Logic Programming and Logarithmic Space

We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455

Modular Nonmonotonic Logic Programming Revisited

Abstract. Recently, enabling modularity aspects in Answer Set Programming (ASP) has gained increasing interest to ease the composition of program parts to an overall program. In this paper, we focus on modular nonmonotonic logic programs (MLP) under the answer set semantics, whose modules may have contextually de-pendent input provided by other modules. Moreover, (mutually) recursive module calls are allowed. We define a model-theoretic semantics for this extended setting, show that many desired properties of ordinary logic programming generalize to our modular ASP, and determine the computational complexity of the new formalism. We investigate the relationship of modular programs to disjunctive logic programs with well-defined input/output interface (DLP-functions) and show that they can be embedded into MLPs

Ranking Services Using Fuzzy HEX Programs

Abstract. The need to reason with knowledge expressed in both Logic Program-ming (LP) and Description Logics (DLs) paradigms on the Semantic Web lead to several integrating formalisms, e.g., Description Logic programs (dl-programs) allow a logic program to retrieve results from and feed results to a DL knowledge base. Two functional extensions of dl-programs are HEX programs and fuzzy dl-programs. The former abstract away from DLs, allowing for general exter-nal queries, the latter deal with the uncertain, vague, and inconsistent nature of knowledge on the Web by means of fuzzy logic mechanisms. In this paper, we generalize both HEX programs and fuzzy dl-programs to fuzzy HEX programs: a LP-based paradigm, supporting both fuzziness as well as reasoning with exter-nal sources. We define basic syntax and semantics and analyze the framework semantically, e.g., by investigating the complexity. Additionally, we provide a translation from fuzzy HEX programs to HEX programs, enabling an implementa-tion via the dlvhex reasoner. Finally, we illustrate the use of fuzzy HEX programs for ranking services by using them to model non-functional properties of services and user preferences.

Query Rewriting with Symmetric Constraints

We address the problem of answering queries using expressive symmetric inter-schema constraints which allow to establish mappings between several heterogeneous information systems. This problem is of high relevance to data integration, as symmetric constraints are essential for dealing with true concept mismatch and are generalizations of the kinds of mappings supported by both local-as-view and global-as-view approaches that were previously studied in the literature. Moreover, the flexibility gained by using such constraints for data integration is essential for virtual enterprise and e-commerce applications. We first discuss resolution-based methods for computing maximally contained rewritings and characterize computability aspects. Then we propose an alternative but semantically equivalent perspective based on a generalization of results relating to the database-theoretic problem of answering queries using views. This leads to a fast query rewriting algorithm, which has been implemented and experimentally evaluated