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

Steen, Alexander; Benzmüller, Christoph

doi:10.12775/llp.2016.021

Cited by 8 publications

(10 citation statements)

References 34 publications

(39 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The practical evidence that quantified modal logics can effectively be modeled as a fragment of classical higher-order logic furthermore suggests that higherorder logic can serve as an universal meta-logic [Ben17b]. This claim is further substantiated by more recent evaluations [BOR12,GSB17,SB18] and the observation that there exist analogous reductions for numerous further non-classical logics [BS16,SB16,Ben17a,Ben17c,BFP18], for many of which there exist none or only few specialized reasoning systems. A strong foundation for the automation of higher-order logic and an effective implementation of a corresponding deduction system thus enables computer-assisted reasoning in even more, practically relevant, logical systems and application areas.…”

Section: Motivationmentioning

confidence: 90%

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

Steen

2019

Künstl Intell

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Motivationmentioning

confidence: 90%

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

Steen

2019

Künstl Intell

Self Cite

View full text Add to dashboard Cite

show abstract

“…Other logics, for which the SSE approach applies, and which are relevant for theoretical philosophy, include quantified conditional logics and multi-valued logic [5,4,35].…”

mentioning

confidence: 99%

Recent Successes with a Meta-Logical Approach to Universal Logical Reasoning (Extended Abstract)

Benzmüller

2017

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

The quest for a most general framework supporting universal reasoning is very prominently represented in the works of Leibniz. He envisioned a scientia generalis founded on a characteristica universalis, that is, a most universal formal language in which all knowledge about the world and the sciences can be encoded. A quick study of the survey literature on logical formalisms suggests that quite the opposite to Leibniz' dream has become reality. Instead of a characteristica universalis, we are today facing a very rich and heterogenous zoo of different logical systems, and instead of converging towards a single superior logic, this logic zoo is further expanding, eventually even at accelerated pace. As a consequence, the unified vision of Leibniz seems farther away than ever before. However, there are also some promising initiatives to counteract these diverging developments. Attempts at unifying approaches to logic include categorial logic algebraic logic and coalgebraic logic.My own research draws on another alternative at universal logical reasoning: the shallow semantical embeddings (SSE) approach. This approach has a very pragmatic motivation, foremost reuse of tools, simplicity and elegance. It utilises classical higherorder logic [22] as a unifying meta-logic in which the syntax and semantics of varying other logics can be explicitly modeled and flexibly combined (cf.[6] and the references therein). Off-the-shelf higher-order interactive and automated theorem provers [7] can then be employed to reason about and within the shallowly embedded logics.Respective experiments have e.g. been conducted in metaphysics. An initial focus thereby has been on computer-supported assessments of modern variants of the ontological argument for the existence of God, where the SSE approach has been utilised in particular for automating variants of higher-order (multi-)modal logics [9].In the course of these experiments (cf. [17,15,16,18,19,14] [20]. Further modern variants of the argument have subsequently been studied with the approach, and theorem provers have even contributed to the clarification of an unsettled philosophical dispute [13].Another, more ambitious study has focused on Ed Zalta's Principia Logico-Metaphysica (PLM) [38], which aims at a foundational logical theory for all of metaphysics and the sciences. This includes mathematics, and in this sense it is more ambitious than Russel's Principia Mathematica. The semantical embedding of PLM in HOL has been very challenging, since in addition to its size, its foundational theory is complicated: the PLM is based on hyperintensional higher-order modal logic S5 defined on top of a relational (as opposed to a functional) type theory that comes with restricted comprehension principles (the use of full comprehension in the PLM has been known to cause para-

show abstract

“…Shallow semantical embeddings into HOL have been studied for various other non-classical logics, including conditional logics [4], hybrid logic [38], intuitionistic logics [7] and more recently, free logics [10] and many-valued logics [33]. All these approaches yield means of automation for the respective non-classical logic.…”

Section: Introductionmentioning

confidence: 99%

Theorem Provers For Every Normal Modal Logic

Gleißner

Steen

Benzmüller

EPiC Series in Computing

Self Cite

View full text Add to dashboard Cite

We present a procedure for algorithmically embedding problems formulated in higherorder modal logic into classical higher-order logic. The procedure was implemented as a stand-alone tool and can be used as a preprocessor for turning TPTP THF-compliant theorem provers into provers for various modal logics. The choice of the concrete modal logic is thereby specified within the problem as a meta-logical statement. This specification format as well as the underlying semantics parameters are discussed, and the implementation and the operation of the tool are outlined.By combining our tool with one or more THF-compliant theorem provers we accomplish the most widely applicable modal logic theorem prover available to date, i.e. no other available prover covers more variants of propositional and quantified modal logics. Despite this generality, our approach remains competitive, at least for quantified modal logics, as our experiments demonstrate.

show abstract

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

Cited by 8 publications

References 34 publications

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

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

Recent Successes with a Meta-Logical Approach to Universal Logical Reasoning (Extended Abstract)

Theorem Provers For Every Normal Modal Logic

Contact Info

Product

Resources

About