[PPT] - Prefix Rewriting and the Pushdown Hierarchy Wolfgang Thomas PowerPoint Presentation

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Prefix Rewriting and the Pushdown Hierarchy

Wolfgang Thomas Francqui Lecture, Mons, April 2013

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Reachability Problem

Wolfgang Thomas

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Overview

1. Prefix Rewriting and the reachability problem
2. Interpretations
3. Unfoldings and Muchnik’s Theorem
4. The pushdown hierarchy

Wolfgang Thomas

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Prefix Rewriting and the Reachability Problem

Wolfgang Thomas

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Rewriting Over Words

Rewriting system: Finite set S of rules u → v Different uses of a rule u → v for the rewrite relation ⊢ Infix rewriting: xuy ⊢ xvy Post’s canonical systems: ux ⊢ xv Prefix rewriting (B¨ uchi’s regular canonical systems):

ux ⊢ vx

Fundamental results: Infix rewriting systems and Post’s canonical systems allow to simulate Turing machines. B¨ uchi 1965: Prefix rewriting systems generate regular sets from regular sets of “axioms”, and the derivability relation is decidable.

Wolfgang Thomas

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The Setting of Pushdown Automata

A pushdown automaton has the form P = (P, Σ, Γ, p0, Z0, ∆) Configurations are words from PΓ∗ A transition induces a move from pγw to quw Write pγw ⊢ quw So pushdown automata are a special from of prefix rewriting systems. Consequence of B¨ uchi’s Theorem: The reachable configurations of a pushdown automaton form a regular set.

Wolfgang Thomas

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The Reachability Sets

Given a pushdown automaton P = (P, Σ, Γ, p0, Z0, ∆) and

T ⊆ PΓ∗ pre∗(T) := {pv ∈ PΓ∗ | ∃qw ∈ T : pv ⊢∗ qw}

Analogously post∗(T). We may suppress Σ and q0, Z0 and obtain a “pusdown system

P = (Q, Γ, ∆) with transitions of the form (p, γ, v, q).

Given a pushdown system P = (P, Γ, ∆) and a finite automaton recognizing a set T ⊆ PΓ∗, one can compute a finite automaton recognizing pre∗(T), similarly for post∗(T). Deciding p1w1 ⊢∗ p2w2: Set T = {p2w2} and check whether the automaton recognizing pre∗(T) accepts p1w1.

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Example

P = (P, Γ, ∆) with P = {p0, p1, p2}, Γ = {a, b, c},

∆ =

{(p0a → p1ba), (p1b → p2ca), (p2c → p0b), (p0b → p0)}

T = {p0aa}. P-automaton for T:

A:

p0 s1 p1 p2 s2 a a

Wolfgang Thomas

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Saturation Algorithm: Idea

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Saturation Algorithm

Input: P-automaton A, pushdown system P = (P, Γ, ∆)

A0 := A, i := 0

REPEAT: IF pa → p′v ∈ ∆ and Ai : p′

v

− → q THEN

add (p, a, q) to Ai and obtain Ai+1

i := i + 1 UNTIL no transition can be added

A := Ai

Output: A′

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Example: Result

A′:

p0 s1 p1 p2 s2 a a b c b a b

So for T = {p0aa}:

pre∗(T) = p0b∗(a + aa) + p1b + p1ba + p2cb∗(a + aa)

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Alternative: Work in the Tree of Words

Consider a prefix rewriting system over {0, 1}. Convert prefix rewriting to suffix rewriting. Then a rewrite step is definable in S2S. Example: Rule R : 11 → 0 leads from a word w11 to w0 Defining formula ϕR(z, z′): ∃x(z = x11 ∧ z′ = x0) For a system S let ϕS(z, z′) :=

R∈S ϕR(z, z′)

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Preservation of Regularity

Let L ⊆ {0, 1}∗ be regular. There is an S2S-formula ϕL(x) defining L in the tree T2 We can write L ⊆ Y for ∀y(ϕL(y) → Y(y)) Then x ∈ post∗(L) iff

∀Y[(L ⊂ Y and ∀z, z′(Y(z) ∧ϕS(z, z′)) → Y(z′)) → Y(x)]

The formula ψ(X) :

∀x(X(x) ↔ “x ∈ post∗(L)′′) is satisfied

by a unique set. By Rabin’s Basis Theorem it must be regular.

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Interpretations

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A First Example

Show Rabin’s Tree Theorem for T3 = ({0, 1, 2}∗, S3

0, S3 1, S3 2).

Idea: Obtain a copy of T3 in T2: Consider T2-vertices in T = (10 + 110 + 1110)∗.

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Interpretation: Details

The element i1 . . . im of T3 is coded by

1i1+10 . . . 1im+10 in T2.

Define the set of codes by

ϕ(x): “x is in the closure of ε under 10-, 110-, and

1110-successors” Define the 0-th, 1-st 2-nd successors by

ψ0(x, y), ψ1(x, y), ψ2(x, y)

The structure (ϕT2, (ψT2

i )i=0,1,2) restricted to ϕT2 is isomorphic

to T3.

Wolfgang Thomas

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Interpretations in General

An MSO-interpretation of a structure A = (A, RA, . . .) in a structure B is given by a “domain formula” ϕ(x) for each relation RA of A, say of arity m, an MSO-formula

ψ(x1, . . . , xm)

such that A is isomorphic to (ϕB, ψB, . . .) Then there is a transformation OF MSO-sentenceS χ (in the signature of A) to sentences χ′ (in the signature of B) such that

A | = χ iff B | = χ′.

Consequence: If A is MSO-interpretable in B and the MSO-theory of B is decidable, then so is the MSO-theory of A.

Wolfgang Thomas

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Pushdown Graphs

Consider A for language L = {anbn | n ≥ 0}:

A = ({q0, q1}, {a, b}, {Z0, Z}, q0, Z0, ∆) with

∆ = (q0, Z0, a, q0, ZZ0),

(q0, Z, a, q0, ZZ), (q0, Z, b, q1, ε), (q1, Z, b, q1, ε)

Initial and final configuration: q0Z0

The associated pushdown graph (of reachable configurations

nly) is:

q1Z0 q1ZZ0 q1ZZZ0 . . . q0Z0 q0ZZ0 q0ZZZ0 . . . b b b b b b a a a

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Interpretation: Second Example

A pushdown graph is MSO-interpretable in T2 Given pushdown automaton A with stack alphabet {1, . . . , k} and states q1, . . . , qm. Let GA = (VA, EA) be the corresponding PD graph.

n := max{k, m}

Find an MSO-interpretation of GA in Tn. Represent configuration (qj, i1 . . . ir) by the vertex ir . . . i1j.

A-steps lead to local moves in Tn.

E.g. a push step from vertex ir . . . i1j to ir . . . i1i0j′. These edges are easily definable in MSO. Hence: The MSO-theory of a PD graph is decidable.

Wolfgang Thomas

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Prefix-Recognizable Graphs

Instead of rules u → v we have rules U → Y wuth regular sets

U, V.

Instead of describing a move from one word wu0 to one wv0 describe all admissible moves from a word wu to a word wv for a rule U → V with u ∈ U, v ∈ V. This can be done by describing successful runs of the automata AU, AV on the path segments from w to wu and from w to wv. A graph is MSO-interpretable in T2 iff its is prefix-recognizable.

Wolfgang Thomas

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Unfolding and Muchnik’s Theorem

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Unfoldings

Given a graph (V, (Ea)a∈Σ, (Pb)b∈Σ′) the unfolding of G from a given vertex v0 is the following tree

TG(v0) = (V′, (E′

a)a∈Σ, (P′ b)b∈Σ′):

V′ consists of the vertices v0a1v1 . . . arvr with

(vi−1, vi) ∈ Eai,

E′

a contains the pairs (v0a1v1 . . . arvr, v0a1v1 . . . arvrav)

with (vr, v) ∈ Ea,

P′

b the vertices v0a1v1 . . . arvr with vr ∈ Pb.

Wolfgang Thomas

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Examples

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Unfolding Preserves Decidability

Theorem (Muchnik, Courcelle/Walukiewicz) If the MSO-theory of G is decidable and v0 is an MSO-definable vertex of G, then the MSO-theory of TG(v0) is decidable. We sketch the proof for pushdown graphs. Their unfoldings are the “algebraic trees”.

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Proof Architecture

Given an unfolding T of a pushdown graph G.

T is finitely branching, with labels say in Σ inherited from G.

For each MSO-formula ϕ(X1, . . . , Xn) find a parity tree automaton Aϕ such that

Aϕ accepts T(P1, . . . , Pn) iff T[P1, . . . , Pn) | = ϕ(X1, . . . , Xn)

The construction of the Aϕ follows precisely the pattern of Rabin’s equivalence theorem. Essential: In the complementation step we use the finite

ut-degree of G.

The general case is more involved.

Wolfgang Thomas

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Muchnik’s Theorem: Continued

Result: For a sentence ϕ we obtain a tree automaton Aϕ, say with state set Q and transition set ∆, with

Aϕ accepts T iff T | = ϕ

The left-hand side says: Automaton has a positional winning strategy in the associated game ΓA,T If G = (V, E, v0) for simplicity, the game graph consists of vertices in V × Q (for Automaton) in V × ∆ (for Pathfinder)

Wolfgang Thomas

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Muchnik’s Theorem Finished

The game ΓA,T is played on a graph

G′ = (V × {1, . . . , k}, E′, (v0, 1))

We use the following fact (shown next Friday): The set of vertices v from where Player Automaton wins in the parity game over G′ = (V′, E′, v′) is MSO-definable by a formula χ(x). Translation Theorem: For each sentence ϕ we can build a sentence ϕ+ such that

G′ |

= ϕ iff G | = ϕ+

Since the MSO theory of G is decidable, we can decide the left-hand side.

Wolfgang Thomas

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Final Step

How to infer decidability of MTh(G × {1, 2}) from decidability

f MTh(G)?

We do not address the definition of the edge relation but just give the idea: Simulate a set quantifier over G × {1, 2} by two set quantifiers

ver G.

Wolfgang Thomas

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Pushdown Hierarchy

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Caucal’s Proposal

We have now two processes which preserve decidability of MSO-theory: interpretation (transforming a tree into a graph) unfolding (transforming a graph into a tree) Let us apply them in alternation! We obtain the Caucal hierarchy or pushdown hierarchy.

Wolfgang Thomas

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Definition

T0 = the class of finite trees Gn = the class of graphs which are MSO-interpretable in a

tree of Tn

Tn+1 = the class of unfoldings of graphs in Gn

Each structure in the pushdown hierarchy has a decidable MSO-theory. Nontrivial fact: The sequence G0, G1, G2, . . . is strictly increasing.

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The First Levels

G0 is the class of finite graphs. T1 contains the regular trees. G1 contains the prefix-recognizable graphs.

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A Finite Graph, a Regular Tree, a PD Graph

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Unfolding Again

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Wolfgang Thomas

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Interpretation of Bottom Line

The sequence of leaves defines a copy of the successor structure of the natural numbers. Domain expression: b + a∗c(d + e)∗ f Successor relation:

bacf+ fe∗cacd∗ f+ fe∗ded∗ f

Predicate “power of 2”: b + a∗cd∗ f Result: (N, Succ, Pow2) is a structure in the Caucal hierarchy.

Wolfgang Thomas

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Factorial Predicate

(N, Succ, Fac)

We start as follows:

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· · ·
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c

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Continuation: Unfolding and Interpretation

· · ·

b b b b c c c c c c

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Another Unfolding

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• • • • •

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a

a a a b b b b c c c c c c c c c

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Scope of Hierarchy?

The pushdown hierarchy is a very rich class of structures all of which have a decidable MSO-theory. Open questions: Understand which structures belong to the hierarchy Compute the smallest level on which a strcuture occurs

Wolfgang Thomas