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Automation of Higher-Order Logic [chapter]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
Besides the readers of this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki for proofreading the document.  ...  ., 2003) , a first-order presentation of (intensional) higher-order logic can be exploited to automate higher-order reasoning .  ...  In the last century, much of the work on the automation of higher-order logic focused on the automation of the elementary type theory.  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

Evaluation of Systems for Higher-order Logic (ESHOL)

Christoph Benzmüller, Florian Rabe, Carsten Schürmann, Geoff Sutcliffe
2008 Conference on Automated Deduction  
-The development of automated reasoning systems for higher-order logic (or fragments of it). -Promising application areas for automated higher-order reasoning systems.  ...  The notion of higher-order included, but was not limited to, ramified type theory, simple type theory, intuitionistic and constructive type theory, and logical frameworks.  ... 
dblp:conf/cade/BenzmullerRSS08 fatcat:dxtlbrmbvng3zadrqbhojnpelm

Self-Formalisation of Higher-Order Logic

Ramana Kumar, Rob Arthan, Magnus O. Myreen, Scott Owens
2016 Journal of automated reasoning  
We present a mechanised semantics for higher-order logic (HOL), and a proof of soundness for the inference system, including the rules for making definitions, implemented by the kernel of the HOL Light  ...  Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system.  ...  Acknowledgements We thank Mike Gordon for helpful comments on drafts of this paper. We are grateful to John Harrison and Freek Wiedijk for providing inspiration for this project.  ... 
doi:10.1007/s10817-015-9357-x fatcat:3dcm7pdklzcrhc37yryjzm3f7u

Analytic Tableaux for Higher-Order Logic with Choice

Julian Backes, Chad Edward Brown
2011 Journal of automated reasoning  
While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers so far have not.  ...  In order to support automated reasoning in the presence of a choice operator, we present a cut-free ground tableau calculus for Church's simple type theory with choice.  ...  Introduction Interactive theorem provers based on classical higher-order logic ( e.g., Isabelle-HOL [26], HOL88 [17], HOL-light [18], ProofPower [22] and HOL4 [28]) build in the axiom of choice by including  ... 
doi:10.1007/s10817-011-9233-2 fatcat:3k5t7oreonhgdmcsepqbb5krv4

Partial and Nested Recursive Function Definitions in Higher-order Logic

Alexander Krauss
2009 Journal of automated reasoning  
Based on inductive definitions, we develop a tool that automates the definition of partial recursive functions in higher-order logic (HOL) and provides appropriate proof rules for reasoning about them.  ...  Termination is modeled by an inductive domain predicate which follows the structure of the recursion.  ...  Jasmin Christian Blanchette, Amine Chaieb, Norbert Schirmer, and Christian Urban commented on drafts of this paper. The anonymous referees gave valuable feedback.  ... 
doi:10.1007/s10817-009-9157-2 fatcat:etpmyhp3c5hmlfs3nqd7q2m76u

Verifying Termination and Reduction Properties about Higher-Order Logic Programs

Brigitte Pientka
2005 Journal of automated reasoning  
We describe two checkers for verifying termination and reduction properties about higher-order logic programs.  ...  At the heart of both checkers lies an inference system to reason about structural properties which are described by higher-order subterm relations.  ...  Several automated methods to prove termination have been developed for first-order functional and logic programs in the past years (for example [3, 20, 9, 18, 31, 35] ).  ... 
doi:10.1007/s10817-005-6534-3 fatcat:aqoexkytzzdqlmvjgghrnd7mzy

Combining and automating classical and non-classical logics in classical higher-order logics

Christoph Benzmüller
2011 Annals of Mathematics and Artificial Intelligence  
We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.  ...  Numerous classical and non-classical logics can be elegantly embedded in Church's simple type theory, also known as classical higher-order logic.  ...  Geoff Sutcliffe, in collaboration with the author and supported by several further contributors, developed the new higher-order TPTP THF infrastructure (EU FP7 grant THFTPTP PIIF-GA-2008-219982).  ... 
doi:10.1007/s10472-011-9249-7 fatcat:4ppv6v36njcshd327mvguuj52a

Embedding and automating conditional logics in classical higher-order logic

Christoph Benzmüller, Dov Gabbay, Valerio Genovese, Daniele Rispoli
2012 Annals of Mathematics and Artificial Intelligence  
A sound and complete embedding of conditional logics into classical higher-order logic is presented.  ...  This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics.  ...  Acknowledgments: We thank the developers and contributors of the TPTP THF infrastructure and the implementors of the theorem provers and model finders employed in this work.  ... 
doi:10.1007/s10472-012-9320-z fatcat:vaqujhbg3nagnm2teabyefrxta

Implementing tactics and tacticals in a higher-order logic programming language

Amy Felty
1993 Journal of automated reasoning  
We present an extensive set of examples that have been implemented in the higher-order logic programming language Prolog.  ...  We argue that a logic programming language with a higher-order intuitionistic logic as its foundation can be used both to naturally specify and implement tactic style theorem provers.  ...  Pfenning 36] adopts LF as the logical foundation of the higher-order logic programming language Elf.  ... 
doi:10.1007/bf00881900 fatcat:plrrvswjzzhntakfcpmfvap75m

Progress in the Development of Automated Theorem Proving for Higher-Order Logic [chapter]

Geoff Sutcliffe, Christoph Benzmüller, Chad E. Brown, Frank Theiss
2009 Lecture Notes in Computer Science  
The Higher-order TPTP The development of the higher-order part of the TPTP involved three of the steps identified in the introduction: development of the Typed Higher-order Form (THF) part of the TPTP  ...  Until recently there has been no corresponding support in higher-order logic.  ...  Thanks to Andrei Tchaltsev and Alexandre Riazanov for developing the first-order parts of the Java parser for the TPTP language.  ... 
doi:10.1007/978-3-642-02959-2_8 fatcat:osm5sbcmnbg6nn3cdwqgunwkle

Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic [chapter]

David Fuenmayor, Christoph Benzmüller
2017 Lecture Notes in Computer Science  
A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented.  ...  IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel's God in order to discuss his emendation of Gödel's ontological argument  ...  Introduction The first part of this paper introduces a shallow semantic embedding of an intensional higher-order modal logic (IHOML) in classical higher-order logic (Isabelle/HOL 3 ).  ... 
doi:10.1007/978-3-319-67190-1_9 fatcat:tiflccsgsndsrgn5cpvweo4tdq

A general formal memory framework in Coq for verifying the properties of programs based on higher-order logic theorem proving with increased automation, consistency, and reusability [article]

Zheng Yang, Hang Lei
2018 arXiv   pre-print
in higher-order logic theorem proving assistant tools.  ...  Higher-order logic theorem proving is one of the most reliable technologies for verifying the properties of programs.  ...  operating environment of the GERM framework to facilitate automated higher-order logic theorem proving.  ... 
arXiv:1803.00403v3 fatcat:jpgpzx5zxnbpbpzfdj7kowgee4

Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure

Sutcliffe, Geoff; University Of Miami, Christoph; Articulate Software Benzmueller, European Community's Seventh Framework Programme FP7/2007-2013
2010 Journal of Formalized Reasoning  
logic, and the use of higher-order ATP in a range of domains.  ...  The extension of the TPTP from first-order form (FOF) logic to typed higher-order form (THF) logic has provided a basis for new development and application of ATP systems for higher-order logic.  ...  Thanks to Chad Brown for contributions of THF problems to the TPTP, developing the automated version of TPS, and implementing Satallax.  ... 
doi:10.6092/issn.1972-5787/1710 dblp:journals/jfrea/SutcliffeB10 fatcat:3j3xtlxvzbdjlofvl4ah62yuru

Model Checking the First-Order Fragment of Higher-Order Fixpoint Logic [chapter]

Roland Axelsson, Martin Lange
Logic for Programming, Artificial Intelligence, and Reasoning  
We present a model checking algorithm for HFL1, the firstorder fragment of Higher-Order Fixpoint Logic.  ...  This logic is capable of expressing many interesting properties which are not regular and, hence, not expressible in the modal μ-calculus.  ...  Here we use the (branching temporal) fixpoint logic HFL1, the first-order fragment of Higher-Order Fixpoint Logic (HFL) [9] .  ... 
doi:10.1007/978-3-540-75560-9_7 dblp:conf/lpar/AxelssonL07 fatcat:rklhn3frujhsljpbbcctlkffiu

Deduction Beyond First-Order Logic (Dagstuhl Seminar 17371)

Jasmin Christian Blanchette, Carsten Fuhs, Viorica Sofronie-Stokkermans, Cesare Tinelli, Marc Herbstritt
2018 Dagstuhl Reports  
The aim of the seminar was to bring together first-order automated reasoning experts and researchers working on deduction methods and tools that go beyond first-order logic.  ...  Other practical problems need a mixture of first-order proof search and some more advanced reasoning (for instance, about higher-order formulas), or simply higher-level reasoning steps.  ...  The proposed approach consists of over-approximations, underapproximations and their combination.  ... 
doi:10.4230/dagrep.7.9.26 dblp:journals/dagstuhl-reports/BlanchetteFST17 fatcat:yfc3wk2fwngc3i2dmjwrz3gsgi
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