## Backward and forward bisimulation minimisation of tree automata (2007)

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Citations: | 11 - 6 self |

### BibTeX

@TECHREPORT{Högberg07backwardand,

author = {Johanna Högberg and Andreas Maletti and Jonathan May},

title = {Backward and forward bisimulation minimisation of tree automata},

institution = {},

year = {2007}

}

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### Abstract

Abstract. We improve an existing bisimulation minimisation algorithm for tree automata by introducing backward and forward bisimulations and developing minimisation algorithms for them. Minimisation via forward bisimulation is also effective for deterministic automata and faster than the previous algorithm. Minimisation via backward bisimulation generalises the previous algorithm and is thus more effective but just as fast. We demonstrate implementations of these algorithms on a typical task in natural language processing.

### Citations

2352 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ... subset of δ that contains entries of the form f(q1, . . .,qk) → q, where f ∈ Σ (k), q ∈ Q, and q1 · · ·qk is in B1 · · ·Bk for some B1 · · ·Bk ∈ L. Our computation model is the random access machine =-=[15]-=-, which supports indirect addressing, and thus allows the use of pointers. This means that we can represent each block in a partition (Q/R) as a record of two-way pointers to its elements, and that we... |

2124 | Building a large annotated corpus of English: The Penn Treebank
- Marcus, Santorini, et al.
- 1993
(Show Context)
Citation Context ...es of a syntactic tree. We prepared a data set by collecting 3-subtrees, i.e. all subtrees of height 3, from sentences taken from the Penn Treebank corpus of syntactically bracketed English news text =-=[11]-=-. An initial nta was constructed by representing each 3-subtree in a single path. We then wrote an implementation of the forward and backward variants of Algorithm 1 in Perl and applied them to data s... |

357 |
Three partition refinement algorithms
- Paige, Tarjan
- 1987
(Show Context)
Citation Context ...guages and are used in applications such as model checking [1] and natural language processing [10]. The minimisation algorithm in [2] was inspired by a partitioning algorithm due to Paige and Tarjan =-=[14]-=-, and relies heavily on bisimulation; a concept introduced by R. Milner as a formal tool for investigating transition systems. Intuitively, two states are bisimilar if they can simulate each other, or... |

287 |
An n log n algorithm for minimizing states in a finite automaton
- Hopcroft
- 1971
(Show Context)
Citation Context ...ural language processing. 1 Introduction Automata minimisation has a long and studied history. For deterministic string automata (dfa) efficient algorithms exist. The well-known algorithm by Hopcroft =-=[8]-=- runs in time O(n log n) where n is the number of states of the input automaton. The situation is worse for nondeterministic automata (nfa). The minimisation problem for nfa is PSPACE-complete [13] an... |

258 | A syntax-based statistical translation model
- Yamada, Knight
- 2001
(Show Context)
Citation Context ...ate sentence in the domain language. Recent research in natural language processing has focused on using treebased models to capture syntactic dependencies in applications such as machine translation =-=[6, 16]-=-. Thus, the problem is elevated to determining whether a given syntactic tree is in a language. Language models are naturally representable as finite-state acceptors. For efficiency and data sparsity ... |

236 |
Tree automata techniques and applications. Available on: http://www.grappa.univ-lille3.fr/tata
- Comon, Dauchet, et al.
- 1997
(Show Context)
Citation Context ...d the approach of [2] in two ways: (i) we relax the constraints for state equivalence, and (ii) we introduce a new bisimulation relation that can be applied to deterministic bottom-up tree 1sautomata =-=[5]-=-, unlike [2], which is ineffective on dta. Thus we are able to find smaller automata than previously possible. The two ways correspond, respectively, to two types of bisimulation: backward and forward... |

219 | What’s in a translation rule
- Galley, Hopkins, et al.
- 2004
(Show Context)
Citation Context ...ta N recognising L is to construct nta N1 and N2 that recognise only f[a, b] and f[a, a], respectively. Then we construct N by disjoint union of N1 and N2. In this manner we could obtain the nta N = (=-=[1, 6]-=-, Σ, δ, {3, 6}) with a() δ → 1 b() δ → 2 f(1, 2) δ → 3 a() δ → 4 a() δ → 5 f(4, 5) δ → 6 . Let P = {1, 4, 5} 2 ∪ {2} 2 ∪ {3} 2 ∪ {6} 2 . We claim that P is a backward bisimulation on N. In fact, we on... |

184 |
The equivalence problem for regular expressions with squaring requires exponential space
- Meyer, Stockmeyer
- 1972
(Show Context)
Citation Context ...oft [8] runs in time O(n log n) where n is the number of states of the input automaton. The situation is worse for nondeterministic automata (nfa). The minimisation problem for nfa is PSPACE-complete =-=[13]-=- and cannot even be efficiently approximated within the factor o(n) unless P = PSPACE [7]. The problem must thus be restricted to allow algorithms of practical value, and one possibility is to settle ... |

183 |
Continuous speech recognition by statistical methods
- Jelinek
- 1976
(Show Context)
Citation Context ...this section we present some experimental results that we obtained by applying a prototype implementation of Algorithm 1 to the problem of language modelling in the natural language processing domain =-=[9]-=-. A language model is a formalism for determining whether a given sentence is in a particular language. Language models are particularly useful in many applications of natural language and speech proc... |

149 | Tree Automata Techniques and Applications
- Comon, Dauchet, et al.
- 2007
(Show Context)
Citation Context ...] in two ways: (i) we relax the constraints for state equivalence, and (ii) we introduce a new bisimulation relation that (with effect) can be applied to deterministic (bottom-up) tree automata (dta) =-=[8]-=-. Note that [4] is ineffective on dta. Thus we are able to find smaller automata than previously possible. The two ways correspond, respectively, to two types of bisimulation: backward and forward bis... |

64 |
An overview of probabilistic tree transducers for natural language processing
- Knight, Graehl
- 2005
(Show Context)
Citation Context ... [2] for non-deterministic tree automata (nta), which are a generalisation of nfa that recognise tree languages and are used in applications such as model checking [1] and natural language processing =-=[10]-=-. The minimisation algorithm in [2] was inspired by a partitioning algorithm due to Paige and Tarjan [14], and relies heavily on bisimulation; a concept introduced by R. Milner as a formal tool for in... |

35 | Regular tree model checking
- Abdulla, Jonsson, et al.
- 2002
(Show Context)
Citation Context ...rtial minimisation. This was done in [2] for non-deterministic tree automata (nta), which are a generalisation of nfa that recognise tree languages and are used in applications such as model checking =-=[1]-=- and natural language processing [10]. The minimisation algorithm in [2] was inspired by a partitioning algorithm due to Paige and Tarjan [14], and relies heavily on bisimulation; a concept introduced... |

31 | 2006b. Tiburon: A weighted tree automata toolkit
- May, Knight
(Show Context)
Citation Context ...t desire a compact representation of large non-deterministic tree automata. We plan to include a refined implementation of this algorithm in a future version of the tree automata toolkit described in =-=[15]-=-. Acknowledgements. The authors acknowledge the support and advice of Frank Drewes and Kevin Knight. We thank Lisa Kaati for providing data and information relevant to the details of [4]. We would als... |

28 | Minimizing NFA’s and regular expressions
- Gramlich, Schnitger
- 2005
(Show Context)
Citation Context ...e situation is worse for nondeterministic automata (nfa). The minimisation problem for nfa is PSPACE-complete [13] and cannot even be efficiently approximated within the factor o(n) unless P = PSPACE =-=[7]-=-. The problem must thus be restricted to allow algorithms of practical value, and one possibility is to settle for a partial minimisation. This was done in [2] for non-deterministic tree automata (nta... |

16 |
An n log n algorithm for minimizing states in a nite automaton
- Hopcroft
(Show Context)
Citation Context ...language processing. 1 Introduction Automata minimisation has a long and studied history. For deterministic nite (string) automata (dfa) e cient algorithms exist. The well-known algorithm by Hopcroft =-=[1]-=- runs in time O (n log n) where n is the number of states of the input automaton. The situation is worse for non-deterministic nite automata (nfa). The minimisation problem for nfa is PSPACE-complete ... |

15 |
Bisimulation relations for weighted automata
- Buchholz
- 2008
(Show Context)
Citation Context ...that [4] is ine ective on dta. Thus we are able to nd smaller automata than previously possible.The two ways correspond, respectively, to two types of bisimulation: backward and forward bisimulation =-=[9]-=-. In a forward bisimulation on an automaton M, bisimilar states are restricted to have identical futures (i.e., the observer can inspect what will happen next). The future of a state q is the set of c... |

14 | Bisimulation Minimization of Tree Automata
- Abdulla, Högberg, et al.
(Show Context)
Citation Context ... also trivial, since (1, 2) /∈ P and the states 3 and 4 only appear on the right hand side of δ →. Thus P is a forward bisimulation. The aggregated nta (N/P) is (Q ′ , Σ, δ ′ , F ′ ) with Q ′ = {[1], =-=[2]-=-, [3]} and F ′ = {[3]} and a() δ′ → [1] b() δ′ → [2] f([1], [2]) δ′ → [3] f([1], [1]) δ′ → [3] . ✷ For the rest of this section we let M be an arbitrary but fixed nta and R be a forward bisimulation o... |

4 |
Bisimulation relations for weighted automata. unpublished
- Buchholz
- 2007
(Show Context)
Citation Context ...which is ineffective on dta. Thus we are able to find smaller automata than previously possible. The two ways correspond, respectively, to two types of bisimulation: backward and forward bisimulation =-=[4]-=-. Let us explain both types on a bottom-up tree automaton. In a forward bisimulation on an automaton M, bisimilar states are restricted to have identical futures (i.e., the observer can inspect what w... |