Super Logic Programs

The Autoepistemic Logic of Knowledge and Belief (AELB) is a powerful nonmonotic formalism introduced by Teodor Przymusinski in 1994. In this paper, we specialize it to a class of theories called `super logic programs'. We argue that these programs form a natural generalization of standard logic programs. In particular, they allow disjunctions and default negation of arbibrary positive objective formulas. Our main results are two new and powerful characterizations of the static semant ics of these programs, one syntactic, and one model-theoretic. The syntactic fixed point characterization is much simpler than the fixed point construction of the static semantics for arbitrary AELB theories. The model-theoretic characterization via Kripke models allows one to construct finite representations of the inherently infinite static expansions. Both characterizations can be used as the basis of algorithms for query answering under the static semantics. We describe a query-answering interpreter for super programs which we developed based on the model-theoretic characterization and which is available on the web.Comment: 47 pages, revised version of the paper submitted 10/200

Belnap's epistemic states and negation-as-failure

Generalizing Belnap's system of epistemic states [Bel77] we obtain the system of disjunctive factbases which is the paradigm for all other kinds of disjunctive knowledge bases. Disjunctive factbases capture the nonmonotonic reasoning based on paraminimal models. In the schema of a disjunctive factbase, certain predicates of the resp. domain are declared to be exact, i.e. two-valued, and in turn some of these exact predicates are declared to be subject to the Closed-World Assumption (CWA). Thus, we distinguish between three kinds of predicates: inexact predicates, exact predicates subject to the CWA, and exact predicates not subject to the CWA

Computing only minimal answers in disjunctive deductive databases

A method is presented for computing minimal answers in disjunctive deductive databases under the disjunctive stable model semantics. Such answers are constructed by repeatedly extending partial answers. Our method is complete (in that every minimal answer can be computed) and does not admit redundancy (in the sense that every partial answer generated can be extended to a minimal answer), whence no non-minimal answer is generated. For stratified databases, the method does not (necessarily) require the computation of models of the database in their entirety. Compilation is proposed as a tool by which problems relating to computational efficiency and the non-existence of disjunctive stable models can be overcome. The extension of our method to other semantics is also considered.Comment: 48 page

Disjunctively incomplete information in relational databases: modeling and related issues

In this dissertation, the issues related to the information incompleteness in relational databases are explored. In general, this dissertation can be divided into two parts. The first part extends the relational natural join operator and the update operations of insertion and deletion to I-tables, an extended relational model representing inclusively indefinite and maybe information, in a semantically correct manner. Rudimentary or naive algorithms for computing natural joins on I-tables require an exponential number of pair-up operations and block accesses proportional to the size of I-tables due to the combinatorial nature of natural joins on I-tables. Thus, the problem becomes intractable for large I-tables. An algorithm for computing natural joins under the extended model which reduces the number of pair-up operations to a linear order of complexity in general and in the worst case to a polynomial order of complexity with respect to the size of I-tables is proposed in this dissertation. In addition, this algorithm also reduces the number of block accesses to a linear order of complexity with respect to the size of I-tables;The second part is related to the modeling aspect of incomplete databases. An extended relational model, called E-table, is proposed. E-table is capable of representing exclusively disjunctive information. That is, disjunctions of the form P[subscript]1\mid P[subscript]2\mid·s\mid P[subscript]n, where ǁ denotes a generalized logical exclusive or indicating that exactly one of the P[subscript]i\u27s can be true. The information content of an E-table is precisely defined and relational operators of selection, projection, difference, union, intersection, and cartisian product are extended to E-tables in a semantically correct manner. Conditions under which redundancies could arise due to the presence of exclusively disjunctive information are characterized and the procedure for resolving redundancies is presented;Finally, this dissertation is concluded with discussions on the directions for further research in the area of incomplete information modeling. In particular, a sketch of a relational model, IE-table (Inclusive and Exclusive table), for representing both inclusively and exclusively disjunctive information is provided

Inconsistency and Incompleteness in Relational Databases and Logic Programs

The aim of this thesis is to study the role played by negation in databases and to develop data models that can handle inconsistent and incomplete information. We develop models that also allow incompleteness through disjunctive information under both the CWA and the OWA in relational databases. In the area of logic programming, extended logic programs allow explicit representation of negative information. As a result, a number of extended logic programs have an inconsistent semantics. We present a translation of extended logic programs to normal logic programs that is more tolerant to inconsistencies. Extended logic programs have also been used widely in order to compute the repairs of an inconsistent database. We present some preliminary ideas on how source information can be incorporated into the repair program in order to produce a subset of the set of all repairs based on a preference for certain sources over others

Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined

Outline of a Theory of Strongly Semantic Information

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