The PIE Environment for First-Order-Based Proving, Interpolating and Eliminating

Abstract The PIE system aims at providing an environment for creating complex applications of automated first-order theorem proving techniques. It is embedded in Prolog. Beyond actual proving tasks, also interpolation and second-order quantifier elimination are supported. A macro feature and a L A T E X formula pretty-printer facilitate the construction of elaborate formalizations from small, understandable and documented units. For use with interpolation and elimination, preprocessing operations allow to preserve the semantics of chosen predicates. The system comes with a built-in default prover that can compute interpolants

Approximate Assertional Reasoning Over Expressive Ontologies

In this thesis, approximate reasoning methods for scalable assertional reasoning are provided whose computational properties can be established in a well-understood way, namely in terms of soundness and completeness, and whose quality can be analyzed in terms of statistical measurements, namely recall and precision. The basic idea of these approximate reasoning methods is to speed up reasoning by trading off the quality of reasoning results against increased speed

Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog

We propose a novel, type-elimination-based method for reasoning in the description logic SHIQbs including DL-safe rules. To this end, we first establish a knowledge compilation method converting the terminological part of an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which represents a canonical model. This OBDD can in turn be transformed into disjunctive Datalog and merged with the assertional part of the knowledge base in order to perform combined reasoning. In order to leverage our technique for full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that preserves satisfiability and entailment of positive and negative ground facts. The proposed technique is shown to be worst case optimal w.r.t. combined and data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for publication in Logical Methods in Computer Scienc

Application of Definability to Query Answering over Knowledge Bases

Answering object queries (i.e. instance retrieval) is a central task in ontology based data access (OBDA). Performing this task involves reasoning with respect to a knowledge base K (i.e. ontology) over some description logic (DL) dialect L. As the expressive power of L grows, so does the complexity of reasoning with respect to K. Therefore, eliminating the need to reason with respect to a knowledge base K is desirable. In this work, we propose an optimization to improve performance of answering object queries by eliminating the need to reason with respect to the knowledge base and, instead, utilizing cached query results when possible. In particular given a DL dialect L, an object query C over some knowledge base K and a set of cached query results S={S1, ..., Sn} obtained from evaluating past queries, we rewrite C into an equivalent query D, that can be evaluated with respect to an empty knowledge base, using cached query results S' = {Si1, ..., Sim}, where S' is a subset of S. The new query D is an interpolant for the original query C with respect to K and S. To find D, we leverage a tool for enumerating interpolants of a given sentence with respect to some theory. We describe a procedure that maps a knowledge base K, expressed in terms of a description logic dialect of first order logic, and object query C into an equivalent theory and query that are input into the interpolant enumerating tool, and resulting interpolants into an object query D that can be evaluated over an empty knowledge base. We show the efficacy of our approach through experimental evaluation on a Lehigh University Benchmark (LUBM) data set, as well as on a synthetic data set, LUBMMOD, that we created by augmenting an LUBM ontology with additional axioms

Independence in CLP Languages

Studying independence of goals has proven very useful in the context of logic programming. In particular, it has provided a formal basis for powerful automatic parallelization tools, since independence ensures that two goals may be evaluated in parallel while preserving correctness and eciency. We extend the concept of independence to constraint logic programs (CLP) and prove that it also ensures the correctness and eciency of the parallel evaluation of independent goals. Independence for CLP languages is more complex than for logic programming as search space preservation is necessary but no longer sucient for ensuring correctness and eciency. Two additional issues arise. The rst is that the cost of constraint solving may depend upon the order constraints are encountered. The second is the need to handle dynamic scheduling. We clarify these issues by proposing various types of search independence and constraint solver independence, and show how they can be combined to allow dierent optimizations, from parallelism to intelligent backtracking. Sucient conditions for independence which can be evaluated \a priori" at run-time are also proposed. Our study also yields new insights into independence in logic programming languages. In particular, we show that search space preservation is not only a sucient but also a necessary condition for ensuring correctness and eciency of parallel execution

Routines and Applications of Symbolic Algebra Software

Computing has become an essential resource in modern research and has found application across a wide range of scientific disciplines. Developments in symbolic algebra tools have been particularly valuable in physics where calculations in fields such as general relativity, quantum field theory and physics beyond the standard model are becoming increasing complex and unpractical to work with by hand. The computer algebra system Cadabra is a tensor-first approach to symbolic algebra based on the programming language Python which has been used extensively in research in these fields while also having a shallow learning curve making it an excellent way to introduce students to methods in computer algebra. The work in this thesis has been concentrated on developing Cadabra, which has involved looking at two different elements which make up a computer algebra program. Firstly, the implementation of algebraic routines is discussed. This has primarily been focused on the introduction of an algorithm for detecting the equivalence of tensorial expressions related by index permutation symmetries. The method employed differs considerably from traditional canonicalisation routines which are commonly used for this purpose by using Young projection operators to make such symmetries manifest. The other element of writing a computer algebra program which is covered is the infrastruc- ture and environment. The importance of this aspect of software design is often overlooked by funding committees and academic software users resulting in an anti-pattern of code not being shared and contributed to in the way in which research itself is published and promulgated. The focus in this area has been on implementing a packaging system for Cadabra which allows the writing of generic libraries which can be shared by the community, and interfacing with other scientific computing packages to increase the capabilities of Cadabra

High Level Efficiency in Database Languages

The subject of this Ph.D. thesis is the design and implementation of database languages. The thesis consists of five articles:  [1] Joan F. Boyar and Kim S. Larsen. Efficient Rebalancing of Chromatic Search Trees. In O. Nurmi and E. Ukkonen, eds., LNCS 621: Algorithm Theory -- SWAT'92 , pp. 151-164. Springer-Verlag, 1992. [2] Kim S. Larsen. On Aggregation and Computation on Domain Values. PB-414, Computer Science Department, Aarhus University, 1992. [3] Kim S. Larsen. Strategies for Expression Evaluation Using Sort-Merge Algorithms. PB-415, Computer Science Department, Aarhus University, 1992. [4] Kim S. Larsen and Michael I. Schwartzbach. Injectivity of Unary Queries With Computation on Domain Values. Computer Science Department, Aarhus University, 1992. Revised version of PB-311. [5] Kim S. Larsen, Michael I. Schwartzbach and Erik M. Schmidt. A New Formalism for Relational Algebra. IPL , 41(3):163-168, 1992. and this survey paper. In [5], a new query language design is proposed. The expressive power of the language is determined in [2] and all reasonable extensions are considered. In [3, 4], we focus on the optimization issue of avoiding unnecessary sorting of relations. The results in these papers are directly applicable to any algebra-based query language. In addition to the query language part, a database system also has to offer update facilities. The theory of standard tuple based updates is quite well developed in the sequential case. In [1], we discuss a new concurrent implementation of balanced search trees for that purpose.This survey paper describes the results of the papers which form the thesis, and relates these results to each other and to the area in a broader sense than is customary in the introductions of individual papers. The paper is intended to be read in combination with the papers on which it is based