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Computability and Complexity 9-1 Computability and Complexity Andrei Bulatov Logic Reminder (Cnt’d)

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Computability and Complexity 9-2 Propositional logic Conjunctive normal forms Predicates, functions and quantifiers Terms Formulas

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Computability and Complexity 9-3 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 on U - to each function symbol f a concrete function on U the equality predicate symbol is always assigned the equality predicate on U

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Computability and Complexity 9-4 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 is defined to be if then is defined to be

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Computability and Complexity 9-5 A model M and a variable assignment T satisfy a formula (written M, T |= ) if if is an atomic formula, then M,T |= if if = then M,T |= if M,T | if then M,T |= if and if then M,T |= if or if = X then M,T |= if, for every u U, where is the assignment that is identical to T, except that if = X then M,T |= if, there exists u U such that

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Computability and Complexity 9-6 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

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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 s a t i s f i a b l e unsatisfiable Formulas and are said to be equivalent, , if they have the same satisfying models and assignments

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Computability and Complexity 9-8 Valid Formulas Boolean tautologies Equality Quantifiers

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Computability and Complexity 9-9 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 then is called a valid consequence of written

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Computability and Complexity 9-10 Examples Let be a first order description of a computer chip, states that a deadlock never occurs is the question “Can the chip be deadlocked?”

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Computability and Complexity 9-11 Proof Systems A proof system consists of a set axioms a collection of proof rules | A proof is a sequence of formulas, where every formula is Either an axiom or obtained from preceding formulas using a rule A theorem is any formula occurring in a proof

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Computability and Complexity 9-12 Propositional Logic Axioms : the main tautologies Proof rules: substitution Let and be propositional formulas and let denote the formula obtained from by replacing every occurrence of X with . Then | is a proof rule modus ponens , | Theorems : the tautologies

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Computability and Complexity 9-13 Predicate Calculus Let be a vocabulary with plenty of predicate and function symbols Axioms: Proof rules : modus ponens Theorems : valid first order formulas AX0 Any Boolean tautology AX1 Any expression of the following forms: AX1a : t=t AX1b : AX1c : AX2 : Any expression of the form AX3 : Any expression of the form X , with X not free in AX4 : Any expression of the form ( X ( ) (( X ) ( X ))

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Computability and Complexity 9-14 Gödel’s Completeness Theorem Theorem Let be a set of formulas and a formula. Then if and only if Theorem Let be a set of formulas and a formula. Then if and only if

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Computability and Complexity 9-15 Resolution Axioms : a set of disjunctions of atomic formulas (a set of clauses) Proof rules: resolution P , P C X C , D X D , C, D C D ( is called the resolvent of and ) R R

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Computability and Complexity 9-16 Satisfiability Instance: A conjunctive normal form . Question: Is satisfiable? Satisfiability Instance: A conjunctive normal form every clause of which contains exactly k literals. Question: Is satisfiable? k- Satisfiability

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Computability and Complexity 9-17 Theorem A Satisfiability instance, , is unsatisfiable if and only if Theorem A Satisfiability instance, , is unsatisfiable if and only if R Example : (1)(2)(3)(4)(5) Proof :

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