Definable sets in tree-like structures Graud Snizergues LaBRI, - - PowerPoint PPT Presentation
Definable sets in tree-like structures Definable sets in tree-like structures Graud Snizergues LaBRI, Bordeaux University, Thursday June 29th 2017 Dedicated to Paul Schupp on his 80th birthday. 1 / 45 Definable sets in tree-like
Definable sets in tree-like structures
Géraud Sénizergues
LaBRI, Bordeaux University,
Thursday June 29th 2017 Dedicated to Paul Schupp on his 80th birthday.
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Definable sets in tree-like structures Introduction
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Definable sets in tree-like structures Introduction
Three fundamental works by D.E. Muller and P.E.Schupp : [Groups, the theory of ends, and context-free languages. Journal of Computer and System Sciences, 1983] [The theory of ends, pushdown automata, and second-order
[Alternating automata on infinite trees. Theoretical Computer Science, 1987]
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Definable sets in tree-like structures Introduction
Decidability : Given a formula Φ, can we decide whether M | = Φ Definability : A subset R is definable iff there exists a formula Φ(X) such that M | = ∃!XΦ(X) and M, R | = Φ(X). Selection : Given a formula Φ(X), a selector (for the formula and the structure M) is a formula ˆ Φ(X) such that M | = (∃X · Φ(X)) → (∃X · ˆ Φ(X)) M | = ∀X · (ˆ Φ(X) → Φ(X)) M | = ∀X · ∀Y · (ˆ Φ(X) ∧ ˆ Φ(Y )) → (X = Y ).
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Definable sets in tree-like structures Introduction
1 Introduction 2 MSO logics 3 Context-free graphs 4 Stupp’s expansion 5 Muchnik’s expansion 6 Decidability of definability
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Definable sets in tree-like structures MSO logics
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Definable sets in tree-like structures MSO logics
Let Sig = {r1, . . . , rn} be a signature containing relational symbols
Let Var = {x, y, z, . . . , X, Y , Z . . .} be a set of variables, where x, y, . . . denote first order variables and X, Y , . . . second order variables. The set of MSO-formulas over Sig, Var is the smallest set such that : for every x, x1, . . . , xρ, X, Y , X1 . . . Xτ in Var and MSO formula Φ, Ψ x ∈ X, Y ⊆ X ¬Φ, Φ ∨ Ψ, Φ ∧ Ψ, Φ → Ψ, ∃x.Φ, ∃X.Φ, ∀x.Φ, ∀X.Φ, are MSO-formulas.
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Definable sets in tree-like structures MSO logics
Let M = DM, r1, . . . , rn be a structure over the signature Sig, and ν : Var → DM ∪ P(DM) a valuation The validity of a MSO-formula in the structure M with valuation ν is then defined by induction on the structure of the formula. Notation : M, ν | = Φ.
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Definable sets in tree-like structures MSO logics
Automata :
Right-action of the monoid A∗ over a set C (set of configurations) : (c, u) → c ⊙ u. Initial configuration c0 and set of final configurations Cf . Right-equivalence over C : c ≡r d ⇔ {u ∈ A∗ | c ⊙ u ∈ Cf } = {u ∈ A∗ | d ⊙ u ∈ Cf }
“small changes” on the right-end of the configuration.
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Definable sets in tree-like structures MSO logics
Definition from [Muller-Schupp 87] An alternating automata over binary trees, labelled on alphabet Σ is a tuple : Q, Σ, q0, δ, Ω where δ : Q × Σ → B+(Q × {ℓ, r}). The tree t : {ℓ, r}∗ → Σ is recognized by the automaton iff player J0 is winning the following game :
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Definable sets in tree-like structures MSO logics
V0 := Q × {ℓ, r}∗ V1 := conjunctive monomials over Q × {ℓ, r} Edges :
by t and δ
monomial.
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Definable sets in tree-like structures MSO logics
The automaton is non-deterministic when every position of J1 accessible in the game has the form (p, ℓ) ∧ (q, r). Theorem (Muller-Schupp 1987) Every alternating finite tree automaton can be simulated by some non-deterministic finite tree-automaton. Key-idea : use Muller deterministic automata over branches. Extension to trees with infinite arity : [Walukiewicz 1996].
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Definable sets in tree-like structures MSO logics
Games :
Arena : a bipartite graph (V0 ∪ V1, E, Ω) where Ω : V0 ∪ V1 → [0, n] is the priority map. Play : v0, v1, . . . , vm, . . . which is a path in the arena ; either it is infinite or its last vertex is a dead-end. The winner is J0 iff max{r | vi = r i.o. } ≡ 0 (mod 2)
J1 is the winner.
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Definable sets in tree-like structures MSO logics
Strategy for Jj : a map Sj : V ∗Vj → V1−j such that Sj ⊆ E and dom(Sj) = dom(E) ∩ Vj It is said positional if Sj(u · v) depends on v only. Theorem (Emerson-Jutla 91) Let G be a parity game. 1- Either player 0 or player 1 has a winning strategy. 2- The winner has a positional winning strategy. Other games :
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Definable sets in tree-like structures MSO logics
We call MSO-interpretation of the structure M into the structure M′ every injective map ϕ : DM → DM′ such that, 1- There exists a formula Φ′(X) ∈ L′, with one free-variable X, which is second-order, fulfilling that, for every subset XM′ ⊆ DM′ XM′ = ϕ(DM) ⇔ M′ | = Φ′(XM′) 2- For every i ∈ [1, n], there exists a formula Φ′
i(x1, . . . , xρi ),
fulfilling that, for every valuation ν (M, ν) | = ri(x1, . . . , xρi ) ⇔ (M′, ϕ ◦ ν) | = Φ′
i(x1, . . . , xρi ).
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Definable sets in tree-like structures MSO logics
Theorem Suppose that there exists a MSO-interpretation of the structure M into the structure M′. Then, there exists a computable map from L to L′ : Φ → Φ′ such that M | = Φ iff M′ | = Φ′. In particular, if M′ has a decidable MSO-theory, then M has a decidable MSO-theory too.
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Definable sets in tree-like structures MSO logics
Theorem (Rabin 1969) Let S be the signature Sa, Sb. The MSO-theory of the structure {a, b}∗, Sa, Sb is decidable. Theorem (Rabin 1969) A subset R ⊂ {a, b}∗ is MSO-definable iff it is recognizable by a deterministic f.automaton. Theorem (Rabin 1969) The MSO-theory of the structure {a, b}∗, Sa, Sb has the selection property.
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Definable sets in tree-like structures MSO logics
Example 1 : [Rabinovitch 05], [Lifsches-Shellah 98].
Example 2 : A structure definable inside an algebraic tree (computation-tree of some pushdown automaton) :
graph)
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Definable sets in tree-like structures Context-free graphs
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Definable sets in tree-like structures Context-free graphs
Let Γ be a graph, labelled over an alphabet X. Given some vertex v ∈ VΓ, and some radius n ∈ N, we call (v, n)-end of Γ (relative to the ball B(v, n)) any connected component of Γ − B(v, n). Definition A graph Γ is said context-free iff it is connected and has only finitely many isomorphism classes of ends.
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Definable sets in tree-like structures Context-free graphs
The canonical automaton (Muller-Schupp) :
the n − end with number i
Theorem (Muller-Schupp 85) If Γ is c.f. then it is isomorphic with the computation-graph of its canonical automaton. Idea : the accessible configurations of the canonical automaton correspond bijectively to the vertices of Γ.
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Definable sets in tree-like structures Context-free graphs
Theorem (Muller-Schupp 1985) For every c.f. graph Γ, the MSO-theory of Γ is decidable Idea : The structure Γ is MSO-interpretable in Q × {zi,j}∗.
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Definable sets in tree-like structures Context-free graphs
A subset R of VΓ is said recognizable iff its set of “coordinates” in the canonical automaton is recognized by some f.automaton Theorem A subset R of VΓ is definable iff it is recognizable Idea : the accessible configurations of the canonical automaton correspond bijectively to the vertices of Γ.
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Definable sets in tree-like structures Context-free graphs
MSO Formula defining the automorphisms. Let us choose a vertex v0 : it defines ends relative to (v0, n) and a canonical automaton. θ(v, V0,0,0, . . . , Vi,j,ℓ, . . . , D0, . . . , Dm, . . . , Dc−1) Property :
ν(v) = v0, ν(Vi,j,ℓ) = {qℓz0,j0 · · · zi,j}, ν(Dm) = {v | d(v0, v) ≡ m (mod c)}
ν(v) = h(v0), ν(Vi,j,ℓ) = h({qℓz0,j0 · · · zi,j}), ν(Dm) = {v | d(v0, v) ≡ m (mod c)} for some automorphism h of Γ.
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Definable sets in tree-like structures Context-free graphs
h → (h(v0), . . . , h({qℓz0,j0 · · · zi,j}), . . . , Dm) is a bijection.
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Definable sets in tree-like structures Context-free graphs
Selection property for c.f. graphs. Theorem For every context-free graph M, whith trivial automorphism-group, the MSO-theory of M has the selection-property. Idea : given a formula Φ(X)
Ψ(Z) over Q × {zi,j}∗
Ψ′(Z) obtained from ˆ Ψ(Z) by : replacing Szi,j(x, y) by
x ∈ Dm ∧ y ∈ Dm+1 ∧ (
y ∈ Vi,j,ℓ) replacing qℓ(x) by
i,j x ∈ Vi,j,ℓ
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Definable sets in tree-like structures Context-free graphs
Φ(X) by : ∃v, . . . , Vi,j,ℓ, . . . , D0, . . . , Dc−1 θ(v, . . . , Vi,j,ℓ, . . . , D0, . . . , Dc−1) ∧ ˆ Ψ′(Z). Extension : when the automorphism group is arbitrary, the set of models of ˆ Φ(X) is one orbit of Aut(Γ) (acting on P(Γ)).
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Definable sets in tree-like structures Stupp’s expansion
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Definable sets in tree-like structures Stupp’s expansion
Definition (S-expansion, Stupp 1975) Let Sig = (r1, . . .) be a signature containing only relational symbols and M = A, r1, . . . a structure over Sig. The S-extended signature is Sig∗ = Sig ∪ {son} The S-expansion is M∗ = A∗, son, r1∗, . . . where A∗ is the set of all finite sequences of elements of A and the relations are defined by : son = {(u, ud) : u ∈ A∗, d ∈ A} r ∗ = {(ud1, . . . , udk) : u ∈ A∗, (d1, . . . , dk) ∈ rM}, ( for all r ∈ Sig, of arity k).
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Definable sets in tree-like structures Stupp’s expansion
Theorem (Stupp 1975) If the MSO theory of M is decidable, then the MSO theory of M∗ is decidable. Idea of proof :
(extends [Rabin 71]) : transitions are defined by logical formulas : ψp,σ(. . . , Xp, . . . , Xq, . . .) expressing that a run, with state p at vertex u, with label t(u) = σ, has a correct choice of states . . . , p, . . . , q, . . . for its sons
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Definable sets in tree-like structures Stupp’s expansion
Let M = A, r1, . . .. Definition A logical finite automaton over A∗ is a tuple A = Q, q0, Qf , δ where Q is a finite set, q0 ∈ Q, Qf ⊆ Q and δ = (Φp,q)(p,q)∈Q×Q is a family of MSO-formulas with one free individual variable.
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Definable sets in tree-like structures Stupp’s expansion
A computation of A is a sequence q0, a1, q1, . . . , an, qn, . . . , aℓ, qℓ such that, for every n ∈ [0, ℓ − 1], M | = Φqn,qn+1(an+1). The language recognized by A is the set of finite sequences a1 · · · an · · · aℓ that admit a computation ending in qℓ ∈ Qf .
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Definable sets in tree-like structures Stupp’s expansion
Theorem (J. Pablo, Master thesis 2016) Let R ⊆ A. It is S-definable iff it is recognized by some finite logical automaton over M. Idea of proof : 1- Suppose R ⊆ A is S-definable.
general position of J0 : (q, u) ∈ Q × A∗ general position of J1 : partition (Sq)q∈Q of u · A.
general position of J0 : q ∈ Q general position of J1 : partition (Sq)q∈Q of A.
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Definable sets in tree-like structures Stupp’s expansion
there exists some partition (Sq)q∈Q of A that can be played from p by J0, and is winning for J0 and x ∈ Sq. 2- Converse : express the language as a least fixpoint.
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Definable sets in tree-like structures Stupp’s expansion
Theorem (J. Pablo and G.S today) If the MSO theory of M has the selection property, then the MSO theory of M∗ has the selection property, Ideas : Game over M. Use the existence of a winning positional strategy for every parity game.
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Definable sets in tree-like structures Muchnik’s expansion
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Definable sets in tree-like structures Muchnik’s expansion
Definition (M-expansion, Muchnik-Semenov 1984) Let Sig = (r1, . . .) be a signature containing only relational symbols and M = A, r1, . . . a structure over Sig. The M-extended signature is Sig∗ = Sig ∪ {son, clone} The M-expansion is M∗ = A∗, son, clone, r ∗
1 , . . .
where the relations are defined by : son = {(u, dd) : u ∈ A∗, d ∈ A} clone = {udd : u ∈ A∗, d ∈ A} r ∗ = {(ud1, . . . , udk) : u ∈ A∗, (d1, . . . , dk) ∈ rM}, ( for all r ∈ Sig, of arity k).
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Definable sets in tree-like structures Muchnik’s expansion
Theorem (Muchnik-Semenov 84, Walukiewicz 96) For every MSO formula Φ over the signature Sig∗ one can effectively find a MSO formula Φ over the signature Sig such that, for every structure M : M | = Φ iff M∗ | = Φ Proof :
ψp,σ(x, . . . , Xp, . . . , Xq, . . .) expressing that a run, with state p at vertex ux, with label t(ux) = σ, has a correct choice of states . . . , p, . . . , q, . . . for its sons
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Definable sets in tree-like structures Muchnik’s expansion
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Definable sets in tree-like structures Muchnik’s expansion
Definition A logical finite di-automaton over A∗ is a tuple A = Q, a0, q0, Qf , δ where Q is a finie set, q0 ∈ Q, Qf ⊆ Q and δ = (Φp,q)(p,q)∈Q×Q is a family of MSO-formulas with two free individual variables. A computation of A is a sequence a0, q0, a1, q1, . . . , an, qn, . . . , aℓ, qℓ such that, for every n ∈ [0, ℓ − 1], M | = Φqn,qn+1(an, an+1).
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Definable sets in tree-like structures Muchnik’s expansion
Theorem (G.S. today) Let R ⊆ A be M-definable. It is M-definable iff it is recognized by some finite logical di-automaton over M. Adapt the case of S-definable subsets to the transitions of the M-tree-automata.
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Definable sets in tree-like structures Decidability of definability
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Definable sets in tree-like structures Decidability of definability
Input : A M-definable subset R of M∗ ? Question : Is R also S-definable ?
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Definable sets in tree-like structures Decidability of definability
A “logical di-automaton” can be turned into a deterministic automaton AQ × A → Q. Proposition Let R ⊆ A be M-definable. It is S-definable iff the minimal automaton of R is finite. Idea : If the minimal automaton is finite then its transitions can be expressed within MSO over M ; hence it is a logical finite automaton.
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Definable sets in tree-like structures Decidability of definability
Theorem (G.S. today) Let us assume that M has a decidable MSO and that the finiteness property for a subset of M is expressible in MSO. Given a M-definable subset of M one can decide whether this subset is S-definable. Ideas :
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