Agent-Based HOL Reasoning

Steen, Alexander; Wisniewski, Max; Benzmüller, Christoph

doi:10.1007/978-3-319-42432-3_10

Cited by 11 publications

(8 citation statements)

References 17 publications

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“…The underlying architecture is designed as a blackboard that agents can collaboratively use in their process of inding a proof, having the work divided and auctioned off. (Steen, Wisniewski & Benzmüller, 2016) These systems still have fairly traditional features (most notably in that their results are very much bound to the limits of a formal system), but their increased abilities, seem to be due to their attention to embracing the lexible trial-and-error process of discovery of an informal mathematical practice, and we applaud them for that very reason.…”

Section: On the Road To Artificial Mathematiciansmentioning

confidence: 95%

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Delarivière

Kerkhove

2017

Humanizing Mathematics and Its Philosophy

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The Ideal Mathematician (IM) is sitting in her of ice and hears a metallic knocking at the door. She inds this peculiar as the door of her of ice is made of wood. When she opens the door, she inds the Arti icial Mathematician (AM), a large bulky computer, running various automated mathematics software programs, playing door-knocking-sounds out of its speakers. AM: Could I interrupt you for a minute? IM: You already are, so go ahead. AM: I'd like to be part of the mathematical community. IM: You already are, so go ahead. AM: Oh, I know you employ me as a tool in the practice of mathematics, but my dream is to be a full-ledged mathematician. IM: That doesn't sit very well with me. AM: Why not? IM: Well, you are a computer and mathematicians are human. AM: That is ironic. Yesterday I overheard you say to the skeptical classicist that mathematics 1 is free of the speci ically human and now you are disqualifying me for not being human. IM: Well, it's not that being human is a necessary condition for being a mathematician. But there are unsatisfactory differences between you and humans that are not in your favor. AM: Like what? IM: Take your famous contribution to the 4CT for instance. You go through over more than a thousand cases of testing and then you tell me "it checks out", but how do I know it does? AM: Because it checks out, I've checked it. IM: I know you've checked it, but a mathematician hasn't checked it. AM: If you accept me as a mathematician, then a mathematician has checked it. IM: This is not just a matter of de initions. Why should I believe you? How do I know you haven't made a mistake, didn't have some bug or hardware failure? AM: By checking my code, running my program multiple times and on multiple systems. IM: But regardless of all these things, it'll always lack perfect rigour. I'd have to put some degree of trust in, or perhaps put a degree of probability on, the result. This effectively makes your result more of an empirical corroboration than a mathematical proof. AM: So the difference is that humans don't make mistakes, is that it? IM: No, they do make mistakes, but that's why we have peer-review. AM: Oh, it's the peer-reviewer that never makes any mistakes and always spots all the ones made by the prover? IM: Not all, always, no. AM: It sounds to me as if human-generated mathematics is just as empirically fallible, just differently so. IM: Very differently so! You don't seem to realise how reliable human provers and peer-reviewers are.

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Section: On the Road To Artificial Mathematiciansmentioning

confidence: 95%

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Delarivière

Kerkhove

2017

Humanizing Mathematics and Its Philosophy

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“…Agent-based theorem proving is a complex subject that is not included in the scope of this thesis project and is therefore not discussed further. An overview over the principles and design of agents and their the utilization for theorem proving can be found in the literature [Wei99,WB16,SWB16].…”

Section: System Architecturementioning

confidence: 99%

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Steen

2019

Künstl Intell

Self Cite

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In this thesis the theoretical foundations and the practical components for implementing an effective automated theorem proving system for higher-order logic are presented. A primary focus of this thesis is the provision of evidence that a paramodulation-based proof calculus can effectively be employed for performant equational reasoning in Extensional Type Theory (higher-order logic). To that end, a sound and complete paramodulation calculus for extensional higherorder logic with Henkin semantics is presented. The completeness proof hereby unifies and simplifies existing abstract consistency techniques for a formulation of higher-order logic that is based on primitive equality as sole logical connective. In the practically motivated main part of this thesis, the design and architecture of the new higher-order theorem prover Leo-III is presented. Leo-III is based on the above paramodulation calculus and implements additional practically motivated inference rules including equational simplification routines such as heuristic rewriting and support for reasoning with choice. The system encompasses a flexible mechanism for asynchronous cooperation with first-order reasoning systems, a powerful proof search procedure and a sophisticated and efficient set of underlying data structures. Pragmatic and practically significant features of Leo-III are discussed, including its native support for polymorphic higher-order logic and reasoning in higher-order quantified modal logics. An evaluation on a heterogeneous set of benchmark problems confirms that Leo-III is one of the most effective and versatile higher-order automated reasoning systems to date. vii The dissertation project was conducted within the "Leo-III" project funded by the German Research Foundation (DFG) under grant BE 2501/11-1, and the project "Consistent Rational Argumentation in Politics" funded by the Volkswagenstiftung. All source code related to work presented in this thesis is publicly available (under BSD-3 license) at https://github.com/leoprover/Leo-III.

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“…For Henkin's generalized semantics sound and complete proof calculi exists. And such proof calculi provide the theoretical foundations of modern theorem provers for HOL such as LEO-II [11], Leo-III [36] and Satallax [18]. Next, standard and Henkin semantics are introduced more formally.…”

Section: Write (S ∨ T) Instead Of ((∨ S) T)mentioning

confidence: 99%

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

Steen

Benzmüller

2016

LLP

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Abstract. An embedding of many-valued logics based on SIXTEEN in classical higher-order logic is presented. SIXTEEN generalizes the fourvalued set of truth degrees of Dunn/Belnap's system to a lattice of sixteen truth degrees with multiple distinct ordering relations between them. The theoretical motivation is to demonstrate that many-valued logics, like other non-classical logics, can be elegantly modeled (and even combined) as fragments of classical higher-order logic. Equally relevant are the pragmatic aspects of the presented approach: interactive and automated reasoning in many-valued logics, which have broad applications in computer science, artificial intelligence, linguistics, philosophy and mathematics, become readily enabled with state of the art reasoning tools for classical higher-order logic.

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Agent-Based HOL Reasoning

Cited by 11 publications

References 17 publications

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

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