Linear automaton transformations

Nerode, Anil

doi:10.1090/s0002-9939-1958-0135681-9

Cited by 387 publications

(164 citation statements)

References 1 publication

(1 reference statement)

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“…The superpolynomial blowup occurs when constructing a DFA from the NFA recognizing the right-hand expression. A lower bound to this blowup is given by the MyhillNerode theorem [11,7]. All the other steps, seen separately, are polynomial-time.…”

Section: Introductionmentioning

confidence: 99%

The Inclusion Problem for Regular Expressions

Hovland

2010

Language and Automata Theory and Applications

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Abstract. This paper presents a new polynomial-time algorithm for the inclusion problem for certain pairs of regular expressions. The algorithm is not based on construction of finite automata, and can therefore be faster than the lower bound implied by the Myhill-Nerode theorem. The algorithm automatically discards unnecessary parts of the right-hand expression. In these cases the right-hand expression might even be 1-ambiguous. For example, if r is a regular expression such that any DFA recognizing r is very large, the algorithm can still, in time independent of r, decide that the language of ab is included in that of (a + r)b. The algorithm is based on a syntax-directed inference system. It takes arbitrary regular expressions as input, and if the 1-ambiguity of the right-hand expression becomes a problem, the algorithm will report this.

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Section: Introductionmentioning

confidence: 99%

The Inclusion Problem for Regular Expressions

Hovland

2010

Language and Automata Theory and Applications

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“…, that is the equivalence relation ∼ coincides with Myhill-Nerode equivalence [11] over the states of the automaton A T E , that can be computed in O(| E |) time and space complexity using Revuz Algorithm [13]. Proof.…”

Section: If E = F Thenmentioning

confidence: 99%

An Efficient Algorithm for the Equation Tree Automaton via the k-C-Continuations

Mignot

Sebti

Ziadi

2014

Language, Life, Limits

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Abstract. Champarnaud and Ziadi, and Khorsi et al. show how to compute the equation automaton of word regular expression E via the k-CContinuations. Kuske and Meinecke extend the computation of the equation automaton to a regular tree expression E over a ranked alphabet Σ and produce a O(R · | E | 2 ) time and space complexity algorithm, where R is the maximal rank of a symbol occurring in Σ and | E | is the size of E. In this paper, we give a full description of the algorithm based on the acyclic minimization of Revuz. Our algorithm, which is performed in an O(|Q| · | E |) time and space complexity, where |Q| is the number of states of the produced automaton, is more efficient than the one obtained by Kuske and Meinecke.

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“…A first attempt to minimize lattice automata would be to follow the classical paradigm for minimizing DFA using the Myhill Nerode theorem [20,21]. In fact, for the case of simple LDFA, the plan proceeds smoothly (see [17,22], where the problem is discussed by means of fuzzy automata) 3 …”

Section: Minimizing Ldfasmentioning

confidence: 99%

“…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 [20,21]. In more detail, given a regular language L over Σ, then the relation y · z ∈ L, is an equivalence relation, its equivalence classes correspond to the states of a minimal automaton for A, and they also uniquely induce the transitions of such an automaton.…”

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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Linear automaton transformations

Cited by 387 publications

References 1 publication

The Inclusion Problem for Regular Expressions

The Inclusion Problem for Regular Expressions

An Efficient Algorithm for the Equation Tree Automaton via the k-C-Continuations

Minimizing Deterministic Lattice Automata

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