502,243 research outputs found

    Caterpillar dualities and regular languages

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    We characterize obstruction sets in caterpillar dualities in terms of regular languages, and give a construction of the dual of a regular family of caterpillars. We show that these duals correspond to the constraint satisfaction problems definable by a monadic linear Datalog program with at most one EDB per rule

    Quotient Complexity of Regular Languages

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    The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

    Non-termination using Regular Languages

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    We describe a method for proving non-looping non-termination, that is, of term rewriting systems that do not admit looping reductions. As certificates of non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201

    Regular Languages meet Prefix Sorting

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    Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with nn states admits an equivalent Wheeler DFA (WDFA) with at most 2n1Σ2n-1-|\Sigma| states that can be computed in O(n3)O(n^3) time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a O(nlogn)O(n\log n)-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in O(nlogn)O(n\log n) time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version with new results (W-MH theorem, linear determinization), added author: Giovanna D'Agostin
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