First-Order Logic Peter Baumgartner http://users.cecs.anu.edu.au/~baumgart/ Data61/CSIRO and ANU August 8, 2019 1 / 79 First-Order Logic (FOL) Recall: propositional logic: variables are statements ranging over ftrue=falseg SocratesIsHuman SocratesIsHuman ! SocratesIsMortal SocratesIsMortal FOL: variables range over individual objects Human(socrates) 8x: (Human(x) ! Mortal(x)) Mortal(socrates) In these lectures: I (Syntax and) semantics of FOL I Normal forms I Reasoning: tableau calculus, resolution calculus 2 / 79 First-Order Logic (FOL) Also called Predicate Logic or Predicate Calculus FOL Syntax variables x; y; z; ··· constants a; b; c; ··· functions f ; g; h; ··· terms variables, constants or n-ary function applied to n terms as arguments a; x; f (a); g(x; b); f (g(x; g(b))) predicates p; q; r; ··· atom >, ?, or an n-ary predicate applied to n terms literal atom or its negation p(f (x); g(x; f (x))); :p(f (x); g(x; f (x))) Note: 0-ary functions: constant 0-ary predicates: P; Q; R;::: 3 / 79 quantifiers existential quantifier 9x:F [x] \there exists an x such that F [x]" universal quantifier 8x:F [x] \for all x, F [x]" FOL formula literal, application of logical connectives (:; _ ; ^ ; ! ; $ ) to formulae, or application of a quantifier to a formula 4 / 79 Example FOL formula 8x: p(f (x); x) ! (9y: p(f (g(x; y)); g(x; y))) ^ q(x; f (x)) | {z } G | {z } F The scope of 8x is F . The scope of 9y is G. The formula reads: \for all x, if p(f (x); x) then there exists a y such that p(f (g(x; y)); g(x; y)) and q(x; f (x))" An occurrence of x within the scope of 8x or 9x is bound, otherwise it is free. 5 / 79 Translations of English Sentences into FOL I The length of one side of a triangle is less than the sum of the lengths of the other two sides 8x; y; z: triangle(x; y; z) ! length(x) < length(y) + length(z) I Fermat's Last Theorem. 8n: integer(n) ^ n > 2 ! 8x; y; z: integer(x) ^ integer(y) ^ integer(z) ^ x > 0 ^ y > 0 ^ z > 0 ! xn + y n 6= zn 6 / 79 FOL Semantics An interpretation I :(DI ; αI ) consists of: I Domain DI non-empty set of values or objects for example DI = playing cards (finite), integers (countably), or reals (uncountably infinite) I Assignment αI I each variable x assigned value αI [x] 2 DI I each n-ary function f assigned n αI [f ]: DI ! DI In particular, each constant a (0-ary function) assigned value αI [a] 2 DI I each n-ary predicate p assigned n αI [p]: DI ! ftrue; falseg In particular, each propositional variable P (0-ary predicate) assigned truth value (true, false) 7 / 79 Example F : p(f (x; y); z) ! p(y; g(z; x)) Interpretation I :(DI ; αI ) DI = Z = {· · · ; −2; −1; 0; 1; 2; · · · g integers 2 2 αI [f ]: DI 7! DI αI [g]: DI 7! DI (x; y) 7! x + y (x; y) 7! x − y 2 αI [p]: D 7! ftrue; falseg I ( true if x < y (x; y) 7! false otherwise Also αI [x] = 13, αI [y] = 42, αI [z] = 1 Compute the truth value of F under I 1: I 6j= p(f (x; y); z) since 13 + 42 ≥ 1 2: I 6j= p(y; g(z; x)) since 42 ≥ 1 − 13 3: I j= F by 1, 2, and ! F is true under I 8 / 79 Semantics: Quantifiers Let x be a variable. An x-variant of interpretation I is an interpretation J :(DJ ; αJ ) such that I DI = DJ I αI [y] = αJ [y] for all symbols y, except possibly x That is, I and J agree on everything except possibly the value of x Denote J : I / fx 7! vg the x-variant of I in which αJ [x] = v for some v 2 DI . Then I I j= 8x: F iff for all v 2 DI , I / fx 7! vg j= F I I j= 9x: F iff there exists v 2 DI s.t. I / fx 7! vg j= F 9 / 79 Example Consider F : 8x: animal(x) ! 9y: (fruit(y) ^ loves(x; y)) and I = (DI ; αI ): DI = f ; ; ; g αI [animal] = f( ) 7! true; ( ) 7! true;:::g (false everywhere else) αI [fruit] = f( ) 7! true; ( ) 7! true;:::g αI [loves] = f( ; ) 7! true; ( ; ) 7! true;:::g Compute the value of F under I : I j= 8x: animal(x) ! 9y: (fruit(y) ^ loves(x; y)) iff for all v 2 f ; ; ; g, I / fx 7! vg j= animal(x) ! 9y: (fruit(y) ^ loves(x; y)) Check all four cases, e.g.: I / fx 7! g j= animal(x) ! 9y: (fruit(y) ^ loves(x; y)) I / fx 7! g j= 9y: (fruit(y) ^ loves(x; y))iff there exists v1 2 f ; ; ; g,iff I / fx 7! g / fy 7! v1g j= loves(x; y) I / fx 7! g / fy 7! g j= loves(x; y) (true)iff 10 / 79 Example Consider F : 8x: 9y: 2 · y = x Here 2 · y is the infix notation of the term ·(2; y), and 2 · y = x is the infix notation of the atom =(·(2; y); x) I 2 is a 0-ary function symbol (a constant). I · is a 2-ary function symbol. I = is a 2-ary predicate symbol. I x; y are variables. What is the truth-value of F ? 11 / 79 Example (Z) F : 8x: 9y: 2 · y = x Let I be the standard interpretation for integers, DI = Z. Compute the value of F under I : I j= 8x: 9y: 2 · y = x iff for all v 2 DI ; I / fx 7! vg j= 9y: 2 · y = x iff for all v 2 DI , there exists v1 2 DI , I / fx 7! vg / fy 7! v1g j= 2 · y = x The latter is false since for 1 2 DI there is no number v1 with 2 · v1 = 1. 12 / 79 Example (Q) F : 8x: 9y: 2 · y = x Let I be the standard interpretation for rational numbers, DI = Q. Compute the value of F under I : I j= 8x: 9y: 2 · y = x iff for all v 2 DI ; I / fx 7! vg j= 9y: 2 · y = x iff for all v 2 DI , there exists v1 2 DI , I / fx 7! vg / fy 7! v1g j= 2 · y = x v The latter is true since for arbitrary v 2 DI we can chose v1 with v1 = 2 . 13 / 79 Satisfiability and Validity F is satisfiable iff there exists an interpretation I such that I j= F . F is valid iff for all interpretations I , I j= F . Note: F is valid iff :F is unsatisfiable. 14 / 79 Example F :(8x: p(x; x)) ! (9x: 8y: p(x; y)) is invalid. How to show this? Find interpretation I such that I j= :((8x: p(x; x)) ! (9x: 8y: p(x; y))) i.e. I j= (8x: p(x; x)) ^ :(9x: 8y: p(x; y)) Choose DI = f0; 1g pI = f(0; 0); (1; 1)g i.e. αI [p] = f(0; 0) 7! true; (1; 1) 7! true; (0; 1) 7! true; (1; 0) 7! falseg I falsifying interpretation ) F is invalid. 15 / 79 Example F :(8x: p(x)) $ (:9x: :p(x)) is valid. How to show this? 1. By expanding definitions. This is easy for this example. 2. By constructing a proof with, e.g., a \semantic argument method" adapted to FOL. Below we will develop such a semantic argument method adapted to FOL. To define it, we first need the concept of \substitutions". 16 / 79 Substitution Suppose we want to replace terms with other terms in formulas, e.g., F : 8y: (p(x; y) ! p(y; x)) should be transformed to G : 8y: (p(a; y) ! p(y; a)) We call the mapping from x to a a substitution, denoted as σ : fx 7! ag. We write F σ for the Formula G. Another convenient notation is F [x] for a formula containing the variable x and F [a] for F σ. 17 / 79 Substitution A substitution σ is a mapping from variables to terms, written as σ : fx1 7! t1;:::; xn 7! tng such that n ≥ 0 and xi 6= xj for all i; j = 1::n with i 6= j. The set dom(σ) = fx1;:::; xng is called the domain of σ. The set cod(σ) = ft1;:::; tng is called the codomain of σ. The set of all variables occurring in cod(σ) is called the variable codomain of σ, denoted by varcod(σ). By F σ we denote the application of σ to the formula F , i.e., the formula F where all free occurrences of xi are replaced by ti . For a formula named F [x] we write F [t] as a shorthand for F [x]fx 7! tg. 18 / 79 Safe Substitution Care has to be taken in presence of quantifiers: F [x]: 9y: y = Succ(x) What is F [y]? We cannot just rename x to y with fx 7! yg: F [y]: 9y: y = Succ(y) Wrong! We need to first rename bound variables occuring in the codomain of the substitution: F [y]: 9y 0: y 0 = Succ(y) Right! Renaming does not change the models of a formula: (9y: y = Succ(x)) , (9y 0: y 0 = Succ(x)) 19 / 79 Recursive Definition of Substitution 8 σ(x) if t = x and x 2 dom(σ) <> tσ = x if t = x and x 2= dom(σ) > :f (t1σ; : : : ; tnσ) if t = f (t1;:::; tn) p(t1;:::; tn)σ = p(t1σ; : : : ; tnσ) (:F )σ = :(F σ) (F ^ G)σ = (F σ ^ Gσ) ··· ( 8x0: (F fx 7! x0g)σ if x 2 dom(σ) [ varcod(σ), x 0 is fresh (8x: F )σ = 8x: F σ otherwise ( 9x0: (F fx 7! x0g)σ if x 2 dom(σ) [ varcod(σ), x 0 is fresh (9x: F )σ = 9x: F σ otherwise 20 / 79 Example: Safe Substitution F σ scope of 8x z }| { F :(8x: p(x; y)) ! q(f (y); x) bound by 8x %- free free %- free σ : fx 7! g(x; y); y 7! f (x)g F σ? 1.