Distance desert automata and the star height problem Daniel Kirsten - - PowerPoint PPT Presentation

Talk given at Journ´ ees Montoises (IRISA, Rennes).

Distance desert automata and the star height problem

Daniel Kirsten1

Dresden University of Technology Institute for Algebra

September 1st, 2006

◮ Distance desert automata and star height substitutions.

Habilitationsschrift, Universit¨ at Leipzig, 2006.

◮ Distance desert automata and the star height problem.

R.A.I.R.O.- I.T.A., 29(3):455–509, 2005.

1Supported by the German Research Community.

Definition: An automaton is a tuple A = [Q, E, I, F], where

◮ Q is a finite set,

(states)

◮ E ⊆ Q × Σ × Q,

(transitions)

◮ I ⊆ Q and

(initial states)

◮ F ⊆ Q.

(accepting states) path, L(A), recognizable languages,. . .

Definition: Hashiguchi 1982 A distance automaton is a tuple A = [Q, E, I, F, θ], where

◮ [Q, E, I, F] is an automaton and ◮ θ : E → {, ∠}. ◮ |A| : Σ∗ → N ∪ {∞} ◮ for paths π let ∆(π) := “number of ∠-transitionsp´

eages”

◮ for w ∈ Σ∗ let |A|(w) := min

• ∆(π)
• π ∈ I

w

❀ F

• .

◮ Limitedness

Choffrut 1979 Is A limited, i.e. ∃ d ∈ N, such that |A|(w) ≤ d for every w ∈ L(A)? decidable Hashiguchi 1982, Leung 1987, Simon 1994 PSPACE-hard Leung 1987 PSPACE-complete Leung/Podolskiy 2004

◮ Linear Limitedness

Does |A|(w) ≤ |w|

2 hold for every w ∈ L(A)?

undecidable Krob 1994

◮ Equivalence

|A1| = |A2|? undecidable Krob 1994

Another cost model: Up to now, ∆(∠∠∠) = 3. For σ ∈ {, }∗ let ∆(σ) := ”maximal length of a factor ∗ in σ”. ∆() = 2, ∆() = 1, ∆() = 2, ∆() = 4, ∆() = 5

A Combined Cost Model: For σ ∈ {, , ∠}∗ let ∆(σ) := “maximum of the lengths of factors ∗ of σ and the number of p´ eages in σ.” ∆(∠) = 1, ∆(∠) = 2, ∆(∠∠∠∠∠) = 5,

Definition: Kirsten 2004 A distance desert automaton is a tuple A = [Q, E, I, F, θ], where

◮ [Q, E, I, F] is an automaton and ◮ θ : E → {, , ∠}. ◮ |A| : Σ∗ → N ∪ {∞} ◮ for w ∈ Σ∗ let |A|(w) := min

• ∆(θ(π))
• π ∈ I

w

❀ F

• .

An Example: q1 q2 a b a b a,b |A|(akbk) = k |A|(akbk) = 1, but for k ≥ 0, we have |A|

• (akbk)k

= k Description by matrices: A = ∠ ∠

• B =

∠ ∠

• AB =

∠ ∠

• A♯ =

∠ ∠ ∠

• B♯ =

∠ ∠ ∠

• A♯B♯ =

∠ ∠ ∠ ∠

• A♯B♯♯ =

ω ω ω ω

• corresponds to (akbk)k for growing k

Theorem 1: Kirsten 2004 Let A = [Q, E, I, F, θ] be a distance desert automaton and let T be the set of transformation matrices of letters: The following assertions are equivalent:

• 1. A is unlimited.
• 2. There is some M ∈ T♯ such that I · M · F = ω.
• 3. There is a k-expression r

(w ∈ Σ∗, rs, rk, (r ks)k) with at most |Q| + 1 nestings of k, such that r(k) ∈ L(A) for every k ∈ N and |A|

• r(k)
• grows unbounded for growing k.

Theorem 2: Kirsten 2004

Limitedness of distance desert automata is PSPACE-complete.

The Star Height Problem: Eggan 1963, Hashiguchi 1988 “Indeed, the existing proof, putting all pieces together, takes more than a hundred pages of very heavy combinatorial reasoning.”

• I. Simon, MFCS’88 Proceedings, 1988

“The proof is very difficult to understand and a lot remains to be done to make it a tutorial presentation.”

• D. Perrin, Finite Automata, Handbook of Theor. Comp. Sc., 1990

“Hashiguchi’s solution for arbitrary star height relies on a complicated induction, which makes the proof very difficult to follow.”

J.-´

• E. Pin, Tropical Semirings, Idempotency, 1998

Theorem 3: Kirsten FoSSaCS’04 It is decidable in 22O(n) space whether the language of a non-deterministic automaton with n states is of star height 1. Proof idea: Let L ⊆ Σ∗.

◮ sh(L) = 0

⇐ ⇒ L is finite.

◮ sh(L) ≤ 1

⇐ ⇒ L =

finite

a1K ∗

1 a2K ∗ 2 . . . akK ∗ k

for a1, . . . , ak ∈ Σ and finite K1, . . . , Kk ∈ Σ+. It suffices to decide whether sh(L) ≤ 1. Let η : Σ∗ → M(L) the syntactic homomorphism. We construct a distance desert automaton A.

Q := P(M(L)) ∪ qI, I := {qI}, F := { R | R ⊆ η(L) } For every P, R ⊆ M(L), a ∈ Σ, we insert a p´ eage (P, a, R), if P · η(a) ⊆ R and P = R (similarly for qI). qI P1 P2 P3 P4 . . . . . . . . . a b b a a c b AP1 AP2 AP3 AP4 For every P ⊆ M(L) we insert an automaton AP satisfying L(AP) = { w | P · η(w) ⊆ P }. . . . A is limited ⇐ ⇒ sh(L) ≤ 1.

Theorem 4: Kirsten 2004 Let h ∈ N. It is decidable in 22O(n) space whether the language of a non-deterministic automaton with n states is of star height h. Proof Ideas:

• 1. h-nested distance desert automata

◮ currencies 0, . . . , h, ◮ One can obtain i-coins at transitions

i.

◮ One has to pay an i-coin at transitions ∠i.

• 2. Their limitedness problem is PSPACE-complete.
• 3. One nests the constructions for star height 1.

Open Questions, . . .

◮ the exact complexity of the star height problem ◮ simplifications ◮ decidability of other hierarchies of recognizable languages ◮ equivalence problem for desert automata ◮ applications