We study abduction in First Order Horn logic theories where all atoms can be abduced and we are looking for preferred solutions with respect to three objective functions: cardinality minimality, coherence, and weighted abduction. We represent this reasoning problem in Answer Set Programming (ASP), in order to obtain a flexible framework for experimenting with global constraints and objective functions, and to test the boundaries of what is possible with ASP. Realizing this problem in ASP is challenging as it requires value invention and equivalence between certain constants, because the Unique Names Assumption does not hold in general. To permit reasoning in cyclic theories, we formally describe fine-grained variations of limiting Skolemization. We identify term equivalence as a main instantiation bottleneck, and improve the efficiency of our approach with on-demand constraints that were used to eliminate the same bottleneck in state-of-the-art solvers. We evaluate our approach experimentally on the ACCEL benchmark for plan recognition in Natural Language Understanding. Our encodings are publicly available, modular, and our approach is more efficient than state-of-the-art solvers on the ACCEL benchmark.
In the area of Automated Theorem Proving, algorithms search for finite models (or theorems, unsatisfiability proofs) in full first order logic without enforcing UNA and including native support for Skolemization (cf. @cite_4 ). These algorithms focus on finding a feasible solution and do not contain support for preferences (optimization criteria). However, the main emphasis of our abduction problems is to find solutions with optimal cost (recall that our problems always have the trivial solution to abduce all input atoms). To tackle our abduction problem with such theorem provers, it would be necessary to transform the optimization problem into a decision problem and perform a search over the optimization criterion, calling the prover several times. Related to theorem proving, a hypertableaux algorithm for coreference resolution is described in @cite_30 . This algorithm is inspired by weighted abduction, however it does not use preferences and relies solely on inconsistency for eliminating undesired solutions.
@cite_30 @cite_4
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Horn clause logic programming can be extended to include abduction with integrity constraints. In the resulting extension of logic programming, negation by failure can be simulated by making negative conditions abducible and by imposing appropriate denials and disjunctions as integrity constraints. This gives an alternative semantics for negation by failure, which generalises the stable model semantics of negation by failure. The abductive extension of logic programming extends negation by failure in three ways: (1) computation can be perfonned in alternative minimal models, (2) positive as well as negative conditions can be made abducible, and (3) other integrity constraints can also be accommodated. * This paper was written while the first author was at Imperial College. 235 Introduction The tenn "abduction" was introduced by the philosopher Charles Peirce [1931] to refer to a particular kind of hypothetical reasoning. In the simplest case, it has the fonn: From A and A fB infer B as a possible "explanation" of A. Abduction has been given prominence in Charniak and McDennot's [1985] "Introduction to Artificial Intelligence", where it has been applied to expert systems and story comprehension. Independently, several authors have developed deductive techniques to drive the generation of abductive hypotheses. Cox and Pietrzykowski [1986] construct hypotheses from the "dead ends" of linear resolution proofs. Finger and Genesereth [1985] generate "deductive solutions to design problems" using the "residue" left behind in resolution proofs. Poole, Goebel and Aleliunas [1987] also use linear resolution to generate hypotheses. All impose the restriction that hypotheses should be consistent with the "knowledge base". Abduction is a fonn of non-monotonic reasoning, because hypotheses which are consistent with one state of a knowledge base may become inconSistent when new knowledge is added. Poole [1988] argues that abduction is preferable to noh-monotonic logics for default reasoning. In this view, defaults are hypotheses fonnulated within classical logic rather than conclusions derived withln some fonn of non-monotonic logic. The similarity between abduction and default reasoning was also pointed out in [Kowalski, 1979]. In this paper we show how abduction can be integrated with logic programming, and we concentrate on the use of abduction to generalise negation by failure. Conditional Answers Compared with Abduction In the simplest case, a logic program consists of a set of Horn Clauses, which are used backward to_reduce goals to sub goals. The initial goal is solved when there are no subgollls left; Much research in the area of constraint processing has recently been focused on extracting small unsatisfiable "cores" from unsatisfiable constraint systems with the goal of finding minimal unsatisfiable subsets (MUSes). While most techniques have provided ways to find an approximation of an MUS (not necessarily minimal), we have developed a sound and complete algorithm for producing all MUSes of an unsatisfiable constraint system. In this paper, we describe a relationship between satisfiable and unsatisfiable subsets of constraints that we subsequently use as the foundation for MUS extraction algorithms, implemented for Boolean satisfiability constraints. The algorithms provide a framework with which many related subproblems can be solved, including relaxations of completeness to handle intractable instances, and we develop several variations of the basic algorithms to illustrate this. Experimental results demonstrate the performance of our algorithms, showing how the base algorithms run quickly on many instances, while the variations are valuable for producing results on instances whose complete results are intractably large. Furthermore, our algorithms are shown to perform better than the existing algorithms for solving either of the two distinct phases of our approach. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.