113 research outputs found

    Unsolvability Cores in Classification Problems

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    Classification problems have been introduced by M. Ziegler as a generalization of promise problems. In this paper we are concerned with solvability and unsolvability questions with respect to a given set or language family, especially with cores of unsolvability. We generalize the results about unsolvability cores in promise problems to classification problems. Our main results are a characterization of unsolvability cores via cohesiveness and existence theorems for such cores in unsolvable classification problems. In contrast to promise problems we have to strengthen the conditions to assert the existence of such cores. In general unsolvable classification problems with more than two components exist, which possess no cores, even if the set family under consideration satisfies the assumptions which are necessary to prove the existence of cores in unsolvable promise problems. But, if one of the components is fixed we can use the results on unsolvability cores in promise problems, to assert the existence of such cores in general. In this case we speak of conditional classification problems and conditional cores. The existence of conditional cores can be related to complexity cores. Using this connection we can prove for language families, that conditional cores with recursive components exist, provided that this family admits an uniform solution for the word problem

    Algebraic Properties of Valued Constraint Satisfaction Problem

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    The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP languages to weighted algebras. We show that the difficulty of VCSP depends only on the weighted variety generated by the associated weighted algebra. Paralleling the results for CSPs we exhibit a reduction to cores and rigid cores which allows us to focus on idempotent weighted varieties. Further, we propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the hardness direction and verify that it agrees with known results for VCSPs on two-element sets [Cohen et al. 2006], finite-valued VCSPs [Thapper and Zivny 2013] and conservative VCSPs [Kolmogorov and Zivny 2013].Comment: arXiv admin note: text overlap with arXiv:1207.6692 by other author

    Optimal Planning Modulo Theories

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    Planning for real-world applications requires algorithms and tools with the ability to handle the complexity such scenarios entail. However, meeting the needs of such applications poses substantial challenges, both representational and algorithmic. On the one hand, expressive languages are needed to build faithful models. On the other hand, efficient solving techniques that can support these languages need to be devised. A response to this challenge is underway, and the past few years witnessed a community effort towards more expressive languages, including decidable fragments of first-order theories. In this work we focus on planning with arithmetic theories and propose Optimal Planning Modulo Theories, a framework that attempts to provide efficient means of dealing with such problems. Leveraging generic Optimization Modulo Theories (OMT) solvers, we first present domain-specific encodings for optimal planning in complex logistic domains. We then present a more general, domain- independent formulation that allows to extend OMT planning to a broader class of well-studied numeric problems in planning. To the best of our knowledge, this is the first time OMT procedures are employed in domain-independent planning

    Calculation and Classification as Dimensions of Social Interaction

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    Cassirer interpreted the transition from medieval to modern theoretical thinking as an epistemological revolution which saw the concept of function to replace that of substance as a guide to knowledge. This article suggests that, in the practical world, classification into classes, types or genres is correlated to the concept of substance, while the concept of function is present in any practical calculation in the form of algorithmic procedures. The way in which algorithmic calculation and classification interact with each other has resulted in the preservation, expansion or loss of social interaction, understood as reciprocity between actors. The action of calculation and classification can then be used as epistemological coordinates to analyze social interaction. Such systematization results in a set of schemes presented in this paper along with ethnological studies supporting the theoretical model

    Resource Bounded Immunity and Simplicity

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    Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in the Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France, August 23-26, 200

    3-manifolds efficiently bound 4-manifolds

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    It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the minimum number of n-simplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of "shadow complexity" of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel
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