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

Certified Computer Algebra on top of an Interactive Theorem Prover

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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 ..

Linking HOL Light to Mathematica using OpenMath

One of the most important benefits of using a theorem prover system is the absolute accuracy of the obtained result. However, solving mathematical problems often requires both deductive reasoning and algebraic computation. This issue is due to the fact that many real-life problems can be described with equations for which we cannot find easily symbolic (or closed-form) solutions and therefore we are not able to formalize them using the theorem prover. In other cases, some applications require well developed libraries and a deep knowledge of the theories to formalize simple expressions. A straightforward way to overcome these issues is the use of computer algebra systems or numerical approaches which are known to be the most efficient tools in symbolic computation. However, to preserve the soundness of the computation, the results of these systems should be formally verified. In this thesis, we present a general architecture to connect HOL Light, a higher-order logic theorem prover, to any mechanized mathematical system that supports the mathematical standard OpenMath. We implemented a prototype, called HolMatica, which links HOL Light to the computer algebra system Mathematica through OpenMath. We describe our implementation of a HOL Light translator which converts HOL Light statements into OpenMath object and vice-versa