Generalising unit-refutation completeness and SLUR via nested input resolution

We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC (Unit refutation Complete), introduced in 1994. The class SLUR, introduced in [Annexstein et al, 1995], is the class of clause-sets for which unit-clause-propagation (denoted by r_1) detects unsatisfiability, or where otherwise iterative assignment, avoiding obviously false assignments by look-ahead, always yields a satisfying assignment. It is natural to consider how to form a hierarchy based on SLUR. Such investigations were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that is, using generalised unit-clause-propagation introduced in [Kullmann, 1999, 2004]. The class UC, studied in [Del Val, 1994], is the class of Unit refutation Complete clause-sets, that is, those clause-sets for which unsatisfiability is decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets F, the minimum k such that r_k determines unsatisfiability of F is exactly the "hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k such that after application of any falsifying partial assignments, r_k determines unsatisfiability. The class UC_k is given by the clause-sets which have hardness <= k. We observe that UC_1 is exactly UC. UC_k has a proof-theoretic character, due to the relations between hardness and tree-resolution, while SLUR_k has an algorithmic character. The correspondence between r_k and k-times nested input resolution (or tree resolution using clause-space k+1) means that r_k has a dual nature: both algorithmic and proof theoretic. This corresponds to a basic result of this paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples, and more details regarding future directions, third version further examples, improved and extended explanations, and more on SLUR, fourth version various additional remarks and editorial improvements, fifth version more explanations and references, typos corrected, improved wordin

Nonmonotonic Reasoning In LDL++

Deductive database systems have made major advances on efficient support for nonmonotonic reasoning. A first generation of deductive database systems supported the notion of stratification for programs with negation and set aggregates. Stratification is simple to understand and efficient to implement but it is too restrictive; therefore, a second generation of systems seeks efficient support for more powerful semantics based on notions such as well-founded models and stable models. In this respect, a particularly powerful set of constructs is provided by the recently enhanced LDL++ system that supports (i) monotonic user-defined aggregates, (ii) XY-stratified programs, and (iii) the nondeterministic choice constructs under stable model semantics. This integrated set of primitives supports a terse formulation and efficient implementation for complex computations, such as greedy algorithms and data mining functions, yielding levels of expressive power unmatched by other deductive..

Complete and Exact Peptide Sequence Analysis Based on Propositional Logic

Peptides are the short polymeric molecules constituting all the proteins. They are formed by the linking of amino acids, and the determination of the amino acid sequence of a peptide is a fundamental issue in many areas of chemistry, medicine and biology. Nowadays, the prevalent approach to this problem consists in using a mass spectrometry analysis. This gives information about the molecular weight of the full peptidic molecule and of its fragments. Such information should be used in order to find the sequence, but this constitutes, in the general case, a difficult mathematical problem. After a brief overview of the approaches proposed in literature, and of their features and limits, the chapter describes in detail a promising one based on propositional logic. Differently from the others, this approach can be proved to be complete and exact

Large-scale genotyping of highly polymorphic loci by next-generation sequencing: how to overcome the challenges to reliably genotype individuals?

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