Algebraic structures for transitive closure

Lehmann, Dirk J.

doi:10.1016/0304-3975(77)90056-1

Cited by 88 publications

(64 citation statements)

References 7 publications

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“…Note that the earlier definition of closed semirings given by Aho et al [1] is not axiomatically correct (see [37,26]). Any complete semiring is a closed semiring.…”

Section: Complete Semiringsmentioning

confidence: 99%

Weighted Automata Algorithms

Mohri

2009

Monographs in Theoretical Computer Science. An EATCS Series

145

149

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This chapter presents several fundamental algorithms for weighted automata and transducers. While the mathematical counterparts of weighted transducers, rational power series, have been extensively studied in the past [22,54,13,36], several essential weighted transducer algorithms, e.g., composition, determinization, minimization, have been devised only in the last decade [38,43], in part motivated by novel applications in speech recognition, speech synthesis, machine translation, other areas of natural language processing, image processing, optical character recognition, and more recently machine learning.These algorithms can be viewed as the generalization to the weighted transducer case of the standard algorithms for unweighted acceptors. However, this generalization is often not straightforward and has required a number of specific studies either because the old schema could not be applied in the presence of weights and a novel technique was required, as in the case of composition [50,46], or because of the analysis of the conditions of application of an algorithm as in the case of determinization [38,3].The chapter favors a presentation of weighted automata and transducers in terms of graphs, the natural concepts for an algorithmic description and complexity analysis. Also, while power series lead to more concise and rigorous proofs in most cases [36], proofs related to questions of ambiguity naturally require the introduction of paths and reasoning on graph concepts. PreliminariesThis section introduces the definitions and notation related to weighted finitestate transducers, weighted transducers for short, and weighted automata.

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“…Note that the earlier definition of closed semirings given by Aho et al [1] is not axiomatically correct (see [37,26]). Any complete semiring is a closed semiring.…”

Section: Complete Semiringsmentioning

confidence: 99%

Weighted Automata Algorithms

Mohri

2009

Monographs in Theoretical Computer Science. An EATCS Series

145

149

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“…Like in [1], we introduce a matrix semiring (S n×n , ⊕, ⊗, * ,Ō,Ī), a set of n × n matrices S n×n over a closed scalar semiring (S , ⊕, ⊗, * ,0,1) with two binary operations, matrix addition ⊕ : S n×n × S n×n → S n×n and matrix multiplication ⊗ : S n×n × S n×n → S n×n , a unary operation called closure of a matrix * : S n×n → S n×n , the zero n × n matrixŌ whose all elements equal to0, and the n × n identity matrixĪ whose all main diagonal elements equal to1 and0 otherwise. Matrix addition and multiplication are defined as usual in linear algebra.…”

Section: The Algebraic Path Problemmentioning

confidence: 99%

“…⊕ = max, ⊗ = +, a * = 0 for all a in S , 0 = −∞, and1 = 0. The closure of a matrix gives the maximum cost (critical) paths, or +∞ if there are paths of unbounded cost [1].…”

Section: The Algebraic Path Problemmentioning

confidence: 99%

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Orbital Systolic Algorithms and Array Processors for Solution of the Algebraic Path Problem

Sedukhin

Miyazaki

Kuroda

2010

IEICE Trans. Inf. & Syst.

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SUMMARYThe algebraic path problem (APP) is a general framework which unifies several solution procedures for a number of well-known matrix and graph problems. In this paper, we present a new 3-dimensional (3-D) orbital algebraic path algorithm and corresponding 2-D toroidal array processors which solve the n × n APP in the theoretically minimal number of 3n time-steps. The coordinated time-space scheduling of the computing and data movement in this 3-D algorithm is based on the modular function which preserves the main technological advantages of systolic processing: simplicity, regularity, locality of communications, pipelining, etc. Our design of the 2-D systolic array processors is based on a classical 3-D→2-D space transformation. We have also shown how a data manipulation (copying and alignment) can be effectively implemented in these array processors in a massively-parallel fashion by using a matrix-matrix multiply-add operation.

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“…The construction of a transitive graph and computation of its edge priorities is a special case of the all-pairs algebraic-path problem Carre, 1971;Lehmann, 1977 , solved by generalized shortest-path algorithms for example, see the textbook by Cormen et al 1990 . If the simple graph has k representations and m s preferences, the complexity of constructing the transitive graph is Ok 2 log k +k m s , which is the most time-consuming step in generating the reduced preference graph.…”

Section: Remove-con Ictsmentioning

confidence: 99%

Automatic Representation Changes in Problem Solving

Fink

1999

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. REPORT DATE JUN 19992. REPORT TYPE 3. DATES COVERED 00-00-1999 to 00-00-1999 TITLE AND SUBTITLE Automatic Representation Changes in Problem Solving5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR (S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)Carnegie Mellon University,School of Computer Science,Pittsurgh,PA,152138. PERFORMING ORGANIZATION REPORT NUMBER SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTES ABSTRACTThe purpose of our research is to enhance the e ciency of AI problem solvers by automating representation changes. We have developed a system that improves description of input problems and selects an appropriate search algorithm for each given problem. Motivation Researchers have accumulated much evidence of the importance of appropriate representations for the e ciency of AI systems. The same problem may be easy or di cult, depending on the way we describe it and on the search algorithm we use. Previous work on automatic improvement of problem description has mostly been limited to the design of individual learning algorithms. The user has traditionally been responsible for the choice of algorithms appropriate for a given problem. We present a system that integrates multiple description-changing and problem-solving algorithms. The purpose of our work is to formalize the concept of representation, explore its role in problem solving, and con rm the following general hypothesis An e ective representation-changing system can be constructed out of three parts a library of problem-solving algorithms a library of algorithms that improve problem description by static analysis and learning a top-level control module that selects appropriate algorithms for each given problem. Representation-changing system We have supported this hypothesis by building a system that improves representations in the prodigy problem-solving architecture. The purpose of our research is to enhance the e ciency of AI problem solvers by automating representation change...

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Algebraic structures for transitive closure

Cited by 88 publications

References 7 publications

Weighted Automata Algorithms

Weighted Automata Algorithms

Orbital Systolic Algorithms and Array Processors for Solution of the Algebraic Path Problem

Automatic Representation Changes in Problem Solving

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