Semirings and Formal Power Series

Droste, Manfred; Kuich, Werner

doi:10.1007/978-3-642-01492-5_1

Cited by 76 publications

(79 citation statements)

References 32 publications

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“…Let R be the underlying semiring of a semimodule monad T. Besides R = B = 2 in which case we obtain nondeterministic FSMs, we obtain standard weighted automata [7] under R = B = N (B is the free T-algebra finitely generated by {1}).…”

Section: T-automata: Examplesmentioning

confidence: 99%

Towards a Coalgebraic Chomsky Hierarchy

Goncharov¹,

Milius²,

Silva³

2014

Lecture Notes in Computer Science

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Abstract. The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance. In this paper we use recent developments on coalgebraic and monadbased semantics to obtain a generic notion of a T-automaton, where T is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, valence automata, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for T-automata, and we show by numerous examples that it correctly instantiates for some known classes of machines/languages captured by the Chomsky hierarchy. Moreover, our approach provides new generic techniques for studying expressivity power of various machine-based models.

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Section: T-automata: Examplesmentioning

confidence: 99%

Towards a Coalgebraic Chomsky Hierarchy

Goncharov¹,

Milius²,

Silva³

2014

Lecture Notes in Computer Science

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“…For various operations to be well-defined, the weight set associated to a weighted transducer must have the structure of a semiring (see [20]). A system (S, ⊕, ⊗, 0, 1) is a semiring if (S, ⊕, 0) is a commutative monoid with identity element 0, (S, ⊗, 1) is a monoid with identity element 1, ⊗ distributes over ⊕, and 0 is an annihilator for ⊗: for all a ∈ S, a ⊗ 0 = 0 ⊗ a = 0.…”

Section: Semiringsmentioning

confidence: 99%

“…A semiring (S, ⊕, ⊗, 0, 1) is said to be complete if for any index set I and any family (a i ) i∈I of elements of S, i∈I a i is an element of S whose definition does not depend on the order of the terms in the ⊕-sum and that has the following properties [22,20]:…”

Section: Complete Semiringsmentioning

confidence: 99%

Weighted Automata Algorithms

Mohri

2009

Monographs in Theoretical Computer Science. An EATCS Series

146

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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“…These novel applications require significant advances in fundamental aspects of automata theory [2]. One such advance is a transition from a Boolean to a multi-valued setting: while traditional automata accept or reject their input, and are therefore Boolean, novel applications, for example in speech recognition and image processing [18], are based on weighted automata, which map an input word to a value from a semiring over a large domain [7].…”

Section: Introductionmentioning

confidence: 99%

Minimizing Deterministic Lattice Automata

Halamish

Kupferman

2011

Foundations of Software Science and Computational Structures

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Abstract. Traditional automata accept or reject their input, and are therefore Boolean. In contrast, weighted automata map each word to a value from a semiring over a large domain. The special case of lattice automata, in which the semiring is a finite lattice, has interesting theoretical properties as well as applications in formal methods. A minimal deterministic automaton captures the combinatoric nature and complexity of a formal language. Deterministic automata are used in run-time monitoring, pattern recognition, and modeling systems. Thus, the minimization problem for deterministic automata is of great interest, both theoretically and in practice.For traditional automata on finite words, a minimization algorithm, based on the Myhill-Nerode right congruence on the set of words, generates in polynomial time a canonical minimal deterministic automaton. A polynomial algorithm is known also for weighted automata over the tropical semiring. For general deterministic weighted automata, the problem of minimization is open. In this paper we study minimization of lattice automata. We show that it is impossible to define a right congruence in the context of lattices, and that no canonical minimal automaton exists. Consequently, the minimization problem is much more complicated, and we prove that it is NP-complete. As good news, we show that while right congruence fails already for finite lattices that are fully ordered, for this setting we are able to combine a finite number of right congruences and generate a minimal deterministic automaton in polynomial time.

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Semirings and Formal Power Series

Cited by 76 publications

References 32 publications

Towards a Coalgebraic Chomsky Hierarchy

Towards a Coalgebraic Chomsky Hierarchy

Weighted Automata Algorithms

Minimizing Deterministic Lattice Automata

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