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B ϋ CHIS MONADIC SECOND ORDER LOGIC Verification Seminar V.Sowjanya Lakshmi ( sowjanya@csa.iisc.ernet.in) Subhasree M. (subha@csa.iisc.ernet.in)

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CONTENTS Introduction Syntax of S1S Semantics of S1S Satisfiability of S1S Proof Conclusion

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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)

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SYNTAX Terms Atomic Formulas Formulas

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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))

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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

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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)

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Remaining Boolean connectives are defined using (not) and (or). Examples: is defined as ( ) is defined as ( ) ( )

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UNIVERSAL QUANTIFIER Universal quantifier is defined using ( x) is defined as (( x) ) ( X) is defined as (( X) )

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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)]

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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 )

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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

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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

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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

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(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}

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Examples (M,N) Sub(X,Y) iff M N M Zero(X) iff 0 M (m,n) Lt(x,y) iff m

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Sentence A sentence is a formula in which no variables occur free A sentence is either true or false Assigning values is not needed X [0 X ( x)(x X succ (x) X )] ( x) (x X)

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SATISFIABILITY An S1S formula is (x 1,x 2,..,x k,X 1,X 2,..,X l ) is said to be satisfiable if we can choose M 1 = (m 1,m 2,..,m k,M 1,M 2,..,M l ) such that M 1 (X 1 ), where X 1= (x 1,x 2,..,x k,X 1,X 2,..,X l )

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Büchi showed that every word in L has an interpretation for the free variables in under which evaluates to true Every interpretation which makes true is represented by some word in L

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Satisfiability... is satisfiable iff there is some interpretation which makes it true iff L is nonempty The language L is defined over the alphabet {0,1} m where m is the number of free variables in

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Language L ({0,1} m ) ) is S1S definable if L= L for some formula Any Language L can be converted into an equivalent language L {0,1} over{0,1} m L ={ α M | M 1 (X 1 )} L {0,1} ={ α {0,1} | α L}

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THEOREM Let be an S1S formula. Then L is an -regular language Let L be an -regular language. Then L {0,1} is S1S definable

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Theorem: Let be an S1S formula. Then L is an -regular language Proof : Proof is by induction on the structure of An equivalent language S1S0 is introduced S1S0 does not have individual variables, x i

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All variables in S1S0 are set variables, X j Atomic formulas are of the form X Y and succ (X,Y ) X Y is true if X is a subset of Y Succ ( X,Y ) is true if X and Y are singletons {x } and {y } respectively and y = x +1

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Converting S1S formula to S1S0 formula 0 such that L = L 0 Removing nested application of successor function succ (succ (x )) X ) can be written as ( y)( z) y =succ(x) z = succ (y) z X

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Eliminating formulas of the form 0 X using the formula Zero ( X ) Eliminating singleton variables, using the formula Sing ( x) ( y) succ(x) = y y Z can be written as ( X) (Sing ( X ) [( y ) Sing ( Y ) succ ( X,Y ) Y Z ] )

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Construct a Büchi Automaton (A,G ) for S1S0 formula = X Y S2S2,, S1S1

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Construct a Büchi Automaton (A,G ) for S1S0 formula = succ (X,Y ) S1S1 S2S2 S3S3

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Induction Step Considering the connectives, and X =Ψ, construct the complement of Ψ = 1 2, construct 1 2

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=( X 1 ) Ψ (X 1,X 2,..,X l ), the language corresponds to the projection of L Ψ via the function Π:{0,1} m {0,1} m-1, erases the first component of each m-tuple in {0,1} m

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Let L be an -regular language. Then L {0,1} is S1S definable. Proof: (A,G) –Büchi Automaton recognizing L = { a 1,a 2,..,a m }, A=(S,, S in ) with S = { s 1,s 2,..,s k } A 1,A 2,..,A m are the free variables A 1 describes the positions in which the input where letter a i occurs S 1,S 2,..,S k describes the runs S j describes the positions in the run where the automaton is in S j

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( S 1 ) ( S 2 )…( S k) ( x) i {1,2,..,m} (x A i ) i {1,2,..,m} (x A i ( j i x A j ) ( x) i {1,2,..,k} (x S i ) i {1,2,..,k} (x S i ( j i x S j ) ( x) Si Sin (0 S i ) ( x) (Si,, ai, sk) (x S i ) (x A j ) (succ (x) S k ) Si G ( x) ( y) (x

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Example a a,b b f e

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( S f ) ( Se) ( x) [(x A a ) (x A b ) (x A a x A b ) (x A b x A a ) ] ( x) [(x S f ) (x S e ) (x S f x S e ) (x S e x S f )] (0 S f ) ( x) [((x S f ) (x A a ) succ (x) S f ) ((x S f ) (x A b ) succ (x) S 2 ) ((x S e ) succ (x) S e )] ( x) ( y) (x

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Conclusion Büchi has proved that Notions of S1S definability and -regularity are equivalent.

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Reference Madhavan Mukund. Linear Time Temporal Logic and B ü chi Automata

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Thank You

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