## A note on join and autointersection of n-ary rational relations (2004)

Venue: | Proc. Eindhoven FASTAR Days, number 04–40 in TU/e CS TR |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Kempe04anote,

author = {Andre Kempe and Jean-marc Champarnaud and Jason Eisner},

title = {A note on join and autointersection of n-ary rational relations},

booktitle = {Proc. Eindhoven FASTAR Days, number 04–40 in TU/e CS TR},

year = {2004},

pages = {64--78}

}

### OpenURL

### Abstract

A finite-state machine with n tapes describes a rational (or regular) relation on n strings. It is more expressive than a relational database table with n columns, which can only describe a finite relation. We describe some basic operations on n-ary rational relations and propose notation for them. (For generality we give the semiring-weighted case in which each tuple has a weight.) Unfortunately, the join operation is problematic: if two rational relations are joined on more than one tape, it can lead to non-rational relations with undecidable properties. We recast join in terms of “auto-intersection” and illustrate some cases in which difficulties arise. We close with the hope that partial or restricted algorithms may be found that are still powerful enough to have practical use.

### Citations

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(Show Context)
Citation Context ...stricted algorithms may be found that are still powerful enough to have practical use. 1 Introduction Multi-tape finite-state machines (FSMs) (Rabin and Scott, 1959; Elgot and Mezei, 1965; Kay, 1987; =-=Kaplan and Kay, 1994-=-) are a natural generalization of the familiar one- and two-tape cases, known respectively as finite-state acceptors and transducers. An n-tape FSM characterizes n-tuples of strings. The set of tuples... |

303 | Finite-State Transducers in Language and Speech Processing
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Citation Context ...child concept〉. An acyclic transducer can represent any finite database of this sort. Shared substrings can make the representation particularly efficient: a hypothesis lattice for speech processing (=-=Mohri, 1997-=-) represents exponentially many pairs in linear space.sUnlike a classical database, a transducer may even define infinitely many pairs. For example, it may characterize the pattern of the spelling-pro... |

233 |
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Citation Context ...iculties arise. We close with the hope that partial or restricted algorithms may be found that are still powerful enough to have practical use. 1 Introduction Multi-tape finite-state machines (FSMs) (=-=Rabin and Scott, 1959-=-; Elgot and Mezei, 1965; Kay, 1987; Kaplan and Kay, 1994) are a natural generalization of the familiar one- and two-tape cases, known respectively as finite-state acceptors and transducers. An n-tape ... |

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Citation Context ...alized composition ⊠{3=1} with a simple weighted machine will produce Figure 4c by replacing all instances of x with w0, etc. 1s4.4 Post’s Correspondence Problem Post’s Correspondence Problem or PCP (=-=Post, 1946-=-) is a classical undecidable problem that is sometimes used to prove the undecidability of other problems. Mark-Jan Nederhof (personal communication) pointed out its relevance to auto-intersection. De... |

67 |
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(Show Context)
Citation Context ...e with the hope that partial or restricted algorithms may be found that are still powerful enough to have practical use. 1 Introduction Multi-tape finite-state machines (FSMs) (Rabin and Scott, 1959; =-=Elgot and Mezei, 1965-=-; Kay, 1987; Kaplan and Kay, 1994) are a natural generalization of the familiar one- and two-tape cases, known respectively as finite-state acceptors and transducers. An n-tape FSM characterizes n-tup... |

64 | A rational design for a weighted finite-state transducer library
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Citation Context ...follow the usual definitions for multi-tape finite-state automata (Elgot and Mezei, 1965; Eilenberg, 1974), with semiring weights added just as for acceptors and transducers (Kuich and Salomaa, 1986; =-=Mohri, Pereira, and Riley, 1998-=-). 2.1 Semirings A monoid is a structure 〈M, ◦, ¯1〉 consisting of a set M, an associative binary operation ◦ on M, and a neutral element ¯1 such that ¯1 ◦ a = a ◦ ¯1 = a for all a ∈ M. A monoid is cal... |

62 | Nonconcatenative Finite-State Morphology
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(Show Context)
Citation Context ...rtial or restricted algorithms may be found that are still powerful enough to have practical use. 1 Introduction Multi-tape finite-state machines (FSMs) (Rabin and Scott, 1959; Elgot and Mezei, 1965; =-=Kay, 1987-=-; Kaplan and Kay, 1994) are a natural generalization of the familiar one- and two-tape cases, known respectively as finite-state acceptors and transducers. An n-tape FSM characterizes n-tuples of stri... |

57 | Finite-state Multimodal Parsing and Understanding
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Citation Context ...th more than 2 columns. In natural language processing, multi-tape machines have recently been used to represent lattices of 〈speech, gesture, interpretation〉 triples for processing multimodal input (=-=Bangalore and Johnston, 2000-=-). They have also been used in the morphological analysis of Semitic languages, using multiple tapes to synchronize the vowels, consonants, and templatic pattern into a surface form (Kay, 1987; Kiraz,... |

35 | Hidden Feature Models for Speech Recognition Using Dynamic Bayesian Network
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Citation Context ...nd templatic pattern into a surface form (Kay, 1987; Kiraz, 2000). They may be similarly useful for coordinating the multiple tiers of autosegmental phonology or articulator-based speech recognition (=-=Livescu, Glass, and Bilmes, 2003-=-). Unfortunately, one pays a price for allowing infinite multi-column databases. Finite-state methods derive their power from a rational algebra, which can combine simple FSMs using operations such as... |

24 |
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Citation Context ...21). One constructs a WFSM containing just those paths (e.g., Γ A (n)(s (n) )), and then sums their weights with an algorithm that generalizes the Kleene-Floyd-Warshall technique to closed semirings (=-=Lehmann, 1977-=-). 2 Divergent sums can be represented by k ∗ = ∞, where ∞ ∈ K is a distinguished value. i=0s3 Operations We now describe some central operations on n-ary weighted relations and their n-tape WFSMs, fo... |

3 |
On n-tape Finite State Acceptors
- Rosenberg
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(Show Context)
Citation Context ...� R (n) (u (n) j ) (12) ℓ=0 ℓ=0 u (n) 1 ,...u(n) ℓ : (∀i∈[[1,n]])si=(u1)i·(u2)i···(uℓ)i These operations can be implemented by simple constructions on the corresponding nondeterministic n-tape WFSMs (=-=Rosenberg, 1964-=-). These n-tape constructions and their semiring-weighted versions are exactly the same as for acceptors (n = 1) and transducers (n = 2), as they are indifferent to the n-tuple transition labels. 3.2 ... |

2 | NLP applications based on weighted multi-tape automata
- Kempe
- 2004
(Show Context)
Citation Context ...ould be replaced by a join cascade, R(4) = R (2) 1 ✶ � {2=1} R (2) 2 ✶ {2=1} R (2) � 3 . The intermediate results are now preserved on tapes 2 and 3 for subsequent inspection or further transduction (=-=Kempe, 2004-=-). In this way, single-tape join is adequate to combine several transducers into any tree topology: R (4) � = R (2) 1 ✶ {2=1} R (2) � 2 ✶ {2=1} R (2) 3 . One can use this technique to implement a tree... |

1 | Franck Guingne, and Florent Nicart. 2004. Algorithms for weighted multi-tape automata - Kempe |