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# Extensional Higher-Order Paramodulation in Leo-III

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## Abstract and Figures

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling, e.g., proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in many quantified normal modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.
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Journal of Automated Reasoning (2021) 65:775–807
https://doi.org/10.1007/s10817-021-09588-x
Extensional Higher-Order Paramodulation in Leo-III
Alexander Steen1·Christoph Benzmüller1,2
Received: 26 July 2019 / Accepted: 8 March 2021 / Published online: 27 March 2021
© The Author(s), under exclusive licence to Springer Nature B.V. 2021
Abstract
Leo-III is an automated theorem prover for extensional type theory with Henkin semantics
and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based
proof search to higher-order logic. The prover may cooperate with multiple external specialist
reasoning systems such as ﬁrst-order provers and SMT solvers. Leo-III is compatible with
the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized
communication between reasoning systems, enabling, e.g., proof reconstruction from within
proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic ﬁrst-order
and higher-order logic, in many quantiﬁed normal modal logics, as well as in different deontic
logics. Its development had initiated the ongoing extension of the TPTP infrastructure to
reasoning within non-classical logics.
Keywords Higher-Order logic ·Henkin semantics ·Extensionality ·Leo-III ·Equational
reasoning ·Automated theorem proving ·Non-classical logics ·Quantiﬁed modal logics
1 Introduction
Leo-III is an automated theorem prover (ATP) for classical higher-order logic (HOL) with
Henkin semantics and choice. In contrast to its predecessors, LEO and LEO-II [25,33], that
were based on resolution proof search, Leo-III implements a higher-order paramodulation
calculus, which aims at improved performance for equational reasoning [89]. In the tradition
of the Leo prover family, Leo-III collaborates with external reasoning systems, in particular,
with ﬁrst-order ATP systems, such as E [85], iProver [70]andVampire[83], and with SMT
solvers such as CVC4 [14]. Cooperation is not restricted to ﬁrst-order systems, and further
specialized systems such as higher-order (counter)model ﬁnders may be utilized by Leo-III.
This work has been supported by the DFG under Grant BE 2501/11-1 (Leo-III) and by the
Volkswagenstiftung (“Consistent Rational Argumentation in Politics”).
BAlexander Steen
alexander.steen@uni.lu
Christoph Benzmüller
c.benzmueller@fu-berlin.de
1University of Luxembourg, FSTM, Esch-sur-Alzette, Luxembourg
2Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
123
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... The basic calculus of Satallax is a complete ground tableau calculus [5,6,2]. In recent years the top systems of the THF division of CASC are primarily based on resolution and superposition [11,3,8]. At the moment it is an open question whether there is a research and development path via which a tableau based prover could again become competitive. ...
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