From multiplicity awareness to computation correlation Jacques - PowerPoint PPT Presentation

From multiplicity awareness to computation correlation Jacques Sakarovitch LTCI – CNRS/ENST

The results presented in this talk are taken from a joint work with Marie-Pierre B´ eal and Sylvain Lombardy, IGM, Universit´ e Paris-Est, Marne-la-Vall´ ee, published in On the equivalence and conjugacy of weighted automata. in Proc. of CSR’06 , LNCS 3967. The complete journal version is still in preparation. Some of the results have been included in the chapter Rational and recognizable series of the Handbook of Weighted Automata , Springer, 2009.

Part I An introductory result

The Rational Bijection Theorem Proposition If two regular languages have the same growth function, then there exists a letter-to-letter rational bijection that maps one language onto the other.

An example: a first language K = ( c + d c + d d ) ∗ \ { c c ( c + d ) ∗ ∪ 1 B ∗ }

An example: a first language K = ( c + d c + d d ) ∗ \ { c c ( c + d ) ∗ ∪ 1 B ∗ } c d c + d r s u t c d d B

An example: a first language K = ( c + d c + d d ) ∗ \ { c c ( c + d ) ∗ ∪ 1 B ∗ } c d c + d r s u t c d d B c cdc cdcc dcdd cdd cddc ddcc dc dcc dccc dddc dd ddc dcdc dddd

An example: a first language K = ( c + d c + d d ) ∗ \ { c c ( c + d ) ∗ ∪ 1 B ∗ } c d c + d r s u t c d d B c cdc cdcc dcdd cdd cddc ddcc dc dcc dccc dddc dd ddc dcdc dddd g K ( n ) = Card ( K ∩ { c , d } n ) = 2 n − 1 ∀ n ∈ N

An example: a second language L = a ( a + b ) ∗

An example: a second language L = a ( a + b ) ∗ a a p q A b

An example: a second language L = a ( a + b ) ∗ a a p q A b a aaa aaaa abaa aab aaab abab aa aba aaba abba ab abb aabb abbb

An example: a second language L = a ( a + b ) ∗ a a p q A b a aaa aaaa abaa aab aaab abab aa aba aaba abba ab abb aabb abbb g L ( n ) = Card ( L ∩ { a , b } n ) = 2 n − 1 ∀ n ∈ N

An example: the rational bijection K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } L = a ( a + b ) ∗ a | c a | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | d a | c b | d b | d a | c a | c a | c b | d b | d qsy quz 11 quz 22 a aaa aaaa abaa c cdc cdcc dcdd aab aaab abab cdd cddc ddcc aa aba aaba abba dc dcc dccc dddc ab abb aabb abbb dd ddc dcdc dddd

An example: the rational bijection K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } L = a ( a + b ) ∗ a | c a | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | d a | c a | c a | c b | d b | d a | c a | c a | c b | d b | d qsy quz 11 quz 22 a aaa aaaa abaa c cdc cdcc dcdd aab aaab abab cdd cddc ddcc aa aba aaba abba dc dcc dccc dddc ab abb aabb abbb dd ddc dcdc dddd

An example: the rational bijection K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } L = a ( a + b ) ∗ a | c a | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | d a | c b | d b | d a | c a | c a | c b | d b | d b | d b | d qsy quz 11 quz 22 a aaa aaaa abaa c cdc cdcc dcdd aab aaab abab cdd cddc ddcc aa aba aaba abba dc dcc dccc dddc ab abb aabb abbb dd ddc dcdc dddd

An example: the rational bijection K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } L = a ( a + b ) ∗ a | c a | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | d b | d b | d a | c a | c a | c b | d b | d a | c a | c a | c b | d b | d b | d b | d qsy quz 11 quz 22 a aaa aaaa abaa c cdc cdcc dcdd aab aaab abab cdd cddc ddcc aa aba aaba abba dc dcc dccc dddc ab abb aabb abbb dd ddc dcdc dddd

An example: the rational bijection K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } L = a ( a + b ) ∗ a | c a | c b | c b | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d a | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | c b | c b | d a | c b | d b | d a | c a | c a | c b | d b | d qsy quz 11 quz 22 a aaa aaaa abaa c cdc cdcc dcdd aab aaab abab cdd cddc ddcc aa aba aaba abba dc dcc dccc dddc ab abb aabb abbb dd ddc dcdc dddd

The result on this example: how to construct the transducer a | c a | c b | c a | d quz 12 quz 21 qty b | c b | d a | d a | d a | d qtz 1 qtz 2 a | d prx b | c b | d a | c b | d b | d a | c a | c a | c b | d b | d qsy quz 11 quz 22 from the automata c a + b d c + d a and p q r s t u c d d A B

Part II The link between growth functions and automata

The generating function K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } A language that is,

The generating function K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } A language that is, c d c + d an unambiguous automaton r s u t c d d B

The generating function K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } A language that is, c d c + d an unambiguous automaton r s u t c d d B is transformed into an automaton over { z } ∗ with weight in N 1 z 1 z 2 z r s u t 1 z 1 z 1 z B ′

The generating function K = ( c + d c + d d ) ∗ \{ c c ( c + d ) ∗ ∪ 1 B ∗ } A language that is, c d c + d an unambiguous automaton r s u t c d d B is transformed into an automaton over { z } ∗ with weight in N 1 z 1 z 2 z r s u t 1 z 1 z 1 z B ′ � g K ( n ) z n which realises the generating function G K ( z ) = n ∈ N

Two regular languages with equal growth functions (i) Two finite automata A and B , preferably unambiguous, c a + b d c + d a p q r s u t c d d A B

Two regular languages with equal growth functions (i) Two finite automata A and B , preferably unambiguous, transformed into A ′ and B ′ , over { z } ∗ with multiplicity in N , (ii) which realise the generating functions G L ( z ) and G K ( z ) : g K ( n ) z n , � g L ( n ) z n � G L ( z ) = and G K ( z ) = n ∈ N n ∈ N c a + b d c + d a p q r s u t c d d A B

Two regular languages with equal growth functions (i) Two finite automata A and B , preferably unambiguous, transformed into A ′ and B ′ , over { z } ∗ with multiplicity in N , (ii) which realise the generating functions G L ( z ) and G K ( z ) : g K ( n ) z n , � g L ( n ) z n � G L ( z ) = and G K ( z ) = n ∈ N n ∈ N 2 z 1 z 1 z 2 z 1 z p q r s u t 1 z 1 z 1 z A ′ B ′

Two regular languages with equal growth functions (i) Two finite automata A and B , preferably unambiguous, transformed into A ′ and B ′ , over { z } ∗ with multiplicity in N , (ii) which realise the generating functions G L ( z ) and G K ( z ) : g K ( n ) z n , � g L ( n ) z n � G L ( z ) = and G K ( z ) = n ∈ N n ∈ N (iii) and whose equivalence is decidable (Sch¨ utzenberger 1961, Eilenberg 1974). 2 z 1 z 1 z 2 z 1 z p q r s u t 1 z 1 z 1 z A ′ B ′

Two regular languages with equal growth functions Generating functions are realised by weighted automata

Weighted automata, a first look a 2 a b p q b 2 b b a b p − − → p − − → p − − → q 2 a 2 b b p − − → q − − → q − − → q ∀ w ∈ A ∗ b ab �− → 5 w �− → � w � 2 s : A ∗ − � A ∗ � → N s : w �− → < s , w > s ∈ N � � s = b + ab + 2 b a + 3 b b + aab + 2 ab a + 3 ab b + 4 b aa + 5 b ab + . . .

Series play the role of languages � A ∗ � � plays the role of P ( A ∗ ) K �

Richness of the model of weighted automata ‘classic’ automata B ◮ ‘usual’ counting N ◮ Z , Q , R numerical multiplicity ◮ M = � N , min , + � Min-plus automata ◮ P ( B ∗ ) = B � � B ∗ � � transducers ◮ N � � B ∗ � � weighted transducers ◮ P ( F ( B )) pushdown automata ◮

Equivalence of weighted automata The equivalence of weighted automata with weights in

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B is decidable

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B decidable a field is decidable

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B decidable a subsemiring of a field is decidable

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B decidable a subsemiring of a field decidable ( Z , min , +) is undecidable

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B decidable a subsemiring of a field decidable ( Z , min , +) undecidable Rat B ∗ is undecidable The equivalence of transducers is undecidable

Equivalence of weighted automata The equivalence of weighted automata with weights in the Boolean semiring B decidable a subsemiring of a field decidable ( Z , min , +) undecidable Rat B ∗ undecidable N Rat B ∗ is decidable The equivalence of transducers undecidable transducers with multiplicity in N is decidable