1. Computability and Complexity 8-1 Computability and Complexity Andrei Bulatov Logic Reminder

2. Computability and Complexity 8-2 Propositional Formulas A propositional formula is an expression built from variables parenthesis (, ) logical connectives -  conjunction “and” -  disjunction “or” -  negation “not” -  implication “if … then …” Examples:

3. Computability and Complexity 8-3 Propositional Formulas Semantics A truth assignment is an assignment of variables in a formula  with truth values 0 and 1 (or “F” and “T”, or “FALSE” and “TRUTH”) Truth assignment T satisfies , written T   : if  is a variable X, then T   if and only if T(X)=1 if  =¬  ’ then T   if and only if T   ’ if then T   if and only if and if then T   if and only if or Examples: T(X)=1, T(Y)=0, T(Z)=0

4. Computability and Complexity 8-4 Types of Propositional Formulas A formula  is said to be valid if T   for any truth assignment T (tautology) A formula  is said to be satisfiable if T   for some truth assignment T A formula  is said to be unsatisfiable if T   for no truth assignment 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 assignments

5. Computability and Complexity 8-5 Main Tautologies

6. Computability and Complexity 8-6 Main Equivalences

7. Computability and Complexity 8-7 Conjunctive Normal Form A literal is a variable or its negation, X or ¬X A clause is a disjunction of literals A Conjunctive Normal Form (CNF) is a conjunction of clauses Examples: Theorem Every propositional formula is equivalent to a CNF.

8. Computability and Complexity 8-8 Predicates and Quantifiers A predicate on a set A is a function A  A  …  A  {0,1} Informally, a predicate expresses some property of its argument Examples: P(X,Y) : X  Y Q(X,Y,Z): Z is in between X and Y, that is X < Z < Y or Y < Z < X Quantifiers: if a set A is fixed  X means “for every X  A ”  X means “there exists X  A ” A function is a function A  A  …  A  A Examples: f(X,Y) = X + Y g(X,Y,Z) = X · log(Y + Z²) h(X) = 3X + X²

9. Computability and Complexity 8-9 First Order Syntax A vocabulary is a collection of predicate and function symbols, each of which is assigned a non-negative number, the arity Example (Number theory): Predicate symbols: =(X,Y), i.e X = Y ; <(X,Y), i.e. X < Y Function symbols: +(X,Y), i.e. X + Y;  (X,Y), i.e. X  Y; ^(X,Y), i.e.  (X), i.e. X + 1 0 Example (Graph theory): Predicate symbols: =(X,Y), i.e X = Y ; E(X,Y), i.e. X is connected to Y Function symbols: no

10. Computability and Complexity 8-10 A term is an expression built from variables and function symbols Examples:  (+(X,Y),+(X,Z)) (X + Y)  (X + Z) We denote 1=  (0), 2 =  (  (0)), … An atomic formula is a predicate symbol followed by a list of terms in parenthesis; the number of terms in the list must match the arity of the predicate symbol =(+(^(X,T),^(Y,T)),^(Z,T)) Examples:

11. Computability and Complexity 8-11 A first order formula is defined as follows: an atomic formula is a formula if  and  are formulas, then (  ) is a formula if  and  are formulas, then (  ) is a formula if  and  are formulas, then (  ) is a formula if  is a formula, then (  ) is a formula if  is a formula, then (  X  ) is a formula if  is a formula, then (  X  ) is a formula

12. Computability and Complexity 8-12 Examples

13. Computability and Complexity 8-13 Free and Bound Variables Any occurrence of X in an expression  X  or  X  is bound Any occurrence which is not bound is free A variable that has a free occurrence is called free A formula without free variables is called a sentence