5,287 research outputs found

    Abduction in Well-Founded Semantics and Generalized Stable Models

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    Abductive logic programming offers a formalism to declaratively express and solve problems in areas such as diagnosis, planning, belief revision and hypothetical reasoning. Tabled logic programming offers a computational mechanism that provides a level of declarativity superior to that of Prolog, and which has supported successful applications in fields such as parsing, program analysis, and model checking. In this paper we show how to use tabled logic programming to evaluate queries to abductive frameworks with integrity constraints when these frameworks contain both default and explicit negation. The result is the ability to compute abduction over well-founded semantics with explicit negation and answer sets. Our approach consists of a transformation and an evaluation method. The transformation adjoins to each objective literal OO in a program, an objective literal not(O)not(O) along with rules that ensure that not(O)not(O) will be true if and only if OO is false. We call the resulting program a {\em dual} program. The evaluation method, \wfsmeth, then operates on the dual program. \wfsmeth{} is sound and complete for evaluating queries to abductive frameworks whose entailment method is based on either the well-founded semantics with explicit negation, or on answer sets. Further, \wfsmeth{} is asymptotically as efficient as any known method for either class of problems. In addition, when abduction is not desired, \wfsmeth{} operating on a dual program provides a novel tabling method for evaluating queries to ground extended programs whose complexity and termination properties are similar to those of the best tabling methods for the well-founded semantics. A publicly available meta-interpreter has been developed for \wfsmeth{} using the XSB system.Comment: 48 pages; To appear in Theory and Practice in Logic Programmin

    Deterministic and Probabilistic Binary Search in Graphs

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    We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node qq, the algorithm learns either that qq is the target, or is given an edge out of qq that lies on a shortest path from qq to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p>12p > \frac{1}{2} (a known constant), and an (adversarial) incorrect one with probability 1p1-p. Our main positive result is that when p=1p = 1 (i.e., all answers are correct), log2n\log_2 n queries are always sufficient. For general pp, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1δ)log2n1H(p)+o(logn)+O(log2(1/δ))(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta)) queries, and identifies the target correctly with probability at leas 1δ1-\delta. Here, H(p)=(plogp+(1p)log(1p))H(p) = -(p \log p + (1-p) \log(1-p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for p=1p = 1, we show several hardness results for the problem of determining whether a target can be found using KK queries. Our upper bound of log2n\log_2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query rr nodes each in kk rounds, we show membership in Σ2k1\Sigma_{2k-1} in the polynomial hierarchy, and hardness for Σ2k5\Sigma_{2k-5}

    Finding Cycles and Trees in Sublinear Time

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    We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k3k\geq 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being CkC_k-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., Ω(1)\Omega(1)-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time \tildeO(\sqrt{N}), where NN denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of NN-vertex graphs can be tested with one-sided error within time complexity \tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known Ω(N)\Omega(\sqrt{N}) query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter \e. For any constant k3k\geq3, we extend this result to testing whether the input graph has a simple cycle of length at least kk. On the other hand, for any fixed tree TT, we show that TT-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter \e. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in o(N)o(\sqrt{N}) complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree Graphs, One-Sided vs Two-Sided Error Probability Updated versio
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