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 Kirsten 1 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. 1 Supported 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,. . .

Hashiguchi 1982 Definition: 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 w � � π ∈ I � � ∆( π ) ❀ 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 |A 1 | = |A 2 | ? 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 w � � π ∈ I � � ∆( θ ( π )) ❀ F .

a a An Example: a , b q 1 q 2 b b |A| ( a k b k ) = k |A| ( a k b k ) = 1 , but for k ≥ 0, we have � ( a k b k ) k � |A| = k Description by matrices: � � � � � � � � � ∠ ∠ ∠ A = B = AB = ∠ � ∠ � ∠ � � ∠ � � � ∠ � � � ∠ ∠ ∠ A ♯ = B ♯ = A ♯ B ♯ = ∠ � ∠ ∠ ∠ ∠ � ω � ω A ♯ B ♯ � ♯ = corresponds to ( a k b k ) k for growing k � ω ω

Kirsten 2004 Theorem 1: 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 . ( w ∈ Σ ∗ , rs , r k , ( r k s ) k ) 3. There is a k -expression r 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.

Eggan 1963, Hashiguchi 1988 The Star Height Problem: “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

Kirsten FoSSaCS’04 Theorem 3: It is decidable in 2 2 O ( 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. a 1 K ∗ 1 a 2 K ∗ 2 . . . a k K ∗ ◮ sh( L ) ≤ 1 L = � ⇐ ⇒ k finite for a 1 , . . . , a k ∈ Σ and finite K 1 , . . . , K k ∈ 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 )) ∪ q I , I := { q I } , 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 q I ) . A P 1 A P 3 . . . b P 1 P 3 a . . . q I a a b b P 2 P 4 . . . c A P 2 A P 4 For every P ⊆ M ( L ) we insert an automaton A P satisfying L ( A P ) = { w | P · η ( w ) ⊆ P } . . . . A is limited ⇐ ⇒ sh( L ) ≤ 1. �

Kirsten 2004 Theorem 4: Let h ∈ N . It is decidable in 2 2 O ( 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