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BϋCHI’S MONADIC SECOND ORDER LOGIC
Verification Seminar
V.Sowjanya Lakshmi ( [email protected])
Subhasree M. ([email protected])
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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, N0={0,1,…..}
Quantification over individual elements of N0 and subsets of N0
Natural ordering of N0 (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 andX 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 () 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 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)]
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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 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
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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
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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 variablesX is free variable
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(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}
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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<n
M ╞ Sing(X) iff M is a singleton {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 neededX [0 X (x)(x X succ (x)
X )] (x) (x X)
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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
)
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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 ={ αM | M1╞ (X1)} 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 languageProof:
Proof is by induction on the structure of An equivalent language S1S0 is introduced S1S0 does not have individual variables, xi
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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
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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
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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
S2
<0,0>, <0,1>, <1,1>
<1,0>S1
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Construct a Büchi Automaton(A ,G ) for S1S0 formula
= succ (X,Y )
S1S2
S3
<0,0>
<1,0> <0,1>
<0,0>
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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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=(∃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
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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
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(∃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)
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Example
aa,b
b f e
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(∃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)
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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 Büchi Automata
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Thank You