Extensions of S1S and the Composition Method Giovanna DAgostino and - PowerPoint PPT Presentation

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Extensions of S1S and the Composition Method Giovanna D’Agostino and Angelo Montanari and Alberto Policriti Department of Mathematics and Computer Science University of Udine, Italy { dagostin,montana,policrit } @dimi.uniud.it

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The composition method 1 MSO logic over expansions of S1S The notion of ¯ k -type Composing ¯ k -types Expansions with unary predicates 2 Ultimately type-periodic words Morphic words Expansions with binary predicates 3 Morphic pictures Linear morphic pictures

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S Let fix a signature Σ = { r 1 , ..., r n } , where each r i is a relational symbol with arity h i . Definition ( Monadic Second-Order (MSO) Logic) We define (a variant) of MSO logic as follows: we have set-variables X , Y , Z , ... only (no first-order variables are used) atomic formulas are of the form X 1 ⊆ X 2 ‘ X 1 is a subset of X 2 ’ r i ( X 1 , ..., X h i ) ‘ r i ( x 1 , ..., x h i ) for some x 1 ∈ X 1 , ..., x h i ∈ X h i ’ NE ( X 1 , X 2 ) ‘ X 1 intersects X 2 ’ ALL ( X 1 , ..., X h ) ‘ the union of X 1 , ..., X h is the universe ’ more complex formulas are build up via the Boolean connectives ∧ , ∨ , ¬ quantifications ∃ X , ∀ X over set-variables.

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S Hereafter, we assume that Σ contains at least a binary relation < . We shall evaluate a given MSO-formula ϕ ( X 1 , ..., X m ) with m free variables ( parameters ) over a linearly ordered structure � � S , P 1 , ..., P m where D om ( S ) is either N or an initial segment { 0 , ..., i } < is the usual ordering of the natural numbers P i is an interpretation for the parameter X i .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions MSO logic over expansions of S1S Hereafter, we assume that Σ contains at least a binary relation < . We shall evaluate a given MSO-formula ϕ ( X 1 , ..., X m ) with m free variables ( parameters ) over a linearly ordered structure � � S , P 1 , ..., P m where D om ( S ) is either N or an initial segment { 0 , ..., i } < is the usual ordering of the natural numbers P i is an interpretation for the parameter X i . Model checking problem We want to find linear structures ( S , ¯ P ) with decidable MSO-theories, namely, effective procedures to decide, for any given MSO-formula ϕ ( ¯ X ), whether ( S , ¯ ϕ ( ¯ P ) � X )

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k -type We now introduce ¯ k -types to tackle the model checking problem... Proposition Every MSO-formula ϕ (¯ X ) can be written in prenex normal form as Q t ¯ Y t ... Q 1 ¯ Y 1 ψ ( ¯ X , ¯ Y 1 , ..., ¯ Y t ), where each Q i is either ∀ or ∃ and each ¯ Y i is a k i -tuple of MSO-variables. Definition (Complexity of an MSO-formula) We define the complexity of an MSO-formula in prenex normal form Q t ¯ Y t ... Q 1 ¯ Y 1 ψ ( ¯ X , ¯ Y 1 , ..., ¯ Y t ) as the tuple ¯ k = ( k 1 , ..., k t ). Definition (¯ k -formula) A ¯ k -formula is any MSO-formula with complexity ¯ k .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k -type Let fix a number m of parameters and a complexity ¯ k . Definition (Type) The ¯ k -type of a structure ( S , ¯ P ) is the set of all ¯ k-formulas ϕ ( ¯ X ) that hold in ( S , ¯ P ) when X i is interpreted as P i .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k -type Let fix a number m of parameters and a complexity ¯ k . Definition (Type) The ¯ k -type of a structure ( S , ¯ P ) is the set of all ¯ k-formulas ϕ ( ¯ X ) that hold in ( S , ¯ P ) when X i is interpreted as P i . ( S , ¯ � � We can inductively define a finite object P ) k representing ¯ the ¯ k -type of a structure ( S , ¯ P ) as follows: ( S , ¯ ϕ (¯ S , ¯ � ϕ (¯ � � � � � � P ) ε = X ) atomic : P X ) �� Q ∈ P ( D om ( S )) k ′ � ( S , ¯ ( S , ¯ P , ¯ k . k ′ : ¯ � � � P ) k . k ′ = Q ) ¯ ¯

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k -type Lemma Given a ¯ k-formula ϕ ( ¯ X ) and the ¯ ( S , ¯ � � k-type P ) k of a structure ¯ ( S , ¯ P ) , one can effectively establish whether ( S , ¯ P ) � ϕ ( ¯ X ) . Proposition The MSO-theory of a structure ( S , ¯ P ) is decidable iff there is a computable function f that maps a complexity ¯ k to the corresponding ¯ ( S , ¯ � � k -type P ) k . ¯

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions The notion of ¯ k -type Lemma Given a ¯ k-formula ϕ ( ¯ X ) and the ¯ ( S , ¯ � � k-type P ) k of a structure ¯ ( S , ¯ P ) , one can effectively establish whether ( S , ¯ P ) � ϕ ( ¯ X ) . Proposition The MSO-theory of a structure ( S , ¯ P ) is decidable iff there is a computable function f that maps a complexity ¯ k to the corresponding ¯ ( S , ¯ � � k -type P ) k . ¯ Note: computing ¯ k -types of finite structures is easy ... How about ¯ k -types of infinite (expanded) structures?

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k -types Idea: use composition method, which allows one to compute the type of a compound structure from the types of its components. Definition (Ordered sum) S i , ¯ � � Let I be an index ordering (e.g., ( N , < )) and let P i i ∈I be a sequence of expanded linear orders. i ∈I ( S i , ¯ The ordered sum � P i ) is the disjoint union of the structures ( S i , ¯ � P i , where i ranges over I . i ∈I ( S i , ¯ (the ordering relation of � P i ) is implicitly given by the ordering relations of I and ( S i , ¯ � P i ).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k -types Composition Theorem - Shelah ’75 Let ¯ k be a complexity and m a number of parameters. We denote by { σ 1 , ..., σ s } the set of all ¯ k -types on m parameters. One can compute a complexity ¯ r such that for any index ordering I S i , ¯ � � for any sequence P i i ∈I of expanded linear orders the ¯ i ∈I ( S i , ¯ k -type of the ordered sum � P i ) is uniquely determined by (and computable from) the ¯ r -type of the expanded index ordering ( I , Q 1 , ..., Q s ) s.t. i ∈ Q j iff [ S i ] ¯ k = σ j .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Composing ¯ k -types Composition Theorem - Shelah ’75 Let ¯ k be a complexity and m a number of parameters. We denote by { σ 1 , ..., σ s } the set of all ¯ k -types on m parameters. One can compute a complexity ¯ r such that for any index ordering I S i , ¯ � � for any sequence P i i ∈I of expanded linear orders the ¯ i ∈I ( S i , ¯ k -type of the ordered sum � P i ) is uniquely determined by (and computable from) the ¯ r -type of the expanded index ordering ( I , Q 1 , ..., Q s ) s.t. i ∈ Q j iff [ S i ] ¯ k = σ j . Example The ¯ k -type of the ordered sum ( S 1 , ¯ P 1 ) + ( S 2 , ¯ P 2 ) can be computed from the ¯ k -types of the two structures ( S 1 , ¯ P 1 ) and ( S 2 , ¯ P 2 ).

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words Now, we briefly present some decidability results for infinite linear orders expanded with unary predicates . Hereafter, we identify a linear order expanded with m unary predicates with its characteristic word w over B m = { 0 , 1 } m .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words Now, we briefly present some decidability results for infinite linear orders expanded with unary predicates . Hereafter, we identify a linear order expanded with m unary predicates with its characteristic word w over B m = { 0 , 1 } m . Definition (Ultimately type-periodic words) A sequence ( w i ) i ∈ N of finite words is ultimately type-periodic if, for every complexity ¯ k , one can compute p , q such that for all i ≥ p , [ w i ] ¯ k = [ w i + q ] ¯ k . An infinite word w is ultimately type-periodic if there is an ultimately type-period factorization w = w 0 · w 1 · w 2 · ... .

Outline The composition method Expansions with unary predicates Expansions with binary predicates Conclusions Ultimately type-periodic words Theorem Any ultimately type-periodic word has a decidable MSO-theory.