## A Kleene theorem for weighted tree automata (2002)

Venue: | Theory of Computing Systems |

Citations: | 19 - 8 self |

### BibTeX

@ARTICLE{Droste02akleene,

author = {Manfred Droste and Heiko Vogler},

title = {A Kleene theorem for weighted tree automata},

journal = {Theory of Computing Systems},

year = {2002},

volume = {38},

pages = {1--38}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we prove Kleene's result for tree series over a commutative and idempotent semiring A (which is not necessarily complete or continuous), i.e., the class of recognizable tree series over A and the class of rational tree series over A are equal. We show the result by direct automata-theoretic constructions and prove their correctness.

### Citations

303 | Finite-State Transducers in Language and Speech Processing
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(Show Context)
Citation Context ...ed much interest due to their applications in image compression (Culik II and Kari [CK93], Hafner [Haf99], Katritzke [Kat01], Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri =-=[Moh97]-=-, [MPR00], Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background on formal power series, we refer the reader to [SS78, KS86, BR88, Kui97b]. In this paper, we wish to extend Thatcher ... |

235 |
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Citation Context ...T and 2 (0) , the -concatenation of L 1 and L 2 is the tree language L 1 L 2 T dened as follows (cf. page 66 of [TW68]; also cf. -product of L 1 and L 2 in Chapter II, Denition 4.5 of [GS84], and on page 16 of [GS97]): L 1 L 2 = [ s2L1 s L 2 where, for every s 2 T and L T , if s has r 0 occurrences of , then s L = fs[s(u 1 ; : : : ; u r )] j u 1 ; : : : ; u r 2 Lg. In... |

179 | Automata-Theoretic Aspects of Formal Power Series - Salomaa, Soittola - 1978 |

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M.: Tree languages
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(Show Context)
Citation Context ...oncatenation of L 1 and L 2 is the tree language L 1 L 2 T dened as follows (cf. page 66 of [TW68]; also cf. -product of L 1 and L 2 in Chapter II, Denition 4.5 of [GS84], and on page 16 of [GS97]): L 1 L 2 = [ s2L1 s L 2 where, for every s 2 T and L T , if s has r 0 occurrences of , then s L = fs[s(u 1 ; : : : ; u r )] j u 1 ; : : : ; u r 2 Lg. Intuitively, t 2 s L i ... |

119 | Mappings and grammars on trees - Rounds - 1970 |

114 |
On the definition of a family of automata
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(Show Context)
Citation Context ... the coincidence of regular and rational languages has been extended in several directions. Thatcher and Wright [TW68] generalized it to tree languages, also cf. [GS84, Eng75b, ES77]. Schutzenberger [=-=Sch61]-=- showed that the formal power series (cost functions) associated with weightedsnite automata over words and an arbitrary semiring for the weights, are precisely the rational formal power series. Weigh... |

92 | Reutenauer: Codes and Automata
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(Show Context)
Citation Context ...t and r are uniquely decomposed by dec U (t; r) = (s; r 0 ; (w 1 ; u 1 ; r 1 ); : : : ; (w k ; u k ; r k )), where U(r) = fw 2 pos(t) j r(w) = pg is a node property of t. Recall, e.g., from p.14 of [=-=Per90]-=- that in the string case one has to take care about the initial states of paths q i ! p 1 ! p 2 ! : : : ! pn ! q j , and form the set X P q i ;q j of all labels of such paths where fp 1 ; : : : ; pn g... |

75 | Methods and applications of (max, +) linear algebra - Gaubert, Plus - 1997 |

70 | Bottom-up and top-down tree transformations, a comparison - Engelfriet - 1975 |

61 |
and Combinatorial Optimization in Ordered Algebraic Structures
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Citation Context .... any subsemigroup of the max-plus semiring of the real numbers), are fundamental in max-plus algebra (Gaubert and Plus [GP97, Gau01], Cuninghame-Green [CG95]) and in algebraic optimization problems (=-=[Zim81]-=-), and also occurred in other investigations on formal power series (e.g., [Kui97b, DG99, DG00]). In [Sei92, Sei94] weigthed tree automata were introduced and decidability results for particular semir... |

56 | Recognizable formal power series on trees
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Citation Context ... extend Thatcher and Wright's and Schutzenberger's approaches to weighted automata on trees, for short: weighted tree automata. In the case where the weight semiring is aseld, Berstel and Reutenauer [=-=BR82]-=- showed that the recognizable series again coincide with the rational ones. Kuich [Kui97a, Kui99] established the analogous result for ordered commutative semirings which are complete and continuous, ... |

42 |
Formal power series over trees
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(Show Context)
Citation Context ...proofs (cf. Theorem 4.1 and Theorem 5.10). Let A be a commutative and idempotent semiring. Then A rec hhT ii S Qsnite set A rat hhT (Q)ii and A rat hhT ii A rec hhT ii: Also in [BR82] and [Kui97=-=a-=-] Kleene's result has been extended to formal tree series. These approaches dier from our one in the requirements on the semiring A. Berstel and Reutenauer useselds. Since Kuich usessxpoints of equati... |

39 | Generalized sequential machine maps - Thatcher - 1970 |

37 | Bottom-up and Topdown Tree Series Transformations - Engelfriet, Fülöp, et al. |

36 |
Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata
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(Show Context)
Citation Context ...s which occur in the literature. 2 Preliminaries 2.1 Semirings and formal power series For a survey paper about the relevance of semirings and formal power series to formal languages and automata cf. =-=[Kui-=-97b] (also cf. [KS86]). Here we only recall the needed denitions. A semiring is an algebraic structure (A; ;s; 0; 1) with two operations sum and productssuch that (A; ; 0) is a commutative monoid, (A... |

35 | Composition of top-down and bottom-up tree transductions - Baker - 1979 |

29 |
Minimax algebra and applications
- Cuninghame-Green
- 1991
(Show Context)
Citation Context ...n). Idempotent semirings exist in abundance (e.g. any subsemigroup of the max-plus semiring of the real numbers), are fundamental in max-plus algebra (Gaubert and Plus [GP97, Gau01], Cuninghame-Green =-=[CG95]-=-) and in algebraic optimization problems ([Zim81]), and also occurred in other investigations on formal power series (e.g., [Kui97b, DG99, DG00]). In [Sei92, Sei94] weigthed tree automata were introdu... |

29 | Tree transducers and formal tree series - Kuich - 1999 |

28 | Tree Automata for Code Selection
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- 1994
(Show Context)
Citation Context ...r investigations on formal power series (e.g., [Kui97b, DG99, DG00]). In [Sei92, Sei94] weigthed tree automata were introduced and decidability results for particular semirings over N were proved. In =-=[FSW94]-=- weighted tree automata were used for ecient code selection. If the weighted tree automata also produce output trees, then tree series transducers are obtained [Kui98, EFV02]; they generalize the conc... |

22 |
Representation of events in nerve nets and automata
- Kleene
- 1956
(Show Context)
Citation Context ...ss of rational tree series over A are equal. We show the result by direct automata-theoretic constructions and prove their correctness. 1 Introduction In automata theory, Kleene's fundamental theorem =-=[Kle56-=-] on the coincidence of regular and rational languages has been extended in several directions. Thatcher and Wright [TW68] generalized it to tree languages, also cf. [GS84, Eng75b, ES77]. Schutzenberg... |

22 |
Generalized automata theory with an application to a decision-problem of second-order logic
- Thatcher, Wright
- 1968
(Show Context)
Citation Context ... correctness. 1 Introduction In automata theory, Kleene's fundamental theorem [Kle56] on the coincidence of regular and rational languages has been extended in several directions. Thatcher and Wright =-=[TW68-=-] generalized it to tree languages, also cf. [GS84, Eng75b, ES77]. Schutzenberger [Sch61] showed that the formal power series (cost functions) associated with weightedsnite automata over words and an ... |

20 | Finite tree automata with cost functions, Theoret - Seidl |

16 |
Tree automata and tree grammars
- Engelfriet
- 1975
(Show Context)
Citation Context ...operations like sum, topconcatenation,s-product and two kinds of -iteration generalizing naturally the classical ones for tree languages investigated by Thatcher and Wright [TW68] and by Engelfriet [E=-=ng75-=-b]. We show that these two kinds of -iterations actually coincide for commutative semirings, leading to an -Kleene star operation for tree series. We also compare our operations with the corresponding... |

11 | Rational Series and Their Languages, volume 12 of EATCS Monographs on Theoretical Computer Science - Berstel, Reutenauer - 1984 |

10 | On aperiodic and star-free formal power series in partially commuting variables, Theory Comput - Droste, Gastin |

8 | The Kleene-Schützenberger theorem for formal power series in partially commuting variables - Droste, Gastin - 1999 |

6 |
Low Bit-Rate Image and Video Coding with Weighted Finite Automata
- Hafner
- 1999
(Show Context)
Citation Context ...or the weights, are precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [CK93], Hafner =-=[Haf99]-=-, Katritzke [Kat01], Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri [Moh97], [MPR00], Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background on formal power... |

4 |
Similarity enrichment in image compression through weighted finite automata
- Jiang, Litow, et al.
(Show Context)
Citation Context ...wer series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [CK93], Hafner [Haf99], Katritzke [Kat01], Jiang, Litow and de Vel =-=[JLdV00]-=-) and in speech-to-text processing (Mohri [Moh97], [MPR00], Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background on formal power series, we refer the reader to [SS78, KS86, BR88, Ku... |

4 |
The design principles of a weighted transducer library. Theoretical Computer Science
- Mohri, Pereira, et al.
- 2000
(Show Context)
Citation Context ...nterest due to their applications in image compression (Culik II and Kari [CK93], Hafner [Haf99], Katritzke [Kat01], Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri [Moh97], =-=[MPR00]-=-, Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background on formal power series, we refer the reader to [SS78, KS86, BR88, Kui97b]. In this paper, we wish to extend Thatcher and Wrigh... |

3 | systems of equations and automata on distributive multioperator monoids. In Contributions to General Algebra 12 - Linear |

2 |
On the determinization of weighted automata
- Buchsbaum, Giancarlo, et al.
- 2000
(Show Context)
Citation Context ...ompression (Culik II and Kari [CK93], Hafner [Haf99], Katritzke [Kat01], Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri [Moh97], [MPR00], Buchsbaum, Giancarlo and Westbrook =-=[BGW00-=-]). For theoretical background on formal power series, we refer the reader to [SS78, KS86, BR88, Kui97b]. In this paper, we wish to extend Thatcher and Wright's and Schutzenberger's approaches to weig... |

2 |
Image compression using weighted automata. Computer and Graphics
- Culik, Kari
- 1993
(Show Context)
Citation Context ...rary semiring for the weights, are precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari =-=[CK93]-=-, Hafner [Haf99], Katritzke [Kat01], Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri [Moh97], [MPR00], Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background... |

1 | Workshop on max-plus-algebras and their applications to discrete-event systems, theoretical computer science, and optimization - Gaubert - 2001 |

1 |
Re of data compression using weighted automata
- Katritzke
- 2001
(Show Context)
Citation Context ... precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [CK93], Hafner [Haf99], Katritzke =-=[Kat01]-=-, Jiang, Litow and de Vel [JLdV00]) and in speech-to-text processing (Mohri [Moh97], [MPR00], Buchsbaum, Giancarlo and Westbrook [BGW00]). For theoretical background on formal power series, we refer t... |