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

A uniform architecture for parsing and generation

The use of a single grammar for both parsing and generation is an idea with a certain elegance, the desirability of which several researchers have noted. In this paper, we discuss a more radical possibility: not only can a single grammar be used by different processes engaged in various "directions" of processing, but one and the same language-processing architecture can be used for processing the grammar in the various modes. In particular, parsing and generation can be viewed as two processes engaged in by a single parameterized theorem prover for the logical interpretation of the formalism. We discuss our current implementation of such an architecture, which is parameterized in such a way that it can be used for either purpose with grammars written in the PATR formalism. Furthermore, the architecture allows fine tuning to reflect different processing strategies, including parsing models intended to mimic psycholinguistic phenomena. This tuning allows the parsing system to operate within the same realm of efficiency as previous architectures for parsing alone, but with much greater flexibility for engaging in other processing regimes.Engineering and Applied Science

An Approach to Assertion Application via Generalised Resolution

In this paper we address assertion retrieval and application in theorem proving systems or proof planning systems for classical first-order logic. Due to Huang the notion of assertion comprises mathematical knowledge such as definitions, theorems, and axioms. We propose a distributed mediator module between a mathematical knowledge base KB and a theorem proving system TP which is independent of the particular proof representation format of TP and which applies generalised resolution in order to analyze the logical consequences of arbitrary assertions for a proof context at hand. Our approach is applicable also to the assumptions which are dynamically created during a proof search process. It therefore realises a crucial first step towards full automation of assertion level reasoning. We discuss the benefits and connection of our approach to proof planning and motivate an application in a project aiming at a tutorial dialogue system for mathematics

A Formally Verified Prover for the ALC Description Logic

The Ontology Web Language (OWL) is a language used for the Semantic Web. OWL is based on Description Logics (DLs), a family of logical formalisms for representing and reasoning about conceptual and terminological knowledge. Among these, the logic ALC is a ground DL used in many practical cases. Moreover, the Semantic Web appears as a new field for the application of formal methods, that could be used to increase its reliability. A starting point could be the formal verification of satisfiability provers for DLs. In this paper, we present the PVS specification of a prover for ALC , as well as the proofs of its termination, soundness and completeness. We also present the formalization of the well–foundedness of the multiset relation induced by a well–founded relation. This result has been used to prove the termination and the completeness of the ALC prover.Ministerio de Educación y Ciencia TIN2004–0388

On the Notion of Interestingness in Automated Mathematical Discovery

Deciding whether something is interesting or not is of central importance in automated mathematical discovery, as it helps determine both the search space and search strategy for finding and evaluating concepts and conjectures

Extracting proofs from documents

Often, theorem checkers like PVS are used to check an existing proof, which is part of some document. Since there is a large difference between the notations used in the documents and the notations used in the theorem checkers, it is usually a laborious task to convert an existing proof into a format which can be checked by a machine. In the system that we propose, the author is assisted in the process of converting an existing proof into the PVS language and having it checked by PVS. 1 Introduction The now-classic ALGOL 60 report [5] recognized three different levels of language: a reference language, a publication language and several hardware representations, whereby the publication language was intended to admit variations on the reference language and was to be used for stating and communicating processes. The importance of publication language ---often referred to nowadays as "pseudo-code"--- is difficult to exaggerate since a publication language is the most effective way..