Partial Word DFAs

Theoretical Computer Science, 2015

Recently, Dassow et al. connected partial words and regular languages. Partial words are sequences in which some positions may be undefined, represented with a "hole" symbol . If we restrict what the symbol can represent, we can use partial words to compress the representation of regular languages. Doing so allows the creation of so-called -DFAs, smaller than the DFAs recognizing the original language L, which recognize the compressed language. However, the -DFAs may be larger than the NFAs recognizing L. In this paper, we investigate a question of Dassow et al. as to how these sizes are related.

2012

Abstract We initiate a study of languages of partial words related to regular languages of full words. First, we study the possibility of expressing a regular language of full words as the image of a partial-words-language through a substitution that only replaces the hole symbols of the partial words with a finite set of letters. Results regarding the structure, uniqueness and succinctness of such a representation, as well as a series of related decidability and computational-hardness results, are presented.

Theoretical Informatics and Applications, 2017

Partial words are sequences of characters from an alphabet in which some positions may be marked with a "hole" symbol,. We can create a-substitution mapping this symbol to a subset of the alphabet, so that applying such a substitution to a partial word results in a set of full words (ones without holes). This setup allows us to compress regular languages into smaller partial languages. Deterministic finite automata for such partial languages, referred to as-DFAs, employ a limited nondeterminism that can allow them to have lower state complexity than the minimal DFAs for the corresponding full languages. Our paper focuses on algorithms for the construction of minimal partial languages, associated with some-substitution, as well as approximation algorithms for the construction of minimal-DFAs.

1991

We show how to turn a regular expression into an O(s) space representation of McNaughton and Yamada's NFA, where s is the number of NFA states. The standard adjacency list representation of McNaughton and Yamada's NFA takes up s+s 2 space in the worst case. The adjacency list representation of the NFA produced by Thompson takes up between 2r and 5r space, where r s in general, and can be arbitrarily larger than s. Given any set T of NFA states, our representation can be used to compute the set N of states one transition away from the states in T in optimal time O(jTj + jNj). McNaughton and Yamada's NFA requires (jTj jNj) in the worst case. Using Thompson's NFA, the equivalent calculation requires (r) time in the worst case. An implementation of our NFA representation con rms that it takes up an order of magnitude less space than McNaughton and Yamada's machine. An implementation to produce a DFA from our NFA representation by subset construction shows linear and quadratic speedups over subset construction starting from both Thompson's and McNaughton and Yamada's NFA's. It also shows that the DFA produced from our NFA is as much as one order of magnitude smaller than DFA's constructed from the two other NFA's.

International Journal of Foundations of Computer Science

It was conjectured by Černý in 1964, that a synchronizing DFA on [Formula: see text] states always has a synchronizing word of length at most [Formula: see text], and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for [Formula: see text], and with bounds on the number of symbols for [Formula: see text]. Here we give the full analysis for [Formula: see text], without bounds on the number of symbols. For PFAs (partial automata) on [Formula: see text] states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding [Formula: see text] for [Formula: see text]. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on [Formula: see text] states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and thr...

Partial words were introduced by Berstel and Boasson in 1999. Since then various of their combinatorial properties have been studied, the foremost being periodicity. Lately Blanchet-Sadri has made a first step in also investigating languages of partial words by introducing the concept of pcodes. With a slightly different approach we define new ways to obtain such languages by puncturing conventional ones. Then we present some first results on properties like the finiteness of their root or the property of being a code for these newly defined punctured languages.

2008

Combinatorics on words, or sequences or strings of symbols over a finite alphabet, is a rather new field although the first papers were published at the beginning of the 20th century [120, 121]. The interest in the study of combinatorics on words has been increasing since it finds applications in various research areas of mathematics, computer science, and biology where the data can be easily represented as words over some alphabet. Such areas may be concerned with algorithms on strings [38, 48, 50, 51, 52, 69, 72, 84, 102, 118], semigroups, automata and languages [2, 45, 55, 75, 82, 92, 93], molecular genetics [78], or codes [5, 73, 79]. Motivated by molecular biology of nucleic acids, Berstel and Boasson introduced in 1999 the notion of partial words which are sequences that may contain a number of “do not know” symbols or “holes” [4]. DNA molecules are the carriers of the genetic information in almost all organisms. Let us look into the structure of such a molecule. A single stra...

2010

Abstract: Regular expressions (res), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA).

Discrete Applied Mathematics, 2009

An unbordered word is a string over a finite alphabet such that none of its proper prefixes is one of its suffixes. In this paper, we extend the results on unbordered words to unbordered partial words. Partial words are strings that may have a number of -do not know‖ symbols. We extend a result of Ehrenfeucht and Silberger which states that if a word u can be written as a concatenation of nonempty prefixes of a word v, then u can be written as a unique concatenation of nonempty unbordered prefixes of v. We study the properties of the longest unbordered prefix of a partial word, investigate the relationship between the minimal weak period of a partial word and the maximal length of its unbordered factors, and also investigate some of the properties of an unbordered partial word and how they relate to its critical factorizations (if any).

Discrete Applied Mathematics, 2005

Primitive words, or strings over a finite alphabet that cannot be written as a power of another string, play an important role in formal language theory, coding theory, and combinatorics on words to name a few. In this paper, we extend some fundamental results about primitive words to primitive partial words. Partial words are strings that may have a number of "do not know" symbols.