## Minimizing Deterministic Weighted Tree Automata (2008)

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Citations: | 5 - 4 self |

### BibTeX

@MISC{Maletti08minimizingdeterministic,

author = {Andreas Maletti},

title = {Minimizing Deterministic Weighted Tree Automata},

year = {2008}

}

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### Abstract

The problem of efficiently minimizing deterministic weighted tree automata (wta) is investigated. Such automata have found promising applications as language models in Natural Language Processing. A polynomial-time algorithm is presented that given a deterministic wta over a commutative semifield, of which all operations including the computation of the inverses are polynomial, constructs an equivalent minimal (with respect to the number of states) deterministic and total wta. If the semifield operations can be performed in constant time, then the algorithm runs in time O(rmn 4) where r is the maximal rank of the input symbols, m is the number of transitions, and n is the number of states of the input wta.

### Citations

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(Show Context)
Citation Context ...that minimization is e ective by providing the analogue to the pumping argument already mentioned above in this more general setting. However, the trivially obtained algorithm is exponential. Angluin =-=[18]-=- learning algorithms exist for both general [19] and deterministic [20, 21] wta. In principle, those polynomial-time learning algorithms could also be used for minimization since they produce minimal ... |

285 |
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Citation Context ...tates. This contrasts the situation encountered with the pushing strategy of [24, 25], which, in general, needs final weights. Finally, let us mention some open problems. Can a Hopcroft-like strategy =-=[29]-=- improve the presented algorithm to run in time O(lm log n)? The author doubts that the presented approach yields to this method, however the approach of [24, 25] for deterministic weighted string aut... |

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Citation Context ...e number of transitions, and n is the number of states of the input wta. 1 Introduction Weighted tree automata (wta) [1{4] are a joint generalization of weighted string automata [5] and tree automata =-=[6, 7]-=-. Weighted string automata have successfully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Ca... |

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Citation Context ...on, minimization, tree series, determinism, partition refinement 1. Introduction Weighted tree automata (wta) [1, 2, 3, 4] are a joint generalization of weighted string automata [5] and tree automata =-=[6, 7]-=-. Weighted string automata have successfully been applied as language models in Natural Language Processing due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel [8]... |

78 |
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Citation Context ...g the taught tree series. However, this also requires us to implement the oracle, which answers coefficient and equivalence queries. Although equivalence is decidable in polynomial time in both cases =-=[23, 17]-=-, a simple implementation would return counterexamples of exponential size, which would again yield an exponential-time minimization algorithm. Clearly, this can be avoided for dwta by the method pres... |

71 | OpenFst: A general and efficient weighted finite-state transducer library
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Citation Context ...successfully been applied as language models in Natural Language Processing due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel [8], Fire Station [9], and OpenFst =-=[10]-=-) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made available by the toolkits. In recent ... |

63 |
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Citation Context ...e of the standardized implementation model and the consolidated algorithms made available by the toolkits. In recent years, the trend toward more syntactical approaches in Natural Language Processing =-=[11]-=- sparked renewed interest in tree-based devices. The weighted tree automaton is the natural tree-based analogue of the weighted string automaton. First experiments with toolkits (e.g., Tiburon [12]) b... |

55 | Minimization Algorithms for Sequential Transducers
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Citation Context ...tial size, which would yield an exponential-time minimization algorithm. Clearly, this can be avoided by the method presented in this contribution. Finally, let us mention the minimization procedures =-=[23, 24]-=- for deterministic weighted string automata. They rely on a weight normal-form obtained by a procedure called pushing. After this normal form is obtained, the weight of a transition is treated as an i... |

42 |
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Citation Context ...le tree series are exactly those recognizable tree series that can be computed by deterministic wta. Recognizable and deterministically recognizable tree series have been thoroughly investigated (see =-=[3, 13]-=- and references provided therein). In fact, [4] and [14] show which recognizable tree series are also deterministically recognizable. In this contribution, we consider deterministically recognizable t... |

31 | Equational elements in additive algebras - Bozapalidis - 1999 |

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Citation Context ...ng [11] sparked renewed interest in tree-based devices. The weighted tree automaton is the natural tree-based analogue of the weighted string automaton. First experiments with toolkits (e.g., Tiburon =-=[12]-=-) based on tree-based devices show that the situation is not as consolidated here. In particular, many basic algorithms are missing in the weighted setting. In general, a wta processes a given input t... |

22 |
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Citation Context ...le tree series are exactly those recognizable tree series that can be computed by deterministic wta. Recognizable and deterministically recognizable tree series have been thoroughly investigated (see =-=[3, 13]-=- and references provided therein). In fact, [4] and [14] show which recognizable tree series are also deterministically recognizable. In this contribution, we consider deterministically recognizable t... |

21 |
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(Show Context)
Citation Context ...hat can be computed by deterministic wta. Recognizable and deterministically recognizable tree series have been thoroughly investigated (see [3, 13] and references provided therein). In fact, [4] and =-=[14]-=- show which recognizable tree series are also deterministically recognizable. In this contribution, we consider deterministically recognizable tree series. To the author's knowledge, we propose the rs... |

20 |
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Citation Context ...e number of transitions, and n is the number of states of the input wta. 1 Introduction Weighted tree automata (wta) [1{4] are a joint generalization of weighted string automata [5] and tree automata =-=[6, 7]-=-. Weighted string automata have successfully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Ca... |

17 |
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Citation Context ...s ? Author on leave from Technische Universitat Dresden, Faculty of Computer Science, 01062 Dresden, Germany with the help of nancial support by a DAAD (German Academic Exchange Service) grant.known =-=[15]-=-. However, it only asserts the existence of a unique, up to slight changes of representation, minimal (with respect to the number of states) deterministic wta recognizing a given tree series. The cons... |

16 |
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Citation Context ...al states. This contrasts the situation encountered with the pushing strategy of [23, 24], which needs nal weights in general. Finally, let us mention some open problems. Can a Hopcroft-like strategy =-=[26]-=- improve the presented algorithm? A more detailed complexity analysis should be conducted to obtain a tighter bound on the time complexity of the algorithm. Can minimization be performed in a similar ... |

16 | Determinization of finite state weighted tree automata
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(Show Context)
Citation Context ...sitions, and n is the number of states of the input dwta. Key words: weighted tree automaton, minimization, tree series, determinism, partition refinement 1. Introduction Weighted tree automata (wta) =-=[1, 2, 3, 4]-=- are a joint generalization of weighted string automata [5] and tree automata [6, 7]. Weighted string automata have successfully been applied as language models in Natural Language Processing due to t... |

11 |
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(Show Context)
Citation Context ..., which given a deterministic wta constructs an equivalent minimal deterministic and total wta, could easily be derived. For (not necessarily deterministic) wta over elds the situation is similar. In =-=[1, 17]-=- the existence of a unique, up to slight changes of representation, minimal wta is proved. Moreover, [17] shows that minimization is e ective by providing the analogue to the pumping argument already ... |

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Citation Context ...esentation. The variables sol and D are set once and then retain their value without further changes. The procedure RefineCong(Π) returns the coarsest congruence on (Q, δ) that refines Π. It is known =-=[28]-=- that it can be implemented to run in time O(lm log n). Our approach is quite the opposite of the approach of [24, 25]. Their approach first handles the transition weights and then uses RefineCong to ... |

8 |
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(Show Context)
Citation Context ...nimal (with respect to the number of states) dwta recognizing a given tree series. The construction of such a dwta, which is given in [16], is not effective, but with the help of the pumping lemma of =-=[17]-=- a simple exponential-time algorithm, which given a dwta constructs an equivalent minimal dwta, could easily be conceived. For (not necessarily deterministic) wta over fields the situation is similar.... |

6 |
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(Show Context)
Citation Context ...argument already mentioned above in this more general setting. However, the trivially obtained algorithm is exponential. Angluin [18] learning algorithms exist for both general [19] and deterministic =-=[20, 21]-=- wta. In principle, those polynomial-time learning algorithms could also be used for minimization since they produce minimal wta recognizing the taught tree series. However, this also requires us to i... |

6 |
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(Show Context)
Citation Context ...he relations defined by these two versions of Definition 3 actually coincide. The main benefits of the version in Definition 3 are that (i) it avoids an exponential blow-up (the one also mentioned in =-=[27]-=-) and (ii) it is consistent with our definition of signs of life. Let us investigate the Myhill-Nerode relation in more detail. We immediately note that (i) all dead states are equivalent, (ii) no dea... |

5 |
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(Show Context)
Citation Context ..., which given a deterministic wta constructs an equivalent minimal deterministic and total wta, could easily be derived. For (not necessarily deterministic) wta over elds the situation is similar. In =-=[1, 17]-=- the existence of a unique, up to slight changes of representation, minimal wta is proved. Moreover, [17] shows that minimization is e ective by providing the analogue to the pumping argument already ... |

5 |
H.: Determinization of nite state weighted tree automata
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- 2003
(Show Context)
Citation Context ...series that can be computed by deterministic wta. Recognizable and deterministically recognizable tree series have been thoroughly investigated (see [3, 13] and references provided therein). In fact, =-=[4]-=- and [14] show which recognizable tree series are also deterministically recognizable. In this contribution, we consider deterministically recognizable tree series. To the author's knowledge, we propo... |

5 |
Equational elements in additive algebras. Theory Comput. Syst
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(Show Context)
Citation Context ...sitions, and n is the number of states of the input dwta. Key words: weighted tree automaton, minimization, tree series, determinism, partition refinement 1. Introduction Weighted tree automata (wta) =-=[1, 2, 3, 4]-=- are a joint generalization of weighted string automata [5] and tree automata [6, 7]. Weighted string automata have successfully been applied as language models in Natural Language Processing due to t... |

4 | Learning multiplicity tree automata
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- 2006
(Show Context)
Citation Context ...nalogue to the pumping argument already mentioned above in this more general setting. However, the trivially obtained algorithm is exponential. Angluin [18] learning algorithms exist for both general =-=[19]-=- and deterministic [20, 21] wta. In principle, those polynomial-time learning algorithms could also be used for minimization since they produce minimal wta recognizing the taught tree series. However,... |

4 |
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(Show Context)
Citation Context ... 7]. Weighted string automata have successfully been applied as language models in Natural Language Processing due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel =-=[8]-=-, Fire Station [9], and OpenFst [10]) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made a... |

4 |
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Citation Context ...ng automata have successfully been applied as language models in Natural Language Processing due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel [8], Fire Station =-=[9]-=-, and OpenFst [10]) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made available by the to... |

4 |
and more general minimization for weighted finite-state automata
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- 2003
(Show Context)
Citation Context ...h would again yield an exponential-time minimization algorithm. Clearly, this can be avoided for dwta by the method presented in this contribution. Finally, let us mention the minimization procedures =-=[24, 25]-=- for deterministic weighted string automata. They rely on a weight normal-form obtained by a procedure called pushing. After this normal form is obtained, the weight of a transition is treated as an i... |

3 |
M.: OpenFst: A general and e cient weighted nite-state transducer library
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(Show Context)
Citation Context ...ully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel [8], Fire Station [9], and OpenFst =-=[10]-=-) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made available by the toolkits. In recent ... |

3 | Learning deterministically recognizable tree series — revisited
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Citation Context ...argument already mentioned above in this more general setting. However, the trivially obtained algorithm is exponential. Angluin [18] learning algorithms exist for both general [19] and deterministic =-=[20, 21]-=- wta. In principle, those polynomial-time learning algorithms could also be used for minimization since they produce minimal wta recognizing the taught tree series. However, this also requires us to i... |

3 |
and more general minimization for weighted nite-state automata
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(Show Context)
Citation Context ...tial size, which would yield an exponential-time minimization algorithm. Clearly, this can be avoided by the method presented in this contribution. Finally, let us mention the minimization procedures =-=[23, 24]-=- for deterministic weighted string automata. They rely on a weight normal-form obtained by a procedure called pushing. After this normal form is obtained, the weight of a transition is treated as an i... |

3 |
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Citation Context ...sitions, and n is the number of states of the input dwta. Key words: weighted tree automaton, minimization, tree series, determinism, partition refinement 1. Introduction Weighted tree automata (wta) =-=[1, 2, 3, 4]-=- are a joint generalization of weighted string automata [5] and tree automata [6, 7]. Weighted string automata have successfully been applied as language models in Natural Language Processing due to t... |

2 |
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Citation Context ...input symbols, m is the number of transitions, and n is the number of states of the input wta. 1 Introduction Weighted tree automata (wta) [1{4] are a joint generalization of weighted string automata =-=[5]-=- and tree automata [6, 7]. Weighted string automata have successfully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. S... |

2 |
Deciding equivalence of nite tree automata
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(Show Context)
Citation Context ...ing the taught tree series. However, this also requires us to implement the oracle, which answers coe cient and equivalence queries. Although equivalence is decidable in polynomial time in both cases =-=[22, 16]-=-, a simple implementation would return counterexamples of exponential size, which would yield an exponential-time minimization algorithm. Clearly, this can be avoided by the method presented in this c... |

2 |
Formal power series over trees, in: S. Bozapalidis (Ed
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(Show Context)
Citation Context |

1 |
Carmel nite-state toolkit
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(Show Context)
Citation Context ...ghted string automata have successfully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel =-=[8]-=-, Fire Station [9], and OpenFst [10]) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made a... |

1 |
B.W.: Fire station: An environment for manipulating nite automata and regular expression views
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- 2004
(Show Context)
Citation Context ...ata have successfully been applied as language models in Natural Language Processing largely due to their ability to easily incorporate n-gram models. Several toolkits (e.g., Carmel [8], Fire Station =-=[9]-=-, and OpenFst [10]) enable language engineers to rapidly prototype and develop language models because of the standardized implementation model and the consolidated algorithms made available by the to... |

1 |
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Citation Context ...n a minimal deterministic wta recognizing the given series after the unweighted minimization. Here we give a direct minimization construction, which uses partition re nement as in the unweighted case =-=[25]-=-. To this end, we rst de ne the Myhill-Nerode relation on states of the deterministic input wta. This de nition, as well as the Myhill-Nerode relation on tree series [15], will include a scaling facto... |