Composition for Orders with an Extra Binary Relation Wolfgang Thomas Bruno’s Workshop, Bordeaux, June 2012
Two Traditions in Effective Logic Orderings with unary predicates MSO-logic Automata, Composition method Graphs Courcelle theory (for MSO) Hanf and Gaifman theorems (for FO) Here: Study of FO-theory of orderings expanded by graphs Structures: ( N , < , R ) with binary R Restriction: R is finite valency R ⊆ A × A is of finite valency if for each a ∈ A there are only finitely many b ∈ A with R ( a , b ) or R ( b , a ) . Injective functions f : N → N provide examples. Wolfgang Thomas
Plan 1. MSO-Th ( N , < , P ) for unary P 2. Homogeneity of colorings is first-order definable 3. Orderings ( N , < , R ) with binary R 4. Conclusion Wolfgang Thomas
Structures ( N , < , P ) with unary P Identify P ⊆ N with 0-1-word α ( P ) uchi’s analysis of MSO-Th ( N , < ) : Consequence of B¨ MSO-Th ( N , < , P ) is decidable iff the following decision problem is decidable: uchi automaton A , decide whether A accepts α ( P ) . Given a B¨ So one only needs to decide whether the word α ( P ) can be cut into pieces u 0 , u 1 , . . . such that u 0 u i A : q 0 → q and A : q → q for i = 1, 2, . . . , with q final. The composition method allows this reduction to periodicity directly, without reference to automata. Wolfgang Thomas
m -Types (for FO and MSO) Given quantifier-depth m define for two words u , v (finite or infinite!): u ≡ m v : ⇐ ⇒ u and v satisfy the same sentences of quantifier-depth m Facts: ≡ m is an equivalence relation of finite index; call the equivalence classes m -types. An m -type τ is definable by a sentence ϕ τ of quantifier-depth m . Each sentence ψ of quantifier-depth m is equivalent to a disjunction of sentences ϕ τ . Wolfgang Thomas
Composition 1. From the m -types of u and v one can compute the m -type of uv . 2. From the m -type of u one can compute the m -type of uuu . . . . Consequence: Given α = uvvv . . . , the m -type ̺ of α is determined by the m -types σ of u and τ of v ; we write ̺ = σ + ∑ ω τ Ramsey’s Theorem guarantees such a decomposition for arbitrary α Wolfgang Thomas
Finite Colorings Given a finite set Col = { c 1 , . . . , c r } of colors. A coloring over N with Col is a map C : { ( m , n ) | m < n } → Col C is additive if from C ( ℓ , m ) = C ( ℓ ′ , m ′ ) and C ( m , n ) = C ( m ′ , n ′ ) we can infer C ( ℓ , n ) = C ( ℓ ′ , n ′ ) . For colors c , d we may write c + d . Example: For quantifier-depth m and ω -word α define C m α ( i , j ) = m -type of α [ i , j − 1 ] (either for FO or MSO) Wolfgang Thomas
Ramsey’s Theorem For any finite additive coloring C there is a ”homogeneous set” H = { h 0 < h 1 < h 2 < . . . } such that all colors C ( h i , h j ) (where i < j ) coincide. Consequence: Then there are two colors c , d such that C ( 0, h 0 ) = c and C ( h i , h i + 1 ) = d Call a color pair ( c , d ) good for C if there is H = { h 0 < h 1 < . . . } such that C ( 0, h 0 ) = c and C ( h i , h j ) = d for i < j , in particular, C ( h i , h i + 1 ) = d , and d = d + d . Wolfgang Thomas
Back to MSOTh ( N , < , P ) MSOTh ( N , < , P ) is decidable iff for each m we can compute the m -type ̺ of α ( P ) iff for each m and the associated coloring C m α ( P ) we can compute those pairs ( σ , τ ) of m -types which are good for C m α ( P ) . In other words, for any P : A sentence ψ of quantifier-depth m is effectively equivalent over ( N , < , P ) to a disjunction of statements ” ( σ , τ ) is good for C m α ( P ) ” [Compare with the automata theoretic periodicity condition.] Wolfgang Thomas
Defining to be Good Let C be the tuple of binary predicates ” C ( i , j ) = c ”. Consider the associated structure ( N , < , C ) . Remark: There is an MSO-sentence ϕ c , d saying in ( N , < , C ) that ( c , d ) is good for C : ∃ X ( X is infinite ∧ C ( 0, x ) = c for the smallest element x of X ∧ C ( x , y ) = d for any x , y ∈ X with x < y ) We show that an FO-sentence suffices. This will also give a proof of Ramsey’s Theorem. Wolfgang Thomas
McNaughton’s Merge-Relation Given α and an additive coloring C . m , n merge at k (short m ∼ C n ( k ) ) if C ( m , k ) = C ( n , k ) If m , n merge at k then also at each k ′ > k . Wolfgang Thomas
Lemma 1 ( c , d ) is good for C iff ( ∗ ) ∃ n [ C ( 0, n ) = c ∧ ∀ m ∃ k > m ( C ( n , k ) = d ∧ n ∼ C k )] Show ⇐ : Take n 0 as the smallest n according to ( ∗ ) . Assume n 0 , . . . , n i are defined, with n 0 ∼ C n j for j = 1, . . . , i . Let n 0 , . . . , n i merge at m . Define n i + 1 as the smallest number k > m guaranteed by ( ∗ ) , namely with C ( n 0 , n i + 1 ) = d and n 0 ∼ n j for all j = 1, . . . , i + 1 . Wolfgang Thomas
Consequence: Ramsey’s Theorem Let M be an infinite ∼ C -equivalence class Let n 0 be its smallest element and set c = C ( 0, n 0 ) . For some d infinitely many n in M exist with C ( n 0 , n ) = d Then ( ∗ ) is satisfied. Hence ( c , d ) is good for C . Wolfgang Thomas
Reducing Quantifier Alternation ( ∗ ) ∃ n [ C ( 0, n ) = c ∧ ∀ m ∃ k > m ( C ( n , k ) = d ∧ n ∼ C k )] is a Σ 3 -condition (w.r.t. unbounded quantifiers). Show that it can be written as a Boolean combination of Σ 2 -conditions. Define a set M ℓ , c ( x ) : Consider the ℓ -tuples of distinct numbers n 1 , . . . , n ℓ ≤ x such that C ( 0, n i ) = c and any two of the n j do not merge at x . If such an ℓ tuple exists let M ℓ , c ( x ) contain the elements of the smallest such tuple (in lexicographical ordering) otherwise let M ℓ , c ( x ) = { x } . Wolfgang Thomas
Lemma 2 Define g ℓ , c ( x ) = max M ℓ , c ( x ) f ℓ , c , d ( x ) = the greatest y < x such that for some z ∈ M ℓ , c C ( z , y ) = d and C ( y , x ) = d and C ( z , x ) = d (take value 0 if such y does not exist) Then ( c , d ) is good for C iff � r ℓ = 1 ( g ℓ , c is bounded and f ℓ , c , d unbounded). Wolfgang Thomas
The ∀∃ ∧ ∃∀ -Lemma Let C be an additive finite coloring and C be the tuple of relations C ( i , j ) = c . There are bounded formulas ϕ c , ℓ ( y ) and ψ ℓ , c , d ( y ) such that ( c , d ) is good for C iff = � | C | ( N , < , C ) | ℓ = 1 ( ∃ x ∀ y > x ϕ c , ℓ ( y ) ∧ ∀ x ∃ y > x ψ c , d , ℓ ( y )) Application: McNaughton’s Theorem Any B¨ uchi automaton can be converted into a deterministic Muller automaton. Use ∼ A -classes as colors: u ∼ A v iff for any states p , q u v A : p → q [passing F ] ⇔ A : p → q [passing F ] Wolfgang Thomas
Other Applications 1. For any P ⊆ N : MSO-Th ( N , < , P ) is decidable iff WMSO-Th ( N , < , P ) is. 2. Any FO-definable ω -language can be recognized by a counter-free Muller automaton. Wolfgang Thomas
Binary relations and composition Consider structures ( N , < , R ) with binary R The m -types of two segments ([ ℓ , m ) , < , R | [ ℓ , m ) ) and ([ m , n ) , < , R | [ m , n ) ) are not sufficient to determine the m -type of ([ ℓ , n ) , < , R | [ ℓ , n ) ) But we can do composition if enough interface information is provided. Wolfgang Thomas
Finite Valency Let R ⊆ N × N be of finite valency: For any a there are at most finitely many b with R ( a , b ) or R ( b , a ) . Call [ a , b ] an R -segment if R ( a , b ) or R ( b , a ) . An R -segment is maximal if it is not properly contained in another R -segment. Remark: If R is of finite valency then each R -segment is contained in a maximal one. Wolfgang Thomas
m -Admissible Segments Define for each b a sequence b ( 0 ) > b ( 1 ) > . . . as follows: b ( 0 ) = b b ( i + 1 ) = biggest c which is below all maximal R -segments [ k , ℓ ] intersecting [ b ( i ) , ∞ ) , if such c exists, 0 otherwise The segment [ a , b ] is m -admissible if b ( 2 m ) > a Write b ∗ for b ( 2 m ) if m is clear. Denote by � b the sequence ( b ( 0 ) , . . . , b ( 2 m )) . For any k there is exist admissible segments [ a , b ] above k . Wolfgang Thomas
T - and D -Types Let [ a , b ] be m -admissible, a 0 , . . . , a r − 1 ∈ [ a , b ] . let T m R [ a , b ]( a 0 , . . . , a r − 1 ) be the FO- m type of the restriction of N to [ a , b ] R [ a ∗ , b ]( � a , � D m R [ a , b ]( a 0 , . . . , a r − 1 ) : = T m b , a 0 , . . . , a r − 1 ) D m R defines an almost total coloring: For each a there are only finitely many b ≥ a such that [ a , b ] is not m -admissible. Wolfgang Thomas
Composition Lemma 1. Given m -admissible segments [ a , b ] and [ b , c ] , D m R [ a , b ] and D m R [ b , c ] determine effectively the type D m R [ a , c ] 2. Given a sequence a 0 , a 1 , . . . such that [ a i , a i + 1 ] is m -admissible and D m R [ a i , a i + 1 ] = τ for some m -type τ , D m R [ a 0 , ∞ ) is determined effectively by τ . If D m R [ 0, a 0 ] = σ we may write D m R [ 0, ∞ ) = σ + τ + τ + . . . Wolfgang Thomas
Nondefinability of + and · Theorem: In a structure ( N , < , R ) with R of finite valency, neither addition nor multiplication is FO-definable. Lemma: Let f : N 2 → N be FO-definable in ( N , < , R ) where R is of finite valency. Then one of the following two sets is finite: X f : = { x ∈ N | λ yf ( x , y ) is injective } Y f : = { y ∈ N | λ x f ( x , y ) is injective } . Note: X + , Y + , X · , Y · are all infinite. Wolfgang Thomas
Recommend
More recommend
