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Page 2190 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews  
This similarity has been exploited, for instance, in order to prove decidability of monadic second- order unification and decidability of linear second-order unifica- tion when no second-order variable  ...  Summary: “Higher order matching consists of solving finite sets of equations of the form u =z, v, where u,v are simply typed terms of A-calculus, and v is a closed term.  ... 

Efficient Full Higher-Order Unification

Petar Vukmirović, Alexander Bentkamp, Visa Nummelin, Zena M. Ariola
2020 International Conference on Formal Structures for Computation and Deduction  
We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski.  ...  Our unification procedure is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.  ...  Acknowledgements We are grateful to the maintainers of StarExec for letting us use their service.  ... 
doi:10.4230/lipics.fscd.2020.5 dblp:conf/fscd/VukmirovicBN20 fatcat:quwuu37yrnfl5jyd5c52kwliti

Linear Higher-Order Matching Is NP-Complete [chapter]

Philippe de Groote
2000 Lecture Notes in Computer Science  
We consider the problem of higher-order matching restricted to the set of linear λ-terms (i.e., λ-terms where each abstraction λx. M is such that there is exactly one free occurrence of x in M ).  ...  We prove that this problem is decidable by showing that it belongs to NP. Then we prove that this problem is in fact NP-complete. Finally, we discuss some heuristics for a practical algorithm.  ...  Rusinowitch for interesting discussions about some of the material in this paper.  ... 
doi:10.1007/10721975_9 fatcat:a6y3c5ebj5exjapfphk4wuhxci

Page 2413 of Mathematical Reviews Vol. , Issue 96d [page]

1996 Mathematical Reviews  
Summary: “Let E be a first-order equational theory. A translation of higher-order E-unification problems into a combinatory logic framework is presented and justified.  ...  higher-order E- unifiers.  ... 

Simplifying the signature in second-order unification

Jordi Levy, Mateu Villaret
2009 Applicable Algebra in Engineering, Communication and Computing  
Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.  ...  In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one binary function symbol and constants.  ...  Bounded Second-Order Unification [21, 29] is a decidable variant of Second-Order Unification.  ... 
doi:10.1007/s00200-009-0106-4 fatcat:ltrscory3nb7bdrfjy5jjoof2q

Higher-order and semantic unification [chapter]

Nachum Dershowitz, Subrata Mitra
1993 Lecture Notes in Computer Science  
in a complete higher-order unification procedure for R.  ...  Furthermore, we use the system of transformation rules to provide a syntactic characterization for R which results in decidability of semantic unification.  ...  Discussion Higher-order semantic unification is of interest in automated theorem proving, type inferencing, higher-order extensions of logic-programming, etc.  ... 
doi:10.1007/3-540-57529-4_49 fatcat:6plogqpu4nbo7p7b6gqqluacm4

Variant-Based Decidable Satisfiability in Initial Algebras with Predicates [article]

Raúl Gutiérrez, José Meseguer
2017 arXiv   pre-print
Variant satisfiability is a theory-generic procedure for quantifier-free satisfiability in the initial algebra of an order-sorted equational theory (Σ,E ∪ B) under two conditions: (i) E ∪ B has the finite  ...  variant property and B has a finitary unification algorithm; and (ii) (Σ,E ∪ B) protects a constructor subtheory (Ω,E_Ω∪ B_Ω) that is OS-compact.  ...  Obviously, the bigger the decidable subtheory T 0 Ď T , the higher the levels of automation and the greater the chances of scaling up the verification effort.  ... 
arXiv:1709.05203v1 fatcat:ew7a2v7ne5fobecruhbiglac64

Page 3465 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews  
Summary: “We study a higher order unification procedure based on proof nets.  ...  Neculai Curteanu (R-IASI-SM;: lasi 99e:68143 68T15 03B35 03F50 Matsuoka, Satoshi (J-NITE-ECE; Nagoya) Towards a higher order unification based on proof nets.  ... 

Page 1117 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
A spe- cial chapter is devoted to higher-order unification, unification in sort theories and constraint solving.  ...  The main emphasis is given to first-order (theory) unification, but the paper also contains substantial information on higher-order unification and unification in sort theories.  ... 

Nominal Unification from a Higher-Order Perspective

Jordi Levy, Mateu Villaret
2012 ACM Transactions on Computational Logic  
From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification.  ...  This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification.  ...  From a higher-order perspective, nominal unification can be seen as a variant of higher-order unification where: 1. variables are all first-order typed, and constants are of order at most three (therefore  ... 
doi:10.1145/2159531.2159532 fatcat:taumfjush5gz7c4on2yyy3aoxu

Variant-Based Decidable Satisfiability in Initial Algebras with Predicates [chapter]

Raúl Gutiérrez, José Meseguer
2018 Lecture Notes in Computer Science  
Keywords: finite variant property (FVP), OS-compactness, user-definable predicates, decidable validity and satisfiability in initial algebras.  ...  variant property and B has a finitary unification algorithm; and (ii) pΣ, E Y Bq protects a constructor subtheory pΩ, EΩ Y BΩq that is OS-compact.  ...  Obviously, the bigger the decidable subtheory T 0 Ď T , the higher the levels of automation and the greater the chances of scaling up the verification effort.  ... 
doi:10.1007/978-3-319-94460-9_18 fatcat:t35d4mbmrbdidk25ywi5vtuloy

Higher-order unification revisited: Complete sets of transformations

Wayne Snyder, Jean Gallier
1989 Journal of symbolic computation  
In this paper, we reexamine the problem of general higher-order unification and develop an approach based on the method of transformations on systems of terms which has its roots in Herbrand's thesis,  ...  proof of the completeness of a method for higher-order unification than has previously been available; and, finally, we have shown the completeness of the strategy of eager variable elimination.  ...  ~(b)) = 0(~), The basic decidability results concerning higher-order unification are as follows.  ... 
doi:10.1016/s0747-7171(89)80023-9 fatcat:v3icruent5eurja2gwclo3pti4

System Description: Twelf — A Meta-Logical Framework for Deductive Systems [chapter]

Frank Pfenning, Carsten Schürmann
1999 Lecture Notes in Computer Science  
Twelf is a meta-logical framework for the specification, implementation, and meta-theory of deductive systems from the theory of programming languages and logics.  ...  It is a significant extension and complete reimplementation of the Elf system [Pfe94]. Twelf is written in Standard ML and runs under SML of New Jersey and MLWorks on Unix and Window platforms.  ...  We would like to thank Iliano Cervesato for his contributions to the logic programming component of Twelf.  ... 
doi:10.1007/3-540-48660-7_14 fatcat:ysgd47xcbzbybccjwjaecilgke

A Generic Framework for Higher-Order Generalizations

David M. Cerna, Temur Kutsia, Michael Wagner
2019 International Conference on Rewriting Techniques and Applications  
Changing the particular values of the parameter, we obtained four new unitary variants of higher-order anti-unification and also showed how the already known pattern generalization fits into the schema  ...  We consider a generic framework for anti-unification of simply typed lambda terms.  ...  Cs-variant of higher-order anti-unification is unitary. Proof.  ... 
doi:10.4230/lipics.fscd.2019.10 dblp:conf/rta/CernaK19 fatcat:q4tv5q5navgz5ndpzapjnoat7q

Towards a Substitution Tree Based Index for Higher-order Resolution Theorem Provers

Tomer Libal, Alexander Steen
2016 Conference on Automated Deduction  
Our attempt tries to handle two difficulties which arise when extending the indexes to higher-order. The first difficulty is the need for higher-order anti-unification.  ...  In this paper we describe an attempt to extend such indexes for the use of higher-order resolution theorem provers.  ...  Acknowledgements Work of the first author was funded by the ERC Advanced Grant ProofCert. The second author has been supported by the DFG under grant BE 2501/11-1 (Leo-III).  ... 
dblp:conf/cade/LibalS16 fatcat:s3vzmt34gbekbaivcja73czzoi
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