. On the Size of the Universal Automaton of a Regular Language - - PowerPoint PPT Presentation

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on the size of the universal automaton of a regular

. On the Size of the Universal Automaton of a Regular Language - - PowerPoint PPT Presentation

Sep 17, 2023 423 likes •798 views

. On the Size of the Universal Automaton of a Regular Language Sylvain Lombardy IGM - Universit e de Marne-la-Vall ee STACS07 - February 22, 2007 p. 1/17 Motivation a b Minimizing deterministic automata a Merging

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. On the Size of the Universal Automaton of a Regular Language

Sylvain Lombardy IGM - Universit´ e de Marne-la-Vall´ ee

STACS’07 - February 22, 2007 – p. 1/17

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Motivation

Minimizing deterministic automata − → Merging states with the same future a b a a b b (Nerode equivalence) − → Hopcroft algorithm O(n log n)

STACS’07 - February 22, 2007 – p. 2/17

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Motivation

Minimizing deterministic automata − → Merging states with the same future a b a a b b (Nerode equivalence) − → Hopcroft algorithm O(n log n)

STACS’07 - February 22, 2007 – p. 2/17

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Motivation

Minimizing deterministic automata − → Merging states with the same future b a b a (Nerode equivalence) − → Hopcroft algorithm O(n log n)

STACS’07 - February 22, 2007 – p. 2/17

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Motivation

Minimizing deterministic automata − → Merging states with the same future b a b a (Nerode equivalence) − → Hopcroft algorithm O(n log n) Minimizing non deterministic automata − → Much harder : PSPACE-complete (Jiang - Ravikumar, 93)

STACS’07 - February 22, 2007 – p. 2/17

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Motivation

Minimizing deterministic automata − → Merging states with the same future b a b a (Nerode equivalence) − → Hopcroft algorithm O(n log n) Minimizing non deterministic automata − → Much harder : PSPACE-complete (Jiang - Ravikumar, 93) − → However merging states may reduce the size of NFAs

STACS’07 - February 22, 2007 – p. 2/17

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Motivation

Problem 1 : How to guarantee that merging does not modify the language ? − → "Equivalences" (Ilie - Yu, 02) "Inequalities" (Champarnaud - Coulon, 04)

STACS’07 - February 22, 2007 – p. 3/17

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Motivation

Problem 1 : How to guarantee that merging does not modify the language ? − → "Equivalences" (Ilie - Yu, 02) "Inequalities" (Champarnaud - Coulon, 04) Problem 2 : What is the difference between this reduce NFA and any minimal NFA for the language ? − → L fixed, ∃EL ∀A s.t. L(A) = L A > EL = ⇒ A has merging states (Câmpeanu - Sântean - Yu, 04)

STACS’07 - February 22, 2007 – p. 3/17

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Motivation

Problem 1 : How to guarantee that merging does not modify the language ? − → "Equivalences" (Ilie - Yu, 02) "Inequalities" (Champarnaud - Coulon, 04) Problem 2 : What is the difference between this reduce NFA and any minimal NFA for the language ? − → L fixed, ∃EL ∀A s.t. L(A) = L A > EL = ⇒ A has merging states (Câmpeanu - Sântean - Yu, 04) − → EL 2k (k : size of the minimal DFA) (Grunsky - Kurganskyy - Potapov, 06)

STACS’07 - February 22, 2007 – p. 3/17

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Motivation

Problem 1 : How to guarantee that merging does not modify the language ? − → "Equivalences" (Ilie - Yu, 02) "Inequalities" (Champarnaud - Coulon, 04) Problem 2 : What is the difference between this reduce NFA and any minimal NFA for the language ? − → L fixed, ∃EL ∀A s.t. L(A) = L A > EL = ⇒ A has merging states (Câmpeanu - Sântean - Yu, 04) − → EL 2k (k : size of the minimal DFA) (Grunsky - Kurganskyy - Potapov, 06) − → EL D(n) (n : size of a minimal NFA)

STACS’07 - February 22, 2007 – p. 3/17

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Factorizations

A accepts L p state of A Past(p) = {w |→ i

w

− → p} Fut(p) = {w | p

w

− → t →} − → Past(p).Fut(p) ⊆ L The universal automaton is an automaton s.t. the pairs (Past(p), Fut(p)) are maximal w.r.t. L − → Factorizations of the language (Conway,71) Example : L = a+b+ : (a+b+, b∗) (a+b∗, b+) (A∗, ∅) (a∗, a+b+) (a+, a∗b+) (∅, A∗)

STACS’07 - February 22, 2007 – p. 4/17

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Universal automaton of L

a+b+, b∗ a+b∗, b+ a+, a∗b+ a∗, a+b+ A∗, ∅ ∅, A∗

a, b a, b a, b a, b a, b a, b a, b a, b a, b a, b a, b

a a a a a a, b b b b b b SL - Sakarovitch, 2000

STACS’07 - February 22, 2007 – p. 5/17

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Universal automaton of L

a+b+, b∗ a+b∗, b+ a+, a∗b+ a∗, a+b+ A∗, ∅ ∅, A∗

a, b a, b a, b a, b a, b a, b a, b a, b a, b a, b a, b

a a a a a a, b b b b b b SL - Sakarovitch, 2000

STACS’07 - February 22, 2007 – p. 5/17

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Universal automaton and reduction

Proposition : The universal automaton is reduced.

STACS’07 - February 22, 2007 – p. 6/17

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Universal automaton and reduction

Proposition : The universal automaton is reduced. Proposition : If L(A) = L, A can be reduced into a subautomaton of UL.

STACS’07 - February 22, 2007 – p. 6/17

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Universal automaton and reduction

Proposition : The universal automaton is reduced. Proposition : If L(A) = L, A can be reduced into a subautomaton of UL. A − → UL p − →

Xp = {u ∈ A∗ | uFut(p) ⊆ L}

Yp = {v ∈ A∗ | Xpv ⊆ L}

STACS’07 - February 22, 2007 – p. 6/17

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Construction of the universal automaton

The co-determinized automaton of A is obtained from A by a “backward” determinization. Example : p q r a b a b p, q q, r r a b a b

STACS’07 - February 22, 2007 – p. 7/17

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Construction of the universal automaton

Theorem : (ICALP 2002) The set of states of the universal automaton of L is

btained from the minimal automaton A of L by computing the intersection of

states of the co-determinized automaton of A p, q, r p, q q, r r q ∅ p q r a b a b p, q q, r r a b a b

STACS’07 - February 22, 2007 – p. 8/17

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Construction of the universal automaton

r ↔ a+b+ q, r ↔ a+b∗ q ↔ a+ p, q ↔ a∗

a a a a a a, b b b b b b

p q r

a b a b

p, q q, r r

a b a b

STACS’07 - February 22, 2007 – p. 9/17

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Bound w.r.t minimal DFA

Proposition : For every n, there exists a deterministic automaton with n states that recognizes a language for which the universal automaton has 2n states.

p q b r b a, b a a

STACS’07 - February 22, 2007 – p. 10/17

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Bound w.r.t minimal DFA

Proposition : For every n, there exists a deterministic automaton with n states that recognizes a language for which the universal automaton has 2n states.

p, q, r a, b q, r p, q b p, r p q r b a a a a a a b b b

STACS’07 - February 22, 2007 – p. 10/17

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Bound w.r.t minimal DFA

Proposition : For every n, there exists a deterministic automaton with n states that recognizes a language for which the universal automaton has 2n states.

p, q, r a q, r b p, q p, r p q b r b ∅ a, b a a a b b b a, b a a

STACS’07 - February 22, 2007 – p. 10/17

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Unary alphabet

p q r a a a

STACS’07 - February 22, 2007 – p. 11/17

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Unary alphabet

p, q q, r p, r a a a

STACS’07 - February 22, 2007 – p. 11/17

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Unary alphabet

p, q, r a q, r p, q p, r p q r ∅ a a a a a a a

STACS’07 - February 22, 2007 – p. 11/17

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From a NFA : unary alphabet

Construction of the universal automaton : first determinization (at most k = G(n) states) then construction of the universal automaton (2k states). − → Upper bound : 2G(n) G(n) = max{lcm(n1, . . . , nr) |

ni = n}

1 n1 − 1 a a

. . .

1 nr − 1 a a

− →

1 n − 1 a a

− → The upper bound is tight.

STACS’07 - February 22, 2007 – p. 12/17

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From a NFA

Construction of the universal automaton : first determinization (at most k = 2n − 1 states) then construction of the universal automaton (2k states). − → Upper bound : 22n−1

p q r a a b b

STACS’07 - February 22, 2007 – p. 13/17

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From a NFA

Construction of the universal automaton : first determinization (at most k = 2n − 1 states) then construction of the universal automaton (2k states). − → Upper bound : 22n−1

p q r pq pr a b a ba a b b p, pr, pq r, pr q, pq p, r, pq, pr p, q, pq, pr a b a b a a b b

STACS’07 - February 22, 2007 – p. 13/17

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Characterization of the states

Proposition : D determinized of A, C co-determinized of D. Let P ⊂ R be two states of D. If P belongs to a state X of C, so does R. − → States of C are upperset. Lemma : The uppersets are closed under intersection. Theorem : The states of the universal automaton are uppersets. Theorem : The universal automaton of a language recognized by a NFA with n states has at most D(n) states (n-th Dedekind number).

STACS’07 - February 22, 2007 – p. 14/17

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D(n) ?

n D(n) 2 1 3 2 6 3 20 4 168 5 7581 6 7828354 7 2414682040998 8 56130437228687557907788

1 25 20 0.2 15 10 5 0.8 0.6 0.4

n − → log2 D(n) 2n

STACS’07 - February 22, 2007 – p. 15/17

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Lower bound

1 2 b b a a a

STACS’07 - February 22, 2007 – p. 16/17

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Lower bound

012 12 02 01 2 1 ∅ a b b a a a b b a a a b b b a, b

STACS’07 - February 22, 2007 – p. 16/17

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Lower bound

∅ 1 2 12 01 02 012 a b b a a a b b a a a b b b a, b

STACS’07 - February 22, 2007 – p. 16/17

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Lower bound

0, 1 1, 2 0, 2 1 2 01 12 02

0, 12 1, 02 2, 01 01, 02, 12 012

02, 12 01, 02 01, 12 0, 1, 2

∅ a a a a a a a a a a a a a a a a b b b b a a a, b a, b

D(3) = 20

STACS’07 - February 22, 2007 – p. 16/17

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.

Thank you for your attention

STACS’07 - February 22, 2007 – p. 17/17