# B ‹ CHIS MONADIC SECOND ORDER LOGIC Verification Seminar V.Sowjanya Lakshmi (...

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B CHIS MONADIC SECOND ORDER LOGIC Verification Seminar V.Sowjanya Lakshmi ( sowjanya@csa.iisc.ernet.in) Subhasree M. (subha@csa.iisc.ernet.in) Slide 2 CONTENTS Introduction Syntax of S1S Semantics of S1S Satisfiability of S1S Proof Conclusion Slide 3 INTRODUCTION Logic interpreted over Natural Numbers, N 0 ={0,1,..} Quantification over individual elements of N 0 and subsets of N 0 Natural ordering of N 0 (unique and one successor) Slide 4 SYNTAX Terms Atomic Formulas Formulas Slide 5 TERM A term is built up from constant 0 and individual variables x,y, by application of successor function succ. Examples of terms: 0,succ(x),succ(succ(succ(67))), succ(succ(y)) Slide 6 ATOMIC FORMULAS An atomic formula is of the form t t or t X where t and t are terms and X is a set variable Slide 7 FORMULAS A formula is built up from atomic formulas using the Boolean connectives (not), (or) with the existential quantifier ( ) Existential quantifier ( ) can be applied to both individual variables and set variables. Examples of formulas:,, ( x), (X) Slide 8 Remaining Boolean connectives are defined using (not) and (or). Examples: is defined as ( ) is defined as ( ) ( ) Slide 9 UNIVERSAL QUANTIFIER Universal quantifier is defined using ( x) is defined as (( x) ) ( X) is defined as (( X) ) Slide 10 EXAMPLES of Formulas x X is defined as x X X Y is defined as x [(x X x Y) (x Y x X )] Sub(X,Y) is defined as ( x) (x X x Y) Zero(x) is defined as ( x) [(x X ) ( y)(y x)] Slide 11 Examples Sing(X ) is defined as ( Y )[Sub(Y,X) (Y X) ( Z ) (Sub (Z,Y ) (Z Y ) )] Lt(x,y) is defined as Z [succ(x) Z ( Z )(z Z succ(z) Z )] (y Z ) Slide 12 SEMANTICS Formulas are interpreted over N 0 Individual variables x,y,..are interpreted as natural numbers ie. elements of N 0 Function Successor corresponding to adding one t t is true provided t and t denote the same natural number Slide 13 Semantics.. Set variables like X,Y,.. are interpreted as subsets of N 0 t X is true iff the number denoted by t belongs to the set denoted by X Slide 14 Free and bound variables A variable is said to occur free in a formula if it is not within the scope of a quantifier Variables which do not occur free are said to be bound Example: ( x) [(x X ) ( y)(y x)] x and y are bound variables X is free variable Slide 15 (x 1,x 2,..,x k,..,X 1,X 2,..,X l ) indicates all the variables which occur free come from {x 1,x 2,..,x k,..,X 1,X 2,..,X l } To assign a truth value to the formula (x 1,x 2,..,x k,..,X 1,X 2,..,X l ),map each individual variable x i to a natural number m i N 0 and each set variable X j to a subset M j N 0 M (X) denote that is true under the interpretation {x i m i } i {1,2,..,k} and {X i M i } i {1,2,.., l} Slide 16 Examples (M,N) Sub(X,Y) iff M N M Zero(X) iff 0 M (m,n) Lt(x,y) iff m