Decidability of MSO Theories of Deterministic Tree Structures - PowerPoint PPT Presentation

Decidability of MSO Theories of Deterministic Tree Structures Gabriele Puppis puppis@dimi.uniud.it (joint work with Angelo Montanari) Department of Mathematics and Computer Science University of Udine, Italy

Outline • MSO logics over tree structures • The automaton-based approach • Reduction to acceptance of regular trees • Structural properties • Application examples • Further work

MSO Logics over tree structures (1) Let Λ = { 1 , . . . , k } be a finite set of edge labels . We consider infinite deterministic trees extended with tuples of ( T , ¯ V ) = (Λ ∗ , ( E l ) l ∈ Λ , ( V i ) i ∈ [1 ,m ] ) unary predicates : Example. V 1 = { ε, 1 , 11 , 111 , . . . } V 2 = { 11 , 12 , . . . , 1 k, 21 , 22 , . . . , 2 k, . . . } { V 1 } 1 k 2 { V 1 } . . . 1 k 1 k 1 k 2 2 2 . . . . . . . . . { V 1 , V 2 } { V 2 } { V 2 } { V 2 }{ V 2 } { V 2 } { V 2 }{ V 2 } { V 2 } . . . . . . . . . . . . . . . . . . . . . . . . . . .

MSO Logics over tree structures (2) MSO formulas over a tree T are built up from atoms: • E l ( X i , X j ) “ X i , X j denote singletons { u } , { v } with ( u, v ) being an l -labeled edge” • X i ⊆ X j “ X i denotes a subset of X j ” ...through connectives ∨ , ¬ and quantifier ∃ over variables. • Each free variable X i in a formula ϕ ( ¯ X ) is interpreted by a designated subset V i . • T � ϕ [¯ ϕ ( ¯ V ] X ) holds in T iff by interpreting V i for X i , for all i . The model-checking problem for ( T , ¯ V ) is to decide whether T � ϕ [¯ V ] , for any given formula ϕ ( ¯ X ) .

MSO Logics over tree structures (3) Example. The formula ϕ ( X ) = X ( ε ) ∧ ∀ x. ∃ y. ( X ( x ) ∧ E 2 ( x, y ) → X ( y )) holds in the binary tree extended with the predicate V represented by black colored vertices: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 . . . . . . . . . . . . . . . . . . . . . . . . Remark. We identify a tree structure ( T , ¯ V ) with its canonical representation T ¯ V (i.e. an infinite complete vertex-colored tree).

The automaton based approach (1) We consider tree automata accepting colored trees in a top-down fashion: • they ‘spread’ states inside the input tree (in accordance to transition relations), • they ‘verify’ that suitable acceptance conditions (envisaging occurrences of states) are satisfied for each path in the tree. We write T ¯ V ∈ L ( M ) to say that the tree T ¯ V is accepted by automaton M . Example. ( Rabin acceptance condition ) Given AC = { ( L 1 , U 1 ) , . . . , ( L n , U n ) } , we require that, for each infinite path, there is a pair ( L i , U i ) ∈ AC such that at least one state in U i , but no state in L i , is visited infinitely often.

The automaton based approach (2) Step 1. We reduce the model checking problem to an acceptance problem by exploiting the correspondence between MSO formulas over tree structures and Rabin tree automata. [Rabin ’69] For every formula ϕ ( ¯ X ) , there is a Rabin tree automaton M (and vice versa) such that for every tree structure ( T , ¯ V ) T � ϕ [¯ V ] ⇔ T ¯ V ∈ L ( M ) ⇒ the decision problem for MTh ( T , ¯ V ) reduces to the acceptance problem Acc ( T ¯ V ) for Rabin tree automata.

The automaton based approach (3) Remark. The problem Acc ( T ¯ V ) can be decided for any regular tree T ¯ V (i.e. a tree with only finitely many distinct subtrees)... 1 2 1 2 1 2 1 2 1 2 1 2 1 2 . . . . . . . . . . . . . . . . . . . . . . . . ...by simply considering the intersection with the tree automaton generating T ¯ V ... 1 2 1 2

The automaton based approach (4) Step 2. We extend the class of trees for which the acceptance problem turns out to be decidable. Idea. Given an automaton M , we define an equivalence ∼ = M that groups together those (finite or infinite) trees on which M ‘behaves’ in a similar way . In particular, for two infinite complete trees T , T ′ , T ∈ L ( M ) ⇔ T ′ ∈ L ( M ) . T ∼ = M T ′ will imply Fact. Many non-regular trees turn out to be equivalent to some (computable) regular trees. ⇒ in such cases we will be able to solve Acc ( T ) by reducing it to the decidable problem Acc ( T ′ )

A digression into Büchi automata (1) Given a Büchi automaton M , we can define an equivalence ∼ = M over finite words s.t. u ∼ = M u ′ iff, for every pair of states r, s , u ′ u • r − − → s ⇔ − − → s r u u ′ • r − ◦→ s ⇔ − ◦→ s r Properties: • ∼ = M has finite index • ∼ = M is a congruence w.r.t. concatenation • ∼ = M -equivalent factorizations are indistinguishable by M , namely, if u i ∼ i for all i ≥ 0 , then = M u ′ u 0 u 1 u 2 . . . ∈ L ( M ) ⇔ u ′ 0 u ′ 1 u ′ 2 . . . ∈ L ( M )

A digression into Büchi automata (2) [Elgot, Rabin, Carton and Thomas...] Let w be an infinite word. If we can provide a factorization u 0 · u 1 · u 2 . . . of w such that, for any congruence ∼ = M there are p, q computable such that ∀ i > p. u i ∼ = M u i + q Then : w ∈ L ( M ) � ( u 0 . . . u p )( u p +1 . . . u p + q )( u p + q +1 . . . u p +2 q ) . . . ∈ L ( M ) � ( u 0 . . . u p )( u p +1 . . . u p + q )( u p +1 . . . u p + q ) . . . ∈ L ( M ) � ( u 0 . . . u p ) · ( u p +1 . . . u p + q ) ω ∈ L ( M ) ⇒ we can decide whether M accepts w .

A Reduction to acceptance of regular trees We define the tree concatenation T 1 · c T 2 of two (finite or infinite) trees T 1 , T 2 as the substitution of all the c -colored leaves in T 1 by T 2 : 1 2 · gray 1 2 1 2 = 1 2 1 2 The notion can be extended to infinite sequences of trees, henceforth called factorizations (e.g. T 0 · c 0 T 1 · c 1 T 2 · c 2 . . . ). Proposition. Any ultimately periodic factorization consisting of only regular trees generates a regular tree.

The notion of equivalence Given an automaton M and a (finite or infinite) tree T , we need to quantify over all the possible partial runs of M on T (i.e. ‘run fragments’). T 1 ∼ = M T 2 Definition. iff ∀ partial run P 1 on T 1 , ∃ a partial run P 2 on T 2 (and vice versa) such that for i = 1 and i = 2 we have the same • pair ( T i ( ε ) , P i ( ε )) (color and state at the root ) • set { ( T i ( u ) , P i ( u )) u } u leaf (pairs color-state at the frontier ) • set {I mg ( P i | π ) } π fin. path (sets of states occurring along finite full paths ) • set {I nf ( P i | π ) } π inf. path (sets of states occurring infinitely often along infinite paths )

Properties of ∼ = M Properties: • ∼ = M has finite index • ∼ = M is a congruence w.r.t. concatenations namely, if T 1 ∼ 1 and T 2 ∼ = M T ′ = M T ′ 2 , then T 1 · c T 2 ∼ = M T ′ 1 · c T ′ 2 • ∼ = M -equivalent factorizations are indistiguishable by M namely, if T i ∼ = M T ′ i for all i ≥ 0 , then T 0 · c 0 T 1 · c 1 . . . ∈ L ( M ) ⇔ T ′ 0 · c 0 T ′ 1 · c 1 . . . ∈ L ( M )

The key ingredient Let T be an infinite complete tree. If we can provide a factorization T 0 · c 0 T 1 · c 1 . . . of T such that, for any congruence ∼ = M there are p, q computable such that ∀ i > p. T i ∼ = M T i + q Then : T ∈ L ( M ) � T 0 · c 0 ... T p · c p T p +1 · c p +1 ... T p + q · c p + q T p + q +1 · c p + q +1 ... ∈ L ( M ) � T 0 · c 0 ... T p · c p T p +1 · c p +1 ... T p + q · c p + q T p +1 · c p + q +1 ... ∈ L ( M ) Remark. The last factorization is ultimately periodic, ⇒ it generates a (decidable) regular tree T ′ provided that T 0 , T 1 , . . . are regular trees.

Residually regular trees Definition. Residually regular trees are defined as follows: • A tree T is level 1 residually regular tree if we can provide a factorization T 0 · c 0 T 1 · c 1 . . . (with T 0 , T 1 , . . . regular trees) which is effectively ultimately periodic w.r.t. any congruence ∼ = M . • We extend the notion to level n > 1 (for n countable ordinal) by allowing the factors to be level n ′ < n residually regular trees . ⇒ this gives rise to a hierarchy that is strictly increasing at least for the initial (finite ordinal) levels.

The main result Theorem. MTh ( T , ¯ V ) is decidable for every residually regular tree T ¯ V . Proof sketch. We decide MTh ( T , ¯ V ) as follows: 1. let S = T 0 · c 0 T 1 · c 1 . . . be a level n residually ultimately periodic factorization for T ¯ V 2. given a formula ϕ , let M be the corresponding automaton 3. compute the prefix p and the period q of S w.r.t. ∼ = M 4. using induction on n , compute the ultimately periodic factorization S ′ consisting of only regular trees 5. compute the regular tree T ′ resulting from S ′ 6. solve Acc ( T ′ ) on automaton M 7. accordingly, return Yes or No to the original problem MTh ( T , ¯ V )

Structural properties (1) Residually regular trees are in general non-regular trees which however exhibit a definite pattern in their structure. 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • We established some structural properties of residually regular trees, such as closure under recursively defined factorizations , iterations , periodical groupings , etc.