Formalizing Abstract Algebra in Constructive Set Theory

We present a machine-checked formalization of elementary abstract algebra in constructive set theory. Our formalization uses an approach where we start by specifying the group axioms as a collection of inference rules, defining a logic for groups. Then we can tell whether a given set with a binary operation is a group or not, and derive all properties of groups constructively from these inference rules as well as the axioms of the set theory. The formalization of all other concepts in abstract algebra is based on that of the group. We give an example of a formalization of a concrete group, the Klein 4-group

Robust Computer Algebra, Theorem Proving, and Oracle AI

In the context of superintelligent AI systems, the term "oracle" has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.Comment: 15 pages, 3 figure

Classification of communication and cooperation mechanisms for logical and symbolic computation systems

The combination of logical and symbolic computation systems has recently emerged from prototype extensions of stand-alone systems to the study of environments allowing interaction among several systems. Communication and cooperation mechanisms of systems performing any kind of mathematical service enable to study and solve new classes of problems and to perform efficient computation by distributed specialized packages. The classification of communication and cooperation methods for logical and symbolic computation systems given in this paper provides and surveys different methodologies for combining mathematical services and their characteristics, capabilities, requirements, and differences. The methods are illustrated by recent well-known examples. We separate the classification into communication and cooperation methods. The former includes all aspects of the physical connection, the flow of mathematical information, the communication language(s) and its encoding, encryption, and ..

Emerging trends proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics: TPHOLs 2004

technical reportThis volume constitutes the proceedings of the Emerging Trends track of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004) held September 14-17, 2004 in Park City, Utah, USA. The TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research cate- gory, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the con- ference and publication in volume 3223 of Springer?s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster session. The work-in-progress papers are held in this volume, which is published as a 2004 technical report of the School of Computing at the University of Utah