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Page 4109 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
In this paper we develop a sorted version of type theory; we ex- tend the syntax and semantics of simple type theory by a higher order sort concept that includes term declarations.  ...  {For the entire collection see MR 94f:68008.} 94g:68097 68T15 03B15 03B35 68Q55 Kohlhase, Michael (D-SAAR; Saarbriicken) Unification in order-sorted type theory.  ... 

Extending the Resolution Method with Sorts

Christoph Weidenbach
1993 International Joint Conference on Artificial Intelligence  
In this paper I extend the standard first-order resolution method with special reasoning mechanisms for sorts. Sorts are unary predicates.  ...  Sorted unification exploits the sort restrictions of variables with respect to the sort theory.  ...  Lemma 6 Sorted unification is undecidable and of unification type infinitary. Lemma 6 is a worst case result.  ... 
dblp:conf/ijcai/Weidenbach93 fatcat:bepvvgezzbe7dd6ajo7t56oqxa

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

1996 Mathematical Reviews  
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.  ...  A spe- cial chapter is devoted to higher-order unification, unification in sort theories and constraint solving.  ... 

Automated Theorem Proving in First-Order Logic Modulo: On the Difference between Type Theory and Set Theory [chapter]

Gilles Dowek
2000 Lecture Notes in Computer Science  
When it is applied to some first-order presentations of type theory, it simulates exactly higherorder resolution. In this note, we compare how it behaves on type theory and on set theory.  ...  Resolution modulo is a first-order theorem proving method that can be applied both to first-order presentations of simple type theory (also called higher-order logic) and to set theory.  ...  Resolution modulo in set theory In set theory, there is no rule rewriting terms to terms. Hence, unification in set theory is simply first-order unification.  ... 
doi:10.1007/3-540-46508-1_1 fatcat:pddj5d3slzbcfbor7f63unxyki

Page 8134 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
the combinatorial maps in the calculus of inductive con- structions (CIC), a variety of type theory well suited for mechaniz- ing mathematics in higher-order logic.  ...  In the third section, we mention some unification results for specific equational theories.  ... 

Unification and Narrowing in Maude 2.4 [chapter]

Manuel Clavel, Francisco Durán, Steven Eker, Santiago Escobar, Patrick Lincoln, Narciso Martí-Oliet, José Meseguer, Carolyn Talcott
2009 Lecture Notes in Computer Science  
Unification is supported in Core Maude, the core rewriting engine of Maude, with commands and metalevel functions for order-sorted unification modulo some frequently occurring equational axioms.  ...  This paper introduces novel features of Maude 2.4 including support for unification and narrowing.  ...  Order-sorted unification is NP-complete in general because Boolean algebra can be encoded as an order-sorted free theory signature and hence satisfiability can be reduced to an order-sorted free theory  ... 
doi:10.1007/978-3-642-02348-4_27 fatcat:adxberqkxrhnviqr4ce2aflcfy

Equational Cryptographic Reasoning in the Maude-NRL Protocol Analyzer

Santiago Escobar, Catherine Meadows, José Meseguer
2007 Electronical Notes in Theoretical Computer Science  
Order-sorted B-unification plays a crucial role; to obtain this functionality we describe a sort propagation algorithm that filters out unsorted B-unifiers provided by the CiME unification tool.  ...  In this paper we extend our framework to include theories of the form ∆ B, where B is the theory of associativity and commutativity and ∆ is convergent modulo B.  ...  Order-sorted Unification Modulo Equations As the protocol is specified using an order-sorted theory with AC axioms, narrowing requires an order-sorted AC unification procedure.  ... 
doi:10.1016/j.entcs.2007.02.053 fatcat:76iv5kvvqzesjclji3zrrizqve

Unification in sort theories and its applications

Christoph Weidenbach
1996 Annals of Mathematics and Artificial Intelligence  
In this language sorted unification in elementary sort theories is of unification type finitary.  ...  Abstract In this article I investigate the properties of unification in sort theories. The usual notion of a sort consisting of a sort symbol is extended to a set of sort symbols.  ...  A sort theory L is of type: The Sorted Unification Algorithm Definition 4.15 (Notions for Sorted Unification) In order to define sorted unification, we have to extend Definition 3.1.  ... 
doi:10.1007/bf02127750 fatcat:gdyscp3ddvepnokebhmwtqs434

Order-sorted Equational Unification Revisited

Joe Hendrix, José Meseguer
2012 Electronical Notes in Theoretical Computer Science  
This paper presents a rule-based algorithm for performing order-sorted E-unification using an unsorted E-unification decision procedure under assumptions about E that are commonly satisfied in practice  ...  In many examples of interest, using order-sorted unification over unsorted unification is able to reduce the total number of unifiers considered, and dramatically improve the performance of the Maude-NRL  ...  When those subterms are well-typed with the same topmost sort k, substitutions generated by the unsorted unification procedure should be well-typed as well Provided that the sorts of fresh variables in  ... 
doi:10.1016/j.entcs.2012.11.010 fatcat:uqqx5qefcbccvnbq2i62lwzptm

Unification theory

Jörg H. Siekmann
1989 Journal of symbolic computation  
This article surveys what is presently known in unification theory and records its early history. 20g  ...  Unification theory provides the formal framework for investigations into the properties of this operation.  ...  theory can be used to solve unification problems in the sorted equational theory.  ... 
doi:10.1016/s0747-7171(89)80012-4 fatcat:mjwsvawtirdvbc5dxdflp4n2zi

Unification in many-sorted algebras as a device for incremental semantic analysis

Gregor Snelting, Wolfgang Henhapl
1986 Proceedings of the 13th ACM SIGACT-SIGPLAN symposium on Principles of programming languages - POPL '86  
During editing, these rules are evaluated using a unification algorithm for many-sorted algebras with semi-lattice ordered subsorts and non-empty equational theories.  ...  Since the general problem is not solvable, we have developed an extension of our unification in order to be able to handle arbitrary equational theories, which works correct with The basic idea is as follows  ...  Nons y n t a c t i c a t t r i b u t e f l o w in languagebased e d i t o r s . Proc. 9th ACH symposium on p r i n c i p l e s of programming languages, 1982, pp,196-205 Habermann, N. et  ... 
doi:10.1145/512644.512665 dblp:conf/popl/SneltingH86 fatcat:vxvbiqhzevhbvf75nehbn2se6e

Special issue on automated deduction: Decidability, complexity, tractability

Silvio Ghilardi, Viorica Sofronie-Stokkermans, Ulrike Sattler, Ashish Tiwari
2010 Journal of symbolic computation  
The main result in this paper is a characterization of a decidable fragment of many-sorted first-order logics important in applications.  ...  In the paper Decidable Fragments of Many-Sorted Logic, Aharon Abadi, Alexander Rabinovich and Mooly Sagiv, study a decidable fragment of many-sorted first-order logic that allows the expression of a large  ... 
doi:10.1016/j.jsc.2009.05.006 fatcat:ojjvruw2jza6tpamkmmgyfk3cq

A Typed Hybrid Description Logic Programming Language with Polymorphic Order-Sorted DL-Typed Unification for Semantic Web Type Systems

Adrian Paschke
2007 arXiv   pre-print
term typing of LP rules based on a polymorphic, order-sorted, hybrid DL-typed unification as procedural semantics of hybrid DLPs.  ...  In this paper we elaborate on a specific application in the context of hybrid description logic programs (hybrid DLPs), namely description logic Semantic Web type systems (DL-types) which are used for  ...  The order-sorted unification yields the term of the two sorts (types) in the given sort hierarchy.  ... 
arXiv:cs/0610006v2 fatcat:szfxxjpbvzej5jq6mjgbrgnmka

Page 2967 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews  
In or- der to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is in- troduced.  ...  We show that saturation under sorted superposi- tion is effective on sorted shallow equational theories.  ... 

Many-Sorted Unification [chapter]

Christoph Walther
1987 A Many-Sorted Calculus Based on Resolution and Paramodulation  
Many-sorted unification is considered; that is, unification in the many-sorted free algebras of terms, where variables, as well as the domains and ranges of functions, are restricted to certain subsets  ...  An algorithm for many-sorted unification is given.  ...  This paper is concerned with unification in many-sorted empty equational theories.  ... 
doi:10.1016/b978-0-273-08718-2.50012-2 fatcat:bbah2qnfvzbahid7x456s6mjpy
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