The Complexity of Compressed Membership Problems for Finite Automata

Jeż, Artur

doi:10.1007/s00224-013-9443-6

Cited by 16 publications

(11 citation statements)

References 30 publications

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“…Recompression was developed for a specific problem concerning compressed data (fully compressed membership problem for finite automata [30]) and was later successfully applied to word equations [36] and other problems related to compressed representations. The usual approach for word equations (and compressed data in general) is that one tries to extract information about the combinatorics of the underlying words from the equation (compressed representation) and use this structure to solve the problem at hand.…”

Section: Recompressionmentioning

confidence: 99%

Recompression: Technique for Word Equations and Compressed Data

Jeż

2020

Language and Automata Theory and Applications

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In this talk I will present the recompression technique on the running example of word equations. In word equation problem we are given an equation u = v, where both u and v are words of letters and variables, and ask for a substitution of variables by words that equalizes the sides of the equation. The recompression technique is based on employing simple compression rules (replacement of two letters ab by a new letter c, replacement of maximal repetitions of a by a new letter), and modifying the equations (replacing a variable X by bX or Xa) so that those operations are sound and complete. The simple analysis focuses on the size of the instance and not on the combinatorial properties of words that are used. The recompression-based algorithm for word equations runs in nondeterministic linear space. The approach turned out to be quite robust and can be applied to various generalized, simplified and related problems, in particular, to problems in the area of grammar compressed words. I will comment on some of those applications.

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Section: Recompressionmentioning

confidence: 99%

Recompression: Technique for Word Equations and Compressed Data

Jeż

2020

Language and Automata Theory and Applications

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“…The wide class of compressed membership problems (deciding eval(P) ∈ L) is studied and discussed in Jeż [16] and Lohrey [20]. In the case of words over the unary alphabet, w ∈ {a} * , expressing w with an SLP is poly-time equivalent to representing it with its length |w| written in binary.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Unary Pushdown Automata and Straight-Line Programs

Chistikov

Majumdar

2014

Automata, Languages, and Programming

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Abstract. We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs-one for the prefix, one for the lasso-that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNPhardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is Π2P-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply Π2P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

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“…The remarkable fact about this latter compressor is that in contrast to [9,23,39,40] it does not use the LZ77 factorization of a string (which makes the compressors from [9,23,39,40] not suitable for a generalization to trees, since LZ77 ignores the tree structure and no good analogue of LZ77 for trees is known), but is based on the recompression technique. This technique was introduced in [19] and successfully applied for a variety of algorithmic problems for SLP-compressed strings [19,22] and word equations [13,24,25]. The basic idea is to compress a string using two operations:…”

Section: Introductionmentioning

confidence: 99%

Approximation of smallest linear tree grammar

Jeż

Lohrey

2016

Information and Computation

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A simple linear-time algorithm for constructing a linear context-free tree grammar of size O(rg + rg log(n/rg)) for a given input tree T of size n is presented, where g is the size of a minimal linear context-free tree grammar for T , and r is the maximal rank of symbols in T (which is a constant in many applications). This is the first example of a grammar-based tree compression algorithm with a good, i.e. logarithmic in terms of the size of the input tree, approximation ratio. The analysis of the algorithm uses an extension of the recompression technique from strings to trees.

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The Complexity of Compressed Membership Problems for Finite Automata

Cited by 16 publications

References 30 publications

Recompression: Technique for Word Equations and Compressed Data

Recompression: Technique for Word Equations and Compressed Data

Unary Pushdown Automata and Straight-Line Programs

Approximation of smallest linear tree grammar

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