12,013 research outputs found

    Dependence Logic vs. Constraint Satisfaction

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    Leibniz international proceedings in informatics. Vol. 62

    The 2CNF Boolean Formula Satisfiability Problem and the Linear Space Hypothesis

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    We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by size parameters using simultaneously polynomial time and sub-linear space on multi-tape deterministic Turing machines. We are particularly focused on a special NL-complete problem, 2SAT---the 2CNF Boolean formula satisfiability problem---parameterized by the number of Boolean variables. It is shown that 2SAT with nn variables and mm clauses can be solved simultaneously polynomial time and (n/2clogn)polylog(m+n)(n/2^{c\sqrt{\log{n}}})\, polylog(m+n) space for an absolute constant c>0c>0. This fact inspires us to propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which states that 2SAT3_3---a restricted variant of 2SAT in which each variable of a given 2CNF formula appears at most 3 times in the form of literals---cannot be solved simultaneously in polynomial time using strictly "sub-linear" (i.e., m(x)εpolylog(x)m(x)^{\varepsilon}\, polylog(|x|) for a certain constant ε(0,1)\varepsilon\in(0,1)) space on all instances xx. An immediate consequence of this working hypothesis is LNL\mathrm{L}\neq\mathrm{NL}. Moreover, we use our hypothesis as a plausible basis to lead to the insolvability of various NL search problems as well as the nonapproximability of NL optimization problems. For our investigation, since standard logarithmic-space reductions may no longer preserve polynomial-time sub-linear-space complexity, we need to introduce a new, practical notion of "short reduction." It turns out that, parameterized with the number of variables, 2SAT3\overline{\mathrm{2SAT}_3} is complete for a syntactically restricted version of NL, called Syntactic NLω_{\omega}, under such short reductions. This fact supports the legitimacy of our working hypothesis.Comment: (A4, 10pt, 25 pages) This current article extends and corrects its preliminary report in the Proc. of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), August 21-25, 2017, Aalborg, Denmark, Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017, vol. 83, pp. 62:1-62:14, 201

    Reachability analysis of first-order definable pushdown systems

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    We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are first-order definable in a fixed countably-infinite structure. We show that the reachability analysis can be addressed with the well-known saturation technique for the wide class of oligomorphic structures. Moreover, for the more restrictive homogeneous structures, we are able to give concrete complexity upper bounds. We show ample applicability of our technique by presenting several concrete examples of homogeneous structures, subsuming, with optimal complexity, known results from the literature. We show that infinitely many such examples of homogeneous structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1

    Meta-Kernelization using Well-Structured Modulators

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    Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters. We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization

    Parity Games of Bounded Tree-Depth

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    The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdr\v{z}\'{a}lek showed that for graphs of bounded tree-width or clique-width, the problem is in P\mathrm{P}, which was later improved by Ganardi, who showed that it is even in LOGCFL\mathrm{LOGCFL} (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform AC0\text{AC}^0. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.Comment: This is the full version of the paper that has been accepted at CSL 2023 and is going to be published in Leibniz International Proceedings in Informatics (LIPIcs

    From algebra to logic: there and back again -- the story of a hierarchy

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    This is an extended survey of the results concerning a hierarchy of languages that is tightly connected with the quantifier alternation hierarchy within the two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian Federation (2014

    Introduction to the 26th International Conference on Logic Programming Special Issue

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    This is the preface to the 26th International Conference on Logic Programming Special IssueComment: 6 page
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