Christoph Wernhard

Modeling the suppression task under weak completion and well-founded semantics

Formal approaches that aim at representing human reasoning should be evaluated based on how humans actually reason. One way of doing so is to investigate whether psychological findings of human reasoning patterns are represented in the theoretical model. The computational logic approach discussed here is the so-called weak completion semantics which is based on the three-valued ᴌukasiewicz logic. We explain how this approach adequately models Byrne’s suppression task, a psychological study where the experimental results show that participants’ conclusions systematically deviate from the classical logically correct answers. As weak completion semantics is a novel technique in the field of computational logic, it is important to examine how it corresponds to other already established non-monotonic approaches. For this purpose we investigate the relation of weak completion with respect to completion and three-valued stable model semantics. In particular, we show that well-founded semantics, a widely accepted approach in the field of non-monotonic reasoning, corresponds to weak completion semantics for a specific class of modified programs.

Soundness of inprocessing in clause sharing SAT solvers

We present a formalism that models the computation of clause sharing portfolio solvers with inprocessing. The soundness of these solvers is not a straightforward property since shared clauses can make a formula unsatisfiable. Therefore, we develop characterizations of simplification techniques and suggest various settings how clause sharing and inprocessing can be combined. Our formalization models most of the recent implemented portfolio systems and we indicate possibilities to improve these. A particular improvement is a novel way to combine clause addition techniques --- like blocked clause addition --- with clause deletion techniques --- like blocked clause elimination or variable elimination.

CIRCUMSCRIPTION AND PROJECTION AS PRIMITIVES OF LOGIC PROGRAMMING

KWe pursue a representation of logic programs as classical first-order sentences. Different semantics for logic programs can then be expressed by the way in which they are wrapped into - semantically defined - operators for circumscription and projection. (Projection is a generalization of second-order quantification.) We demonstrate this for the stable model semantics, Clarks completion and a three-valued semantics based on the Fitting operator. To represent the latter, we utilize the polarity sensitiveness of projection, in contrast to second-order quantification, and a variant of circumscription that allows to express predicate minimization in parallel with maximization. In accord with the aim of an integrated view on different logic-based representation techniques, the material is worked out on the basis of first-order logic with a Herbrand semantics.

Abduction in Logic Programming as Second-Order Quantifier Elimination

It is known that skeptical abductive explanations with respect to classical logic can be characterized semantically in a natural way as formulas with second-order quantifiers. Computing explanations is then just elimination of the second-order quantifiers. By using application patterns and generalizations of second-order quantification, like literal projection, the globally weakest sufficient condition and circumscription, we transfer these principles in a unifying framework to abduction with three non-classical semantics of logic programming: stable model, partial stable model and well-founded semantics. New insights are revealed about abduction with the partial stable model semantics.

The Boolean Solution Problem from the Perspective of Predicate Logic

Finding solution values for unknowns in Boolean equations was a principal reasoning mode in the Algebra of Logic of the 19th century. Schroder investigated it as Auflosungsproblem (solution problem). It is closely related to the modern notion of Boolean unification. Today it is commonly presented in an algebraic setting, but seems potentially useful also in knowledge representation based on predicate logic. We show that it can be modeled on the basis of first-order logic extended by second-order quantification. A wealth of classical results transfers, foundations for algorithms unfold, and connections with second-order quantifier elimination and Craig interpolation show up.

Second-Order Quantifier Elimination on Relational Monadic Formulas --- A Basic Method and Some Less Expected Applications

For relational monadic formulas the Lowenheim class second-order quantifier elimination, which is closely related to computation of uniform interpolants, forgetting and projection, always succeeds. The decidability proof for this class by Behmann from 1922 explicitly proceeds by elimination with equivalence preserving formula rewriting. We reconstruct Behmanns method, relate it to the modern DLS elimination algorithm and show some applications where the essential monadicity becomes apparent only at second sight. In particular, deciding

An Abductive Model for Human Reasoning

mathcal{ALCOQH}

The Boolean Solution Problem from the Perspective of Predicate Logic - Extended Version.

knowledge bases, elimination in DL-Lite knowledge bases, and the justification of the success of elimination methods for Sahlqvist formulas.