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

User-friendly Support for Common Concepts in a Lightweight Verifier

Machine verification of formal arguments can only increase our confidence in the correctness of those arguments, but the costs of employing machine verification still outweigh the benefits for some common kinds of formal reasoning activities. As a result, usability is becoming increasingly important in the design of formal verification tools. We describe the "aartifact" lightweight verification system, designed for processing formal arguments involving basic, ubiquitous mathematical concepts. The system is a prototype for investigating potential techniques for improving the usability of formal verification systems. It leverages techniques drawn both from existing work and from our own efforts. In addition to a parser for a familiar concrete syntax and a mechanism for automated syntax lookup, the system integrates (1) a basic logical inference algorithm, (2) a database of propositions governing common mathematical concepts, and (3) a data structure that computes congruence closures of expressions involving relations found in this database. Together, these components allow the system to better accommodate the expectations of users interested in verifying formal arguments involving algebraic and logical manipulations of numbers, sets, vectors, and related operators and predicates. We demonstrate the reasonable performance of this system on typical formal arguments and briefly discuss how the system's design contributed to its usability in two case studies

A novel approach to symbolic algebra

A prototype for an extensible interactive graphical term manipulation system is presented that combines pattern matching and nondeterministic evaluation to provide a convenient framework for doing tedious algebraic manipulations that so far had to be done manually in a semi-automatic fashion.Comment: 15 page

A Survey of User Interfaces for Computer Algebra Systems

AbstractThis paper surveys work within the Computer Algebra community (and elsewhere) directed towards improving user interfaces for scientific computation during the period 1963–1994. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. Work related to improving human interaction with computer algebra systems is the main focus of the paper. However, the paper includes additional materials on some closely related issues such as structured document editing, graphics, and communication protocols