Classical Predicative Logic-Enriched Type Theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes correcte

Complexity Results and Practical Algorithms for Logics in Knowledge Representation

Description Logics (DLs) are used in knowledge-based systems to represent and reason about terminological knowledge of the application domain in a semantically well-defined manner. In this thesis, we establish a number of novel complexity results and give practical algorithms for expressive DLs that provide different forms of counting quantifiers. We show that, in many cases, adding local counting in the form of qualifying number restrictions to DLs does not increase the complexity of the inference problems, even if binary coding of numbers in the input is assumed. On the other hand, we show that adding different forms of global counting restrictions to a logic may increase the complexity of the inference problems dramatically. We provide exact complexity results and a practical, tableau based algorithm for the DL SHIQ, which forms the basis of the highly optimized DL system iFaCT. Finally, we describe a tableau algorithm for the clique guarded fragment (CGF), which we hope will serve as the basis for an efficient implementation of a CGF reasoner.Comment: Ph.D. Thesi

On the Hybrid Extension of CTL and CTL+

The paper studies the expressivity, relative succinctness and complexity of satisfiability for hybrid extensions of the branching-time logics CTL and CTL+ by variables. Previous complexity results show that only fragments with one variable do have elementary complexity. It is shown that H1CTL+ and H1CTL, the hybrid extensions with one variable of CTL+ and CTL, respectively, are expressively equivalent but H1CTL+ is exponentially more succinct than H1CTL. On the other hand, HCTL+, the hybrid extension of CTL with arbitrarily many variables does not capture CTL*, as it even cannot express the simple CTL* property EGFp. The satisfiability problem for H1CTL+ is complete for triply exponential time, this remains true for quite weak fragments and quite strong extensions of the logic