arity n > 0, and t 1,..., t n are terms, then f (t 1,..., t n ) is a term Ground terms Terms not containing any variable!

12 17 / 43 Terms (2) Examples socrates _xiknve father(socrates) father(father(socrates)) father(father(father(socrates))) firstson(father(socrates), _xiknve) supercalifragilistichespiralidoso287(x, y, z, socrates, _xiknve, x) Intuitively, each term represents an object

13 19 / 43 Predicate symbols Let P be a countable set of predicate symbols An arity is associated with each predicate symbol Examples Human (arity 1) Mortal (arity 1) Married (arity 2) Yggdrasil (arity 18)

14 20 / 43 Atoms Atomic formula, or atom A structure of the form P(t 1,..., t n ), where P is a predicate symbol of arity n 0 t 1,..., t n are terms Examples Human(socrates) Mortal(father(_xiknve)) Married(socrates, x) Predicates with arity 0 are like propositional variables!

15 21 / 43 Formulas Similar to propositional logic with different atoms! φ is a wff if and only if φ is an atomic formula, or φ =, or φ =, or φ = ( ψ) where ψ is a formula, or φ = (ψ ψ ) where ψ, ψ are formulas, or φ = (ψ ψ ) where ψ, ψ are formulas, or φ = (ψ ψ ) where ψ, ψ are formulas, or φ = (ψ ψ ) where ψ, ψ are formulas, or φ = ( x ψ) where x is a variable and ψ is a formula, or φ = ( x ψ) where x is a variable and ψ is a formula

16 22 / 43 Quantified variables is the universal quantifier is the existential quantifier In x ψ and x ψ ψ is the scope of x x is bound in ψ Variables which are not bound in a formula are free Formulas without free variables are closed, also called sentences

17 23 / 43 Binding strength As for propositional logic, we usually minimize parentheses,, and have the same binding strength Then come,,, and, in this order Examples Human(socrates) Mortal(socrates) ( x (Human(x) Mortal(x))) ( x (Human(firstSonOf (socrates, x)) Married(socrates, x))) ( x (( y Human(firstSonOf (x, y))) ( z Married(x, z)))) (( y Human(firstSonOf (x, y))) ( z Married(x, z)))

18 24 / 43 Formulas as trees Every formula can be written as a formula tree x P(x) Q(y) P(y) R(y) x Q(y) R(y) P(x) P(y) Similarly, we can define (immediate) subformulas How to determine the scope of a variable? How to determine bound variables?

19 Examples Example 1 1 N(o) 2 x (N(x) N(s(x))) 3 x y ( E(x, y) E(s(x), s(y))) 4 x E(s(x), o) Example 2 1 x E(s(x), o) 2 x y (E(s(x), s(y)) E(x, y)) 3 x E(a(x, o), x) 4 x y E(a(x, s(y)), s(a(x, y))) 5 x E(m(x, o), o) 6 x y E(m(x, s(y)), a(m(x, y), x)) 25 / 43

20 Free variables 26 / 43 Given a term t, the set free(t) of its free variables is defined as: free(t) = {x} if t is a variable x free(t) = if t is a constant free(t) = n i=1 free(t i) if t is f (t 1,..., t n ) Given a formula φ, the set free(φ) of its free variables is defined as: free(φ) = n i=1 free(t i) if φ is an atom P(t 1,..., t n ) free(φ) = if φ is or free(φ) = free(ψ) if φ is ψ free(φ) = free(ψ) free(ψ ) if φ is ψ ψ for {,,, } free(φ) = free(ψ) \ {x} if φ is x ψ or x ψ

21 27 / 43 Term substitution (1) Substitute variable x by term t in term s, denoted s[t/x]: s[t/x] = t if s is a variable and s = x s[t/x] = s if s is a variable and s x s[t/x] = f (t 1 [t/x],..., t n [t/x]) if s = f (t 1,..., t n ) Substitute variable x by term t in a formula φ, denoted φ[t/x]: φ[t/x] = φ if φ is or φ[t/x] = P(t 1 [t/x],..., t n [t/x]) if φ is an atom P(t 1,..., t n ) φ[t/x] = ψ[t/x] if φ = ψ φ[t/x] = ψ[t/x] ψ [t/x] if φ = ψ ψ for {,,, } if φ = Qy ψ for Q {, } then φ[t/x] = Qy ψ[t/x] if x y and y / free(t) φ[t/x] = Qz ψ[z/y][t/x] if x y and y free(t), where z is a variable not occurring in ψ and t φ[t/x] = Qy ψ if x = y

22 28 / 43 Term substitution (2) t[t 1 /x 1,..., t n /x n ] denotes the simultaneous substitution of x 1,..., x n by t 1,..., t n in the term t φ[t 1 /x 1,..., t n /x n ] denotes the simultaneous substitution of x 1,..., x n by t 1,..., t n in the formula φ Example Compare the followings: 1 A(x, y)[y/x, x/y] 2 A(x, y)[y/x][x/y] 3 A(x, y)[x/y][y/x]

23 29 / 43 Formula substitution In propositional logic we replaced atoms by formulas Can we do the same in first-order logic? Yes, we can! Definition For wffs γ, φ and atom A, let γ[φ/a] denote the formula in which φ replaces all occurrences of A

24 Semantics Requirements Domain We want to speak about objects Each term represents an object We need a set of objects called domain (or universe) of discourse Interpretation function A constant symbol should be associated to an object in the domain A function symbol of arity n should be associated to a function mapping each n-tuple of objects to an object A predicate of arity n should be associated to a function mapping each n-tuple of objects to a Boolean value We need an interpretation function for this purpose 32 / 43

25 Structures Definition A structure is a pair A = (D A, I A ), where D A is a nonempty set, called domain I A is an interpretation function, i.e., a function associating each constant symbol c with an object I A (c) D A each function symbol f of arity n with a function I A (f ) : D n A D A each predicate symbol P of arity n with a relation I A (P) : D n A {0, 1} Notation. We will usually write c A, f A and P A instead of I A (c), I A (f ) and I A (P), respectively Observation. Constants are actually associated with functions of arity 0 33 / 43

26 Variable assignments What about (free) variables? We need a variable assignment, also called environment Definition A variable assignment ξ A for a structure A is a function from the set of variables to the domain, i.e., ξ A : V D A Notation We extend substitutions to variable assignments. Let ξ A be a variable assignment, x be a variable and d be an object in D A. ξ A [d/x] is the variable assignment such that ξ A [d/x](y) = d if y = x ξ A [d/x](y) = ξ A (y) if y x 34 / 43

27 The valuation ν (A,ξA) (t) of a term t wrt (A, ξ A ) is ν (A,ξA) (t) = ξ A (t) if t is a variable ν (A,ξA) (t) = f A (ν (A,ξA) (t 1 ),..., ν (A,ξA) (t n )) if t is f (t 1,..., t n ) The valuation of formulas is also defined inductively ν (A,ξA) ( ) = 1 and ν (A,ξA) ( ) = 0 ν (A,ξA) (P(t 1,..., t n )) = P A (ν (A,ξA) (t 1 ),..., ν (A,ξA) (t n )) ν (A,ξA) ( φ) = ν (A,ξA) (φ) using propositional logic ν (A,ξA) (φ ψ) = ν (A,ξA) (φ) ν (A,ξA) (ψ) using propositional logic, for {,,, } ν (A,ξA) ( x φ) = 1 iff ν (A,ξA [d/x]) (φ) = 1 for all d D A ν (A,ξA) ( x φ) = 1 iff ν (A,ξA [d/x]) (φ) = 1 for some d D A 36 / 43 Valuation Interpretation An interpretation for a first-order language is a pair (A, ξ A ), where A = (D A, I A ) is a structure ξ A is a variable assignment for A

28 37 / 43 Examples (1) Example 1 1 N(o) 2 x (N(x) N(s(x))) 3 x y ( E(x, y) E(s(x), s(y))) 4 x E(s(x), o) D A = N (natural numbers, including 0) o A = 0 s A : Successor of a number N A : True for natural numbers E A : True for equal numbers ξ A? Not relevant because there are no free variables

29 Examples (2) Example 2 1 x E(s(x), o) 2 x y (E(s(x), s(y)) E(x, y)) 3 x E(a(x, o), x) 4 x y E(a(x, s(y)), s(a(x, y))) 5 x E(m(x, o), o) 6 x y E(m(x, s(y)), a(m(x, y), x)) D A = N (natural numbers, including 0) o A = 0 s A : Successor of a number a A : Sum of two numbers m A : Product of two numbers E A : True for equal numbers ξ A? Not relevant because there are no free variables 38 / 43

30 39 / 43 Socrates example 1 Human(socrates) 2 x (Human(x) Mortal(x)) 3 Mortal(socrates) Can we conclude 3 from 1 and 2? Not yet! We must still define the notions of model and entailment

31 40 / 43 Peano (first-order) axioms 1 N(0) 2 x E(x, x) 3 x y (E(x, y) E(y, x)) 4 x y z (E(x, y) E(y, z) E(x, z)) 5 x y (N(x) E(x, y) N(y)) 6 x (N(x) N(s(x))) 7 x E(s(x), 0) 8 x y (E(s(x), s(y)) E(x, y)) 9 If P is a predicate of arity 1 then (P(0) x (P(x) P(s(x)))) x P(x)

32 Simple observations 41 / 43 Lemma Let A be a structure, and ξ1 A, ξa 2 be variable assignments such that ξ1 A(x i) = ξ2 A(x i) for i = 1,..., n. If t is a term such that free(t) = {x 1,..., x n } then ν (A,ξA 1 ) (t) = ν (A,ξA 2 ) (t) If φ is a formula such that free(φ) = {x 1,..., x n } then ν (A,ξA 1 ) (φ) = ν (A,ξA 2 ) (φ)

33 Exercises 42 / 43 1 Find the free variables in the following wffs 1 y x A(x, y) B(x, y) 2 x y(a(x, y) B(x, y)) 3 y x A(y) (B(x, y) z C(x, z)) 4 x y(a(x, y) B(x)) z C(z) D(z) 2 For each formula φ above 1 find an interpretation valuating φ as 0 2 find an interpretation valuating φ as 1 3 find a structure A such that ν (A,ξA) (φ) = 1 for every ξ A

34 END OF THE LECTURE 43 / 43