BϋCHI’S MONADIC SECOND ORDER LOGIC

Verification Seminar

V.Sowjanya Lakshmi ( sowjanya@csa.iisc.ernet.in)

Subhasree M. (subha@csa.iisc.ernet.in)

CONTENTS

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

INTRODUCTION

Logic interpreted over Natural Numbers, N0={0,1,…..}

Quantification over individual elements of N0 and subsets of N0

Natural ordering of N0 (unique and one successor)

SYNTAX

Terms

Atomic Formulas

Formulas

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))

ATOMIC FORMULAS

An atomic formula is of the form t t’ or t X

where t and t’ are terms andX is a set variable

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)

Remaining Boolean connectives are defined using (not) and (or).

Examples:

is defined as ()

is defined as () is defined as () ()

UNIVERSAL QUANTIFIER

Universal quantifier is defined using

(x) is defined as ((x) )

(X) is defined as ((X) )

EXAMPLES of Formulas xX 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)]

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 )

SEMANTICS Formulas are interpreted over N0

Individual variables x,y,..are interpreted as natural numbers ie. elements of N0

Function Successor corresponding to adding one

t t’ is true provided t and t’ denote the same natural number

Semantics .. Set variables like X,Y,.. are

interpreted as subsets of N0

t X is true iff the number denoted by t belongs to the set denoted by X

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 variablesX is free variable

(x1,x2,..,xk,..,X1,X2,..,Xl) indicates all the variables which occur free come from {x1,x2,..,xk,..,X1,X2,..,Xl}

To assign a truth value to the formula(x1,x2,..,xk,..,X1,X2,..,Xl) ,map each individual variable xi to a natural number miN0

and each set variable Xj to a subset MjN0

M╞ (X) denote that is true under the interpretation {xi→mi} i {1,2,..,k} and

{Xi→Mi } i {1,2,.., l}

Examples (M,N) ╞ Sub(X,Y) iff M N

M ╞ Zero(X) iff 0 M

(m,n) ╞ Lt(x,y) iff m<n

M ╞ Sing(X) iff M is a singleton {m}

Sentence A sentence is a formula in which

no variables occur free A sentence is either true or false Assigning values is not neededX [0 X (x)(x X succ (x)

X )] (x) (x X)

SATISFIABILITY

An S1S formula is (x1,x2,..,xk,X1,X2,..,Xl ) is said to be satisfiable if we can choose

M1= (m1,m2,..,mk,M1,M2,..,Ml ) such that

M1╞ (X1), where X1= (x1,x2,..,xk,X1,X2,..,Xl

)

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

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

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 ={ αM | M1╞ (X1)} L {0,1}={α {0,1} | α L}

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

Theorem: Let be an S1S formula. Then L is an -regular languageProof:

Proof is by induction on the structure of An equivalent language S1S0 is introduced S1S0 does not have individual variables, xi

All variables in S1S0 are set variables, Xj

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

Converting S1S formula to S1S0 formula 0 such that L = L0

Removing nested application of successor

function

succ (succ (x ))X ) can be written as

(∃y)(∃z) y =succ(x) ∧z = succ (y)∧z X

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 ] )

Construct a Büchi Automaton(A ,G ) for S1S0 formula

= X ⊆Y

S2

<0,0>, <0,1>, <1,1>

<1,0>S1

Construct a Büchi Automaton(A ,G ) for S1S0 formula

= succ (X,Y )

S1S2

S3

<0,0>

<1,0> <0,1>

<0,0>

Induction Step

Considering the connectives ⌐,∨ and ∃X

= ⌐Ψ, construct the complement of Ψ

= 1∨ 2 ,construct 1 2

=(∃X1 ) Ψ(X1,X2,..,Xl ) , the language corresponds to the projection of LΨ via the function

Π:{0,1}m →{0,1} m-1, erases the first component of each m-tuple in {0,1}m

Let L be an -regular language.

Then L{0,1} is S1S definable.Proof:

(A,G) –Büchi Automaton recognizing L⊆

= {a1,a2,..,am} , A=(S,→, Sin ) with S = {s1,s2,..,sk}

A1,A2,..,Am are the free variables

A1 describes the positions in which the input where letter ai occurs

S1,S2,..,Sk describes the runs

Sj describes the positions in the run where the automaton is

in Sj

(∃S1) (∃ S2)…(∃ Sk)

( x) i {1,2,..,m} (x Ai) i {1,2,..,m}

(x Ai (j i x Aj )

( x) i {1,2,..,k} (x Si) i {1,2,..,k} (x Si

(j i x Sj )

( x) Si Sin (0 Si)

( x) (Si,, ai, sk)→ (x Si) (x Aj) (succ (x) Sk)

Si G ( x) (∃y) (x<y) (y Si)

Example

aa,b

b f e

(∃Sf) (∃ Se)

( x) [(x Aa) (x Ab) (x Aa x Ab)

(x Ab x Aa) ]

( x) [(x Sf) (x Se) (x Sf x Se)

(x Se x Sf)]

(0 Sf)

( x) [((x Sf) (x Aa) succ (x) Sf)

((x Sf) (x Ab) succ (x) S2) ((x Se) succ (x) Se)]

( x) (∃y) (x<y) (y Sf)

Conclusion

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

Reference

Madhavan Mukund. Linear Time Temporal

Logic and Büchi Automata

Thank You