The paper presents two equivalent definitions of answer sets for logic programs with aggregates. These definitions build on the notion of unfolding of aggregates, and they are aimed at creating methodologies to translate logic programs with aggregates to normal logic programs or positive programs, whose answer set semantics can be used to defined the semantics of the original programs. The first definition provides an alternative view of the semantics for logic programming with aggregates described by The second definition is similar to the traditional answer set definition for normal logic programs, in that, given a logic program with aggregates and an interpretation, the unfolding process produces a positive program. The paper shows how this definition can be extended to consider aggregates in the head of the rules. The proposed views of logic programming with aggregates are simple and coincide with the ultimate stable model semantics, and with other semantic characterizations for large classes of program (e.g., programs with monotone aggregates and programs that are aggregate-stratified). Moreover, it can be directly employed to support an implementation using available answer set solvers. The paper describes a system, called ASP^A, that is capable of computing answer sets of programs with arbitrary (e.g., recursively defined) aggregates.
[ @math ''] Let us assume @math is a stable model of @math according to @cite_34 . A result in @cite_34 shows that @math where [ ] We will show that @math is a answer set of @math by proving that @math where @math and @math the aggregate-free head reduct of @math with respect to @math (Definition ).
@cite_34
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. We characterize the set of properties of Boolean-valued functions on a finite domain @math that are testable with a constant number of samples. Specifically, we show that a property @math is testable with a constant number of samples if and only if it is (essentially) a @math -part symmetric property for some constant @math , where a property is @math -part symmetric if there is a partition @math of @math such that whether @math satisfies the property is determined solely by the densities of @math on @math . We use this characterization to obtain a number of corollaries, namely: (i) A graph property @math is testable with a constant number of samples if and only if whether a graph @math satisfies @math is (essentially) determined by the edge density of @math . (ii) An affine-invariant property @math of functions @math is testable with a constant number of samples if and only if whether @math satisfies @math is (essentially) determined by the density of @math . (iii) For every constant @math , monotonicity of functions @math on the @math -dimensional hypergrid is testable with a constant number of samples. Let @math be a partially ordered set. If the Boolean lattice @math can be partitioned into copies of @math for some positive integer @math , then @math must satisfy the following two trivial conditions: (1) the size of @math is a power of @math , (2) @math has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if @math only satisfies condition (2), we can still almost partition @math into copies of @math . We prove that if @math has a unique maximal and minimal element, then there exists a constant @math such that all but at most @math elements of @math can be covered by disjoint copies of @math . Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large @math -dimensional array @math and a fixed @math -dimensional pattern @math over a finite alphabet, we say that @math is @math -free if it does not contain a copy of the forbidden pattern @math as a consecutive subarray. The distance of @math to @math -freeness is the fraction of entries of @math that need to be modified to make it @math -free. For any @math and any large enough pattern @math over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least @math and the case that it is at most @math , with query complexity and running time @math , where @math and @math depend only on @math . To analyze the testers we establish several combinatorial results, including the following @math -dimensional modification lemma, which might be of independent interest: for any large enough pattern @math over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array @math containing a copy of @math , one can delete this copy by modifying one of its locations without creating new @math -copies in @math . Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns.