From Logic Programming to Human Reasoning:: How to be Artificially Human

Results of psychological experiments have shown that humans make assumptions, which are not necessarily valid, that they are influenced by their background knowledge and that they reason non-monotonically. These observations show that classical logic does not seem to be adequate for modeling human reasoning. Instead of assuming that humans do not reason logically at all, we take the view that humans do not reason classical logically. Our goal is to model episodes of human reasoning and for this purpose we investigate the so-called Weak Completion Semantics. The Weak Completion Semantics is a Logic Programming approach and considers the least model of the weak completion of logic programs under the three-valued Łukasiewicz logic. As the Weak Completion Semantics is relatively new and has not yet been extensively investigated, we first motivate why this approach is interesting for modeling human reasoning. After that, we show the formal correspondence to the already established Stable Model Semantics and Well-founded Semantics. Next, we present an extension with an additional context operator, that allows us to express negation as failure. Finally, we propose a contextual abductive reasoning approach, in which the context of observations is relevant. Some properties do not hold anymore under this extension. Besides discussing the well-known psychological experiments Byrne’s suppression task and Wason’s selection task, we investigate an experiment in spatial reasoning, an experiment in syllogistic reasoning and an experiment that examines the belief-bias effect. We show that the results of these experiments can be adequately modeled under the Weak Completion Semantics. A result which stands out here, is the outcome of modeling the syllogistic reasoning experiment, as we have a higher prediction match with the participants’ answers than any of twelve current cognitive theories. We present an abstract evaluation system for conditionals and discuss well-known examples from the literature. We show that in this system, conditionals can be evaluated in various ways and we put up the hypothesis that humans use a particular evaluation strategy, namely that they prefer abduction to revision. We also discuss how relevance plays a role in the evaluation process of conditionals. For this purpose we propose a semantic definition of relevance and justify why this is preferable to a exclusively syntactic definition. Finally, we show that our system is more general than another system, which has recently been presented in the literature. Altogether, this thesis shows one possible path on bridging the gap between Cognitive Science and Computational Logic. We investigated findings from psychological experiments and modeled their results within one formal approach, the Weak Completion Semantics. Furthermore, we proposed a general evaluation system for conditionals, for which we suggest a specific evaluation strategy. Yet, the outcome cannot be seen as the ultimate solution but delivers a starting point for new open questions in both areas

Loop Formulas for Description Logic Programs

Description Logic Programs (dl-programs) proposed by Eiter et al. constitute an elegant yet powerful formalism for the integration of answer set programming with description logics, for the Semantic Web. In this paper, we generalize the notions of completion and loop formulas of logic programs to description logic programs and show that the answer sets of a dl-program can be precisely captured by the models of its completion and loop formulas. Furthermore, we propose a new, alternative semantics for dl-programs, called the {\em canonical answer set semantics}, which is defined by the models of completion that satisfy what are called canonical loop formulas. A desirable property of canonical answer sets is that they are free of circular justifications. Some properties of canonical answer sets are also explored.Comment: 29 pages, 1 figures (in pdf), a short version appeared in ICLP'1

Mackey-complete spaces and power series -- A topological model of Differential Linear Logic

In this paper, we have described a denotational model of Intuitionist Linear Logic which is also a differential category. Formulas are interpreted as Mackey-complete topological vector space and linear proofs are interpreted by bounded linear functions. So as to interpret non-linear proofs of Linear Logic, we have used a notion of power series between Mackey-complete spaces, generalizing the notion of entire functions in C. Finally, we have obtained a quantitative model of Intuitionist Differential Linear Logic, where the syntactic differentiation correspond to the usual one and where the interpretations of proofs satisfy a Taylor expansion decomposition

A process-algebraic semantics for generalised nonblocking.

Generalised nonblocking is a weak liveness property to express the ability of a system to terminate under given preconditions. This paper studies the notions of equivalence and refinement that preserve generalised nonblocking and proposes a semantic model that characterises generalised nonblocking equivalence. The model can be constructed from the transition structure of an automaton, and has a finite representation for every finite-state automaton. It is used to construct a unique automaton representation for all generalised nonblocking equivalent automata. This gives rise to effective decision procedures to verify generalised nonblocking equivalence and refinement, and to a method to simplify automata while preserving generalised nonblocking equivalence. The results of this paper provide for better understanding of nonblocking in a compositional framework, with possible applications in compositional verification

Strongly Complete Logics for Coalgebras

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts. Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor. Part II investigates algebras for a functor over ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical extensions of Boolean algebras with operators to this setting. Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T. Based on Part II we prove the logic to be strongly complete under a reasonable condition on T