A Spectrum of Applications of Automated Reasoning

The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it offers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include finding a first proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert's twenty-fourth problem--which is extremely amenable to attack with the appropriate automated reasoning program--a problem concerned with proof simplification. The methodologies include those for seeking shorter proofs and for finding proofs that avoid unwanted lemmas or classes of term, a specific option for seeking proofs with smaller equational or formula complexity, and a different option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbert-style axiomatic, including details that permit one sometimes to gain new insights. We include questions still open and challenges that merit consideration.Comment: 13 page

Double-Negation Elimination in Some Propositional Logics

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Lukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program OTTER played an indispensable role in this study.Comment: 32 pages, no figure

Graph Based Reduction of Program Verification Conditions

Increasing the automaticity of proofs in deductive verification of C programs is a challenging task. When applied to industrial C programs known heuristics to generate simpler verification conditions are not efficient enough. This is mainly due to their size and a high number of irrelevant hypotheses. This work presents a strategy to reduce program verification conditions by selecting their relevant hypotheses. The relevance of a hypothesis is determined by the combination of a syntactic analysis and two graph traversals. The first graph is labeled by constants and the second one by the predicates in the axioms. The approach is applied on a benchmark arising in industrial program verification

Lemmas: Generation, Selection, Application

Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates useful lemmas for automated theorem provers, demonstrating improvement for several representative systems and solving a hard problem not solved by any system for twenty years. By focusing on condensed detachment problems we simplify the setting considerably, allowing us to get at the essence of lemmas and their role in proof search

Investigations into Proof Structures

We introduce and elaborate a novel formalism for the manipulation and analysis of proofs as objects in a global manner. In this first approach the formalism is restricted to first-order problems characterized by condensed detachment. It is applied in an exemplary manner to a coherent and comprehensive formal reconstruction and analysis of historical proofs of a widely-studied problem due to {\L}ukasiewicz. The underlying approach opens the door towards new systematic ways of generating lemmas in the course of proof search to the effects of reducing the search effort and finding shorter proofs. Among the numerous reported experiments along this line, a proof of {\L}ukasiewicz's problem was automatically discovered that is much shorter than any proof found before by man or machine.Comment: This article is a continuation of arXiv:2104.1364

Larry Wos - Visions of automated reasoning

This paper celebrates the scientific discoveries and the service to the automated reasoning community of Lawrence (Larry) T. Wos, who passed away in August 2020. The narrative covers Larry's most long-lasting ideas about inference rules and search strategies for theorem proving, his work on applications of theorem proving, and a collection of personal memories and anecdotes that let readers appreciate Larry's personality and enthusiasm for automated reasoning

Ideological Interpretation and the Aesthetic Nature of Literature

Recent literary theory has assumed that literary works are reflections of the dominant ideological thought present within culture. This thesis addresses the question of what role the aesthetic nature of a literary work plays in determining its own ideological meaning. Terry Eagleton claims that literature and literary theory have reinforced the dominant political modes of thought, as the aesthetic properties have merely masked this ideological function, and must be disregarded in order to reveal the true ideological nature contained within a work. Deriving his framework from Jaques Lacan, Slavoj Zizek argues that ideology is pervasive through all levels of culture, inseparable from fundamental understanding. He further reveals a divide between the Symbolic order which contains conceptualized understanding in language, and the Real which escapes such definition. Umberto Eco argues that interpretation must remain coherent with the totality of the work in order to avoid overinterpretation, which uses the work as an example of something external to itself. Alan Sinfield argues that literature reveals “faultlines” within a culture, and allows dissident views to engage with dominant ideologies. Mikhail Bakhtin’s view of dialogism suggests that any literary expression will contain multiple views or ideologies, including internally persuasive ones which can be contained within aesthetic expression, and external authoritative ones which remain detached from artistic nature. This thesis argues that defining a work of literature as a particular example of an ideological position disregards the aesthetic qualities it possesses and reduces it to an indistinguishable cultural artifact which is used as an example of culture without recognizing the unique experience that it provides as a narrative. Such attempts diminish the continuing possibilities of meaning that a work can generate through its ambiguous nature. Rather, the aesthetic qualities of literature allow it to provide multiple meanings which surpass reduction to a specific ideological position, even as it necessarily possesses ideological content. Final ideological conclusions are deferred, and interpretative freedom is provided for the reader to reach individual conclusions in the aesthetic space beyond Symbolic conceptualized thought which allows a remainder of meaning to escape ideology as a hint of the Real

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

OpenCog Hyperon: A Framework for AGI at the Human Level and Beyond

An introduction to the OpenCog Hyperon framework for Artificiai General Intelligence is presented. Hyperon is a new, mostly from-the-ground-up rewrite/redesign of the OpenCog AGI framework, based on similar conceptual and cognitive principles to the previous OpenCog version, but incorporating a variety of new ideas at the mathematical, software architecture and AI-algorithm level. This review lightly summarizes: 1) some of the history behind OpenCog and Hyperon, 2) the core structures and processes underlying Hyperon as a software system, 3) the integration of this software system with the SingularityNET ecosystem's decentralized infrastructure, 4) the cognitive model(s) being experimentally pursued within Hyperon on the hopeful path to advanced AGI, 5) the prospects seen for advanced aspects like reflective self-modification and self-improvement of the codebase, 6) the tentative development roadmap and various challenges expected to be faced, 7) the thinking of the Hyperon team regarding how to guide this sort of work in a beneficial direction ... and gives links and references for readers who wish to delve further into any of these aspects