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Computability and Complexity 9-1 Logic Reminder (Cnt’d) Computability and Complexity Andrei Bulatov Computability and Complexity • Propositional logic • Conjunctive normal forms • Predicates, functions and quantifiers • Terms • Formulas 9-2 Computability and Complexity Models (First Order Semantics) Let be a vocabulary. A model appropriate to is a pair M=(U,) consisting of • the universe of M, a non-empty set U • the interpretation, a function that assigns - to each predicate symbol P a concrete predicate P M on U M - to each function symbol f a concrete function f on U • the equality predicate symbol is always assigned the equality predicate on U 9-3 Computability and Complexity Meaning of terms Let t be a term and let T be an assignment of variables in t with values from U • if t is a variable, say, t = X, then t T is defined to be X T • if t f (t1 ,, tk ) then t T is defined to be f M (t1T , , t T ) k 9-4 Computability and Complexity A model M and a variable assignment T satisfy a formula (written M, T |= ) if • if P(t1 ,, tk ) is an atomic formula, then M,T |= if P M (t1T , , t kT ) 1 • if = then M,T |= if M,T | • if 1 2 then M,T |= if M , T | 1 and M , T | 2 • if 1 2 then M,T |= if M , T | 1 or M , T | 2 • if = X then M,T |= if, for every u U, T X u | where T X u is the assignment that is identical to T, except that T X u( X ) u • if = X then M,T |= if, there exists u U such that T X u | 9-5 Computability and Complexity Examples A model of the vocabulary of graph theory is a pair G=(U,E), where U is a set and E is a binary relation, that is a graph A model of the vocabulary of number theory U = N, the predicate and function symbols have their usual sense U = Z, the predicate and function symbols have their usual sense U = Q, the predicate and function symbols have their usual sense U = R, the predicate and function symbols have their usual sense U = C, the predicate and function symbols have their usual sense U = {0,…,n-1}, the predicate symbols have their usual sense, the function symbols are interpreted by operations modulo n 9-6 Computability and Complexity 9-7 Types of First Order Formulas A formula is said to be valid if M,T for any model M and assignment T A formula is said to be satisfiable if M,T for some M and T A formula is said to be unsatisfiable if M,T for no M and T valid unsatisfiable satisfiable Formulas and are said to be equivalent, , if they have the same satisfying models and assignments Computability and Complexity Valid Formulas Boolean tautologies X Y ( E ( X ,Y ) E ( X ,Y )) Equality X ( X 1 X 1) ( X X 2 Y ) (( Z X X 2 ) ( Z Y )) Quantifiers P(1, X ) (Y P(Y , X )) (Y P(Y , X )) P(1, X ) 9-8 Computability and Complexity Models and Theories Let be a vocabulary. For a sentence (set of sentences) , Mod() denotes the class of all models satisfying For a model (set of models) M, Th(M) denotes the set all sentences satisfied by M (each model from M) It is called the elementary theory of M If Th(Mod(1 ,, k )) then is called a valid consequence of 1 ,, k written 1 ,, k | 9-9 Computability and Complexity Examples (X ( X X X )) | (X ( X X X X )) (X Y (( X Y ) (( E ( X , Y ) E (Y , X )))) | ((X 1 X 2 X n (( X 1 X 2 ) ( X 1 X 3 ) ( X n1 X n ))) (Y1 Y2 Yn ((Y1 Y2 ) (Y1 Y3 ) (Yn1 Yn ) E (Y1 , Y2 ) E (Yn1 , Yn )))) Let be a first order description of a computer chip, states that a deadlock never occurs is the question “Can the chip be deadlocked?” 9-10 Computability and Complexity Proof Systems A proof system consists of • a set axioms 1 , 2 , • a collection of proof rules 1 , 2 ,, k | A proof is a sequence of formulas 1 , 2 , , where every formula is Either an axiom or obtained from preceding formulas using a rule A theorem is any formula occurring in a proof 9-11 Computability and Complexity Propositional Logic Axioms: the main tautologies Proof rules: • substitution Let and be propositional formulas and let | X denote the formula obtained from by replacing every occurrence of X with . Then | | X is a proof rule • modus ponens , | Theorems: the tautologies 9-12 Computability and Complexity 9-13 Predicate Calculus Let be a vocabulary with plenty of predicate and function symbols Axioms: AX0 Any Boolean tautology AX1 Any expression of the following forms: AX1a: t=t AX1b: (t1 t '1 tk t 'k ) ( f (t1 ,, tk ) f (t '1 ,, t 'k )) AX1c: (t1 t '1 tk t 'k ) ( P(t1 ,, tk ) P(t '1 ,, t 'k )) AX2: Any expression of the form (X ) |X t AX3: Any expression of the form X , with X not free in AX4: Any expression of the form (X ( ) ((X ) (X)) Proof rules: Theorems: modus ponens valid first order formulas Computability and Complexity Gödel’s Completeness Theorem Theorem Let be a set of formulas and a formula. Then if and only if 9-14 Computability and Complexity 9-15 Resolution Axioms: a set of disjunctions of atomic formulas (a set of clauses) Proof rules: • resolution (P ), (P ) , R ( ) C (X C), D (X D) C, D R (C D) (( ) is called the resolvent of and ) Computability and Complexity Satisfiability Satisfiability Instance: A conjunctive normal form . Question: Is satisfiable? k-Satisfiability Instance: A conjunctive normal form every clause of which contains exactly k literals. Question: Is satisfiable? 9-16 Computability and Complexity 9-17 Theorem A Satisfiability instance, , is unsatisfiable if and only if R Example: (1) (2) (3) (4) (5) X Y , X Z , Y Z , X Z , X Y Proof: X Z , Y Z , X Y , X Y , X Z , X X X , X Y , X X X,