Restricted Combinatory Unification

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First-order theorem provers are commonly utilised as backends to
proof assistants. In order to improve efficiency, it is desirable that such provers
can carry out some higher-order reasoning. In his 1991 paper, Dougherty proposed a combinatory unification algorithm for higher-order logic. The algorithm removes the need to deal with ∧-binders and α-renaming, making it attractive to implement in first-order provers. However, since publication it has garnered little interest due to its poor characteristics. It fails to terminate on many trivial instances and requires polymorphism.We present a restricted version of Dougherty’s algorithm that is incomplete, terminating and does not require polymorphism. Further, we describe its implementation in the Vampire theorem prover, including a novel use of a substitution tree as a filtering index for higher-order unification. Finally, we analyse the performance of the algorithm on two benchmark sets and show that it is a significant step forward.
Original languageEnglish
Title of host publication Automated Deduction – CADE 27
ISBN (Electronic)978-3-030-29436-6
Publication statusE-pub ahead of print - 20 Aug 2019
EventThe 27th International Conference on Automated Deduction - UFRN/Hotel Praiamar, Natal, Brazil
Duration: 23 Aug 201930 Aug 2019

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


ConferenceThe 27th International Conference on Automated Deduction
Abbreviated titleCADE27


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