Manfred Kerber

Computational Logic — CL 2000

Syntax for Variable Binders: An Overview . . . . . . . . . . . . . . . . . . . . 239 Dale Miller Goal-Directed Proof Search in Multiple-Conclusioned Intuitionistic Logic . . 254 James Harland, Tatjana Lutovac, and Michael Winikoff Efficient EM Learning with Tabulation for Parameterized Logic Programs . 269 Yoshitaka Kameya and Taisuke Sato Model Generation Theorem Proving with Finite Interval Constraints . . . . . 285 Reiner Hähnle, Ryuzo Hasegawa, and Yasuyuki Shirai Combining Mobile Processes and Declarative Programming . . . . . . . . . . . . . 300 Rachid Echahed and Wendelin Serwe

Omega: Towards a Mathematical Assistant

Ωmega is a mixed-initiative system with the ultimate purpose of supporting theorem proving in main-stream mathematics and mathematics education. The current system consists of a proof planner and an integrated collection of tools for formulating problems, proving subproblems, and proof presentation.

A Mechanization of Strong Kleene Logic for Partial Functions

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

Combined reasoning by automated cooperation

Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of first-order and higher-order techniques. First-order reasoning systems, on the one hand, have reached considerable strength in some niches, but in many areas of mathematics they still cannot reliably solve relatively simple problems, for example, when reasoning about sets, relations, or functions. Higher-order reasoning systems, on the other hand, can solve problems of this kind automatically. But the complexity inherent in their calculi prevents them from solving a whole range of problems. However, while many problems cannot be solved by any one system alone, they can be solved by a combination of these systems. We present a general agent-based methodology for integrating different reasoning systems. It provides a generic integration framework which facilitates the cooperation between diverse reasoners, but can also be refined to enable more efficient, specialist integrations. We empirically evaluate its usefulness, effectiveness and efficiency by case studies involving the integration of first-order and higher-order automated theorem provers, computer algebra systems, and model generators.

Ω-MKRP: A proof development environment

This report presents the main ideas underlyingtheOmegaGamma mkrp-system, an environmentfor the development of mathematical proofs. The motivation for the development ofthis system comes from our extensive experience with traditional first-order theoremprovers and aims to overcome some of their shortcomings. After comparing the benefitsand drawbacks of existing systems, we propose a system architecture that combinesthe positive features of different types of theorem-proving systems, most notably theadvantages of human-oriented systems based on methods (our version of tactics) andthe deductive strength of traditional automated theorem provers.In OmegaGamma mkrp a user first states a problem to be solved in a typed and sorted higher-order language (called POST ) and then applies natural deduction inference rules inorder to prove it. He can also insert a mathematical fact from an integrated data-base into the current partial proof, he can apply a domain-specific problem-solvingmethod, or he can call an integrated automated theorem prover to solve a subprob-lem. The user can also pass the control to a planning component that supports andpartially automates his long-range planning of a proof. Toward the important goal ofuser-friendliness, machine-generated proofs are transformed in several steps into muchshorter, better-structured proofs that are finally translated into natural language.This work was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D2, D3)

Intuitive and Formal Representations: The Case of Matrices

A major obstacle for bridging the gap between textbook mathematics and formalising it on a computer is the problem how to adequately capture the intuition inherent in the mathematical notation when formalising mathematical concepts. While logic is an excellent tool to represent certain mathematical concepts it often fails to retain all the information implicitly given in the representation of some mathematical objects. In this paper we concern ourselves with matrices, whose representation can be particularly rich in implicit information. We analyse different types of matrices and present a mechanism that can represent them very close to their textbook style appearance and captures the information contained in this representation but that nevertheless allows for their compilation into a formal logical framework. This firstly allows for a more human-oriented interface and secondly enables efficient reasoning with matrices.

What Makes a Problem Hard for XCS

Despite two decades of work learning classifier systems researchers have had relatively little to say on the subject of what makes a problem difficult for a classifier system. Wilsons accuracy-based XCS, a promising and increasingly popular classifier system, is, we feel, the natural first choice of classifier system with which to address this issue. To make the task more tractable we limit our considerations to a restricted, but very important, class of problems. Most significantly, we consider only single step reinforcement learning problems and the use of the standard binary/ternary classifier systems language. In addition to distinguishing several dimensions of problem complexity for XCS, we consider their interactions, identify bounding cases of difficulty, and consider complexity metrics for XCS. Based on these results we suggest a simple template for ternary single step test suites to more comprehensively evaluate classifier systems.

Experiments with an Agent-Oriented Reasoning System

This paper discusses experiments with an agent oriented approach to automated and interactive reasoning. The approach combines ideas from two subfields of AI (theorem proving/proof planning and multi-agent systems) and makes use of state of the art distribution techniques to decentralise and spread its reasoning agents over the internet. It particularly supports cooperative proofs between reasoning systems which are strong in different application areas, e.g., higher-order and first-order theorem provers and computer algebra systems.

A Tableau Calculus for Partial Functions

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

A qualitative comparison of the suitability of four theorem provers for basic auction theory

Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machine-checked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickreys 1961 theorem on the properties of second-price auctions. Based on our formalisation experience, taking an auction designers perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox.