B ϋ CHI’S MONADIC SECOND ORDER LOGIC

1. BϋCHI’S MONADIC SECOND ORDER LOGIC Verification Seminar V.Sowjanya Lakshmi ( sowjanya@csa.iisc.ernet.in) Subhasree M. (subha@csa.iisc.ernet.in)

2. CONTENTS • Introduction • Syntax of S1S • Semantics of S1S • Satisfiability of S1S • Proof • Conclusion

3. 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)

4. SYNTAX • Terms • Atomic Formulas • Formulas

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))

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

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)

8. Remaining Boolean connectives are defined using (not) and (or). Examples:  is defined as ()   is defined as ()  is defined as ()  ()

9. UNIVERSAL QUANTIFIER Universal quantifier  is defined using  (x)  is defined as  ((x)   ) (X)  is defined as  ((X)   )

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)]

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 )

12. 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

13. 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

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 isfree variable

15. (x1,x2,..,xk,..,X1,X2,..,Xl) indicates all the variableswhich 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 miN0 and each set variable Xj to a subset MjN0 • M╞  (X) denote that  is true under theinterpretation {xi→mi} i {1,2,..,k} and {Xi→Mi } i {1,2,.., l}

16. 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}

17. 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)

18. 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 )

19. Büchi showed that every word in L hasan interpretation for the free variables in  under which  evaluates to true • Every interpretation which makes  true is represented by some word in L

20. 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}mwhere m is the number of free variables in 

21. 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}

22. 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

23. 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, xi

24. 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

25. 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

26. 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 ] )

27. Construct a Büchi Automaton(A ,G ) for S1S0 formula  •  = X ⊆Y <0,0>, <0,1>, <1,1> <1,0> S1 S2

28. Construct a Büchi Automaton(A ,G ) for S1S0 formula   = succ (X,Y ) <0,0> <0,0> <1,0> <0,1> S3 S2 S1

29. Induction Step Considering the connectives ⌐,∨ and ∃X •  = ⌐Ψ, construct the complement of Ψ •  = 1∨ 2 ,construct 1  2

30.  =(∃X1 )Ψ(X1,X2,..,Xl ) , the language corresponds to the projection of LΨvia the function Π:{0,1}m →{0,1} m-1,erases the first component of each m-tuple in{0,1}m

31. 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 )withS = {s1,s2,..,sk} A1,A2,..,Am are the free variables A1 describes the positions in which the input where letter ai occurs S1,S2,..,Skdescribes the runs Sj describes the positions in the run where the automaton is in Sj

32. (∃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)

33. Example a a,b b f e

34. (∃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)

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

36. Reference Madhavan Mukund. Linear Time Temporal Logic and Büchi Automata

37. Thank You