Incremental Recompilation of Knowledge

Approximating a general formula from above and below by Horn formulas (its Horn envelope and Horn core, respectively) was proposed by Selman and Kautz (1991, 1996) as a form of ``knowledge compilation,'' supporting rapid approximate reasoning; on the negative side, this scheme is static in that it supports no updates, and has certain complexity drawbacks pointed out by Kavvadias, Papadimitriou and Sideri (1993). On the other hand, the many frameworks and schemes proposed in the literature for theory update and revision are plagued by serious complexity-theoretic impediments, even in the Horn case, as was pointed out by Eiter and Gottlob (1992), and is further demonstrated in the present paper. More fundamentally, these schemes are not inductive, in that they may lose in a single update any positive properties of the represented sets of formulas (small size, Horn structure, etc.). In this paper we propose a new scheme, incremental recompilation, which combines Horn approximation and model-based updates; this scheme is inductive and very efficient, free of the problems facing its constituents. A set of formulas is represented by an upper and lower Horn approximation. To update, we replace the upper Horn formula by the Horn envelope of its minimum-change update, and similarly the lower one by the Horn core of its update; the key fact which enables this scheme is that Horn envelopes and cores are easy to compute when the underlying formula is the result of a minimum-change update of a Horn formula by a clause. We conjecture that efficient algorithms are possible for more complex updates.Comment: See http://www.jair.org/ for any accompanying file

Disjunctions of Horn Theories and their Cores

In this paper, we study issues on disjunctions of propositional Horn theories. In particular, we consider the problems of deciding whether a disjunction of Horn theories is Horn, and, if not, computing a Horn core (i.e., a maximal Horn theory included in this disjunction) and the Horn envelope (i.e., the minimum Horn theory including the disjunction), where a Horn core and the Horn envelope are important approximations of the original theory in articial intelligence. The problems are investigated for two dierent representations of Horn theories, namely for Horn CNFs and characteristic models. While the problems are shown to be intractable in general, in the case of bounded disjunctions, we present polynomial time algorithms for testing the Horn property in both representations, and for computing a Horn core in the CNF representation. Even in the case of bounded disjunction, no polynomial algorithm exists (unless P=NP) for computing a Horn core in the characteristic model representation. Computing the Horn envelope is polynomial in the characteristic model representation, while it is exponential in the CNF representation, even for bounded disjunction. Running head. Disjunctions of Horn Theories and their Cores key words. Computational issues in articial intelligence, Logic in computer science, Horn theory. AMS subject classication. 68Q25, 68T27.

Hypergraph Transversal Computation and Related Problems in Logic and AI

Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science. In the present paper, we address this problem and its decisional variant, i.e., the recognition of the transversal hypergraph for another hypergraph

Association of late-onset Alzheimer's disease with genetic variation in multiple members of the GAPD gene family

Although several genes have been implicated in the development of the early-onset autosomal dominant form of Alzheimer's disease (AD), the genetics of late-onset AD (LOAD) is complex. Loci on several chromosomes have been linked to the disease, but so far only the apolipoprotein E gene has been consistently shown to be a risk factor. We have performed a large-scale single-nucleotide polymorphism (SNP)-based association study, across the region of linkage on chromosome 12, in multiple case-control series totaling 1,089 LOAD patients and 1,196 control subjects and report association with SNPs in the glyceraldehyde-3-phosphate dehydrogenase (GAPD) gene. Subsequent analysis of GAPD paralogs on other chromosomes demonstrated association with two other paralogs. A significant association between LOAD and a compound genotype of the three GAPD genes was observed in all three sample sets. Individually, these SNPs make differential contributions to disease risk in each of the casecontrol series, suggesting that variants in functionally similar genes may account for series-to-series heterogeneity of disease risk. Our observations raise the possibility that GAPD genes are AD risk factors, a hypothesis that is consistent with the role of GAPD in neuronal apoptosis