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. We proceed by proving several results related to Wheeler automata: (i) We show that every Wheeler NFA (WNFA) with n states admits an equivalent Wheeler DFA (WDFA) with at most 2n − 1 − |Σ| states (Σ being the alphabet) that can be computed in O(n 3 ) time. This is in sharp contrast with general NFAs (where the blow-up could be exponential). (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a 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(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. Our contributions imply new results of independent interest. Contributions (i-iii) extend the universe of known regular languages for which membership can be tested efficiently [Backurs and Indyk, FOCS 2016] and provide a new class of NFAs for which the minimization problem can be approximated within constant factor in polynomial time. In general, the NFA minimization problem does not admit a polynomial-time o(n)-approximation unless P=PSPACE. Contribution (iv) is a big step towards a complete solution to the well-studied problem of indexing graphs for linear-time pattern matching queries: our algorithm provides a provably minimum-size solution for the deterministic-acyclic case.We wish to thank Travis Gagie for introducing us to the problem and for stimulating discussions. Corresponding author. Supported by the project MIUR-SIR CMACBioSeq ("Combinatorial methods for analysis and compression of biological sequences") grant n. RBSI146R5L.
The (modal) μ-calculus ([14]) is a very powerful extension of modal logic with least and greatest fixed point operators. It is of great interest to computer science for expressing properties of processes such as termination (every run is finite) and fairness (on every infinite run, no action is repeated infinitely often to the exclusion of all others).The power of the μ-calculus is also evident from a more theoretical perspective. The μ-calculus is a fragment of monadic second-order logic (MSO) containing only formulae that are invariant for bisimulation, in the sense that they cannot distinguish between bisimilar states. Janin and Walukiewicz prove the converse: any property which is invariant for bisimulation and MSO-expressible is already expressible in the μ-calculus ([13]). Yet the μ-calculus enjoys many desirable properties which MSO lacks, like a complete sequent-calculus ([29]), an exponential-time decision procedure, and the finite model property ([25]). Switching from MSO to its bisimulation-invariant fragment gives us these desirable properties.In this paper we take a classical logician's view of the μ-calculus. As far as we are concerned a new logic should not be allowed into the community of logics without at least considering the standard questions that any logic is bothered with. In this paper we perform this rite of passage for the μ-calculus. The questions we will be concerned with are the following.
In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially di erent universes of such s e t s. The unisets of the rst universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We w i l l h a ve a closer look into the connection of the iterated circular multisets and arbitrary trees.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.