## Decision Problems For Patterns (1995)

Venue: | Journal of Computer and System Sciences |

Citations: | 17 - 3 self |

### BibTeX

@ARTICLE{Jiang95decisionproblems,

author = {Tao Jiang and Arto Salomaa and Kai Salomaa and Sheng Yu},

title = {Decision Problems For Patterns},

journal = {Journal of Computer and System Sciences},

year = {1995},

volume = {50},

pages = {53--63}

}

### OpenURL

### Abstract

We settle an open problem, the inclusion problem for pattern languages [1, 2]. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue [16] and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns. Categories and Subject Descriptors: F.4.3 [Mathematical Logic and Formal Languages ]: Formal Languages --- Decision problems, Algebraic language theory; F.4.1 [Mathe- matical Logic and Formal Languages]: Mathematical Logic --- Computability theory. General Terms: Theory, Formal Languages Additional Key Words and Phrases: Patterns, Inclusion problems, Equivalence problems, Descriptive patterns, Unavoidable patterns 1 Introduction. The main result Instead of an exhaustive definition for a language, [7], it is sometimes better to give more leeway in the defi...

### Citations

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Citation Context ...Words and Phrases: Patterns, Inclusion problems, Equivalence problems, Descriptive patterns, Unavoidable patterns 2 1 Introduction. The main result Instead of an exhaustive definition for a language, =-=[7]-=-, it is sometimes better to give more leeway in the definition, and try to find patterns common to all words in a sample set. Such an approach is especially appropriate if the sample set is growing, f... |

296 | Inductive inference of formal languages from positive data - Angluin - 1980 |

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Citation Context ...ada grants OGP0041630 and OGP0046613 and the Project 11281 of the Academy of Finland. All correspondence to Sheng Yu. 1 Abstract We settle an open problem, the inclusion problem for pattern languages =-=[1, 2]-=-. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue [16] ... |

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Citation Context ... and the internal state. The fact that the stacks can be restricted to be over a unary alphabet is essential for our technical constructions. A nondeterministic 2-counter automaton without input, cf. =-=[8]-=-, is denoted as a quintuple M = (Q; q 0 ; Q F ; 0; ffi ); (1) where Q is the finite set of states, q 0 2 Q is the initial state, Q F ` Q is the set of final states, 0 62 Q is the single symbol of the ... |

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Citation Context ... , C i 2 conf(M ), i = 2; : : : ; m \Gamma 1, and C i )M C i+1 , i = 1; : : : ; m \Gamma 1. It is well known that the emptiness problem for deterministic 7 2-counter automata is undecidable, cf. e.g. =-=[4, 8, 13]-=-. Thus it is also clearly undecidable whether a nondeterministic 2-counter automaton without input has an accepting computation. If u; v 2 \Sigma and u is a subword of v this is denoted as ussub v. No... |

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Citation Context ... letters then, for all n, LNE (x 1 x 2 : : : x n ) contains words not belonging to LNE (xyyz). The above results concerning squares and cubes have been extended to arbitrary terminal-free patterns in =-=[5]-=-. We present the definitions in our terminology. A terminal-free pattern ff is termed unavoidable (on an alphabet \Sigma) iff, for some n, LNE (x 1 x 2 : : : x n ) ` LNE (xffz); where the x i 's are d... |

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Citation Context ... , C i 2 conf(M ), i = 2; : : : ; m \Gamma 1, and C i )M C i+1 , i = 1; : : : ; m \Gamma 1. It is well known that the emptiness problem for deterministic 7 2-counter automata is undecidable, cf. e.g. =-=[4, 8, 13]-=-. Thus it is also clearly undecidable whether a nondeterministic 2-counter automaton without input has an accepting computation. If u; v 2 \Sigma and u is a subword of v this is denoted as ussub v. No... |

23 | The equivalence problem of multitape finite automata, Theoret
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Citation Context ... but the equivalence problem is decidable, [7]. (iii) The inclusion problem for languages accepted by finite deterministic multi-tape automata is undecidable but the equivalence problem is decidable, =-=[6]-=-. (iv) The inclusion problem for nonerasing pattern languages is undecidable but the equivalence problem is decidable. Assume that for some language family (a) the inclusion problem is undecidable, wh... |

16 |
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Citation Context ...enquire whether or not some specified words belong to the set. For instance, either one of the queries 0 2 1110 2 or 01100110 3 would decide between the patterns 0x10 2 and 0x0x0 2 . We refer to [3], =-=[9]-=-, and [15] for such interrelations and to [10] for some background information and interconnections with random numbers and Kolmogorov complexity. Our main result can be stated very simply, without as... |

4 |
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Citation Context ...hether or not some specified words belong to the set. For instance, either one of the queries 0 2 1110 2 or 01100110 3 would decide between the patterns 0x10 2 and 0x0x0 2 . We refer to [3], [9], and =-=[15]-=- for such interrelations and to [10] for some background information and interconnections with random numbers and Kolmogorov complexity. Our main result can be stated very simply, without assuming any... |

2 |
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(Show Context)
Citation Context ...1, 2]. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue =-=[16]-=- and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns. Categories and Subject Descriptors: F.4.3 [M... |

1 |
Pattern languages with and without erasing", submitted to the International Journal of Computer Mathematics
- Jiang, Kinber, et al.
(Show Context)
Citation Context ...elong to the set. For instance, either one of the queries 0 2 1110 2 or 01100110 3 would decide between the patterns 0x10 2 and 0x0x0 2 . We refer to [3], [9], and [15] for such interrelations and to =-=[10]-=- for some background information and interconnections with random numbers and Kolmogorov complexity. Our main result can be stated very simply, without assuming any previous knowledge on the part of t... |

1 |
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Citation Context ... variables of other types, such as the variables x P considered in Section 1, then also problems of different kinds can be reduced to the inclusion problem. For instance, the celebrated new result of =-=[11]-=- can be expressed as follows. If \Sigma contains at least four letters then, for all n, LNE (x 1 x 2 : : : x n ) contains words not belonging to LNE (xyy P z). The above result is not valid for smalle... |