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Rule-Based Unification in Combined Theories and the Finite Variant Property [chapter]

Ajay K. Eeralla, Serdar Erbatur, Andrew M. Marshall, Christophe Ringeissen
2019 Lecture Notes in Computer Science  
These theories are also known in the context of protocol analysis as theories with the finite variant property and admit a variant-based unification algorithm.  ...  In this paper, we present a new rule-based unification algorithm which can be seen as an alternative to the variant-based approach.  ...  When the equational theory E has the Finite Variant Property (FVP) [8] , there exists a reduction from E-unification to syntactic unification via the computation of finitely many variants of the unification  ... 
doi:10.1007/978-3-030-13435-8_26 fatcat:cjiw2gix55dbjk7aljows4ib5e

Variants, Unification, Narrowing, and Symbolic Reachability in Maude 2.6

Francisco Duran, Steven Eker, Santiago Escobar, Jose Meseguer, Carolyn Talcott, Marc Herbstritt
2011 International Conference on Rewriting Techniques and Applications  
The equational theory (Ax ∪ E) has the finite variant property iff there is a finite set of most general variants.  ...  , and (iv) narrowing-based symbolic reachability modulo finite variant theories.  ...  We are very thankful to the other members of the Maude team, namely, Manuel Clavel, Patrick Lincoln, and Narciso Martí-Oliet, with whom we have designed and built the Maude language and system.  ... 
doi:10.4230/lipics.rta.2011.31 dblp:conf/rta/DuranEEMT11 fatcat:zqh55brfcna45bt43enmxm3rli

Theories of Homomorphic Encryption, Unification, and the Finite Variant Property

Fan Yang, Santiago Escobar, Catherine Meadows, José Meseguer, Paliath Narendran
2014 Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming - PPDP '14  
This requires, among other things, that the theory be decomposable into a set of axioms B and a set of rewrite rules R such that R has the finite variant property with respect to B.  ...  However, this application requires unification algorithms that can be easily implemented and easily extended to combinations of different theories of interest.  ...  Both the first and second desiderata of unification-based cryptographic can be achieved, if the decomposition (R, B) has the finite variant property, via variant unification as described in Section 2.  ... 
doi:10.1145/2643135.2643154 dblp:conf/ppdp/YangEMMN14 fatcat:ss5awwz6wffnnlp6i67rg5svcu

Most General Variant Unifiers

Santiago Escobar, Julia Sapiña
2019 Electronic Proceedings in Theoretical Computer Science  
Variant-based unification is decidable when the equational theory satisfies the finite variant property.  ...  Our experiments suggest that this new adaptation of the variant-based unification is more efficient both in execution time and in the number of computed variant unifiers than the original algorithm available  ...  An example is the unification under homomorphism (or one-side distributivity), where there is a finite number of unifiers of two terms but the theory does not satisfy the finite variant property (see  ... 
doi:10.4204/eptcs.306.21 fatcat:fg744co2bff5nm5cme5nbao2ru

Asymmetric Unification: A New Unification Paradigm for Cryptographic Protocol Analysis [chapter]

Serdar Erbatur, Santiago Escobar, Deepak Kapur, Zhiqiang Liu, Christopher A. Lynch, Catherine Meadows, José Meseguer, Paliath Narendran, Sonia Santiago, Ralf Sasse
2013 Lecture Notes in Computer Science  
We present a new paradigm for unification arising out of a technique commonly used in cryptographic protocol analysis tools that employ unification modulo equational theories.  ...  We also give results on the complexity and decidability of asymmetric unification.  ...  analysis that combines both cancellation rules and a non-trivial theory E in the decomposition (Σ, E, R).  ... 
doi:10.1007/978-3-642-38574-2_16 fatcat:yom3pf444rbntievnplrn4zo6i

Variant Narrowing and Equational Unification

Santiago Escobar, José Meseguer, Ralf Sasse
2009 Electronical Notes in Theoretical Computer Science  
, and yields a finitary E-unification algorithm when E has the finite variant property.  ...  We furthermore identify a class of equational theories for which the finite bound ensuring the finite variant property can be effectively computed by a generic algorithm.  ...  We would like to thank Hubert Comon and Stephanie Delaune for useful discussion on the finite variant problem.  ... 
doi:10.1016/j.entcs.2009.05.015 fatcat:r6idb5eyz5bd7bt3gi6wxgnomi

Folding Variant Narrowing and Optimal Variant Termination [chapter]

Santiago Escobar, Ralf Sasse, José Meseguer
2010 Lecture Notes in Computer Science  
And it is optimally variant terminating in the sense of terminating for an input term t iff t has a finite, complete set of variants.  ...  In this work we propose an effective strategy based on the idea of the E ∪ Ax-variants of a term that we call folding variant narrowing.  ...  Definition 8 (Finite variant property In [8] we developed a technique to check whether an equational theory has the finite variant property.  ... 
doi:10.1007/978-3-642-16310-4_5 fatcat:6iiy27jwwneqbaomsnjkgdeb7u

Folding variant narrowing and optimal variant termination

Santiago Escobar, Ralf Sasse, José Meseguer
2012 The Journal of Logic and Algebraic Programming  
And it is optimally variant terminating in the sense of terminating for an input term t iff t has a finite, complete set of variants.  ...  In this work we propose an effective strategy based on the idea of the E ∪ Ax-variants of a term that we call folding variant narrowing.  ...  Definition 8 (Finite variant property In [8] we developed a technique to check whether an equational theory has the finite variant property.  ... 
doi:10.1016/j.jlap.2012.01.002 fatcat:4bphqzvazjdgzbpfmcb6o63oqm

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.  ...  user and open-source developer base.  ...  exponentiation, and so on, which often have the finite variant property [9] .  ... 
doi:10.1007/978-3-642-02348-4_27 fatcat:adxberqkxrhnviqr4ce2aflcfy

The Finite Variant Property: How to Get Rid of Some Algebraic Properties [chapter]

Hubert Comon-Lundh, Stéphanie Delaune
2005 Lecture Notes in Computer Science  
For instance, we prove that the finite variant property holds for Abelian Groups, and a theory of modular exponentiation and does not hold for the theory ACUNh (Associativity, Commutativity, Unit, Nilpotence  ...  The goal of this paper is to give equivalent (resp. sufficient) conditions for the finite variant property and to systematically investigate this property for equational theories, which are relevant to  ...  Narendran and the anonymous referees who gave relevant comments which helped in improving the paper.  ... 
doi:10.1007/978-3-540-32033-3_22 fatcat:vlv7h2etlvh25gjmrjpaub2swm

On Asymmetric Unification and the Combination Problem in Disjoint Theories [chapter]

Serdar Erbatur, Deepak Kapur, Andrew M. Marshall, Catherine Meadows, Paliath Narendran, Christophe Ringeissen
2014 Lecture Notes in Computer Science  
If the combination of two theories with the finite variant property also has the finite variant property, then one applies variant narrowing.  ...  One important question is the problem of asymmetric unification in a combination of theories, in particular how to produce an algorithm for the combined theory by combining algorithms for the separate  ... 
doi:10.1007/978-3-642-54830-7_18 fatcat:l23piq2shbb6lcevfd53wxjuka

Termination Modulo Combinations of Equational Theories [chapter]

Francisco Durán, Salvador Lucas, José Meseguer
2009 Lecture Notes in Computer Science  
Our method is based on the notion of variants of a term recently proposed by Comon and Delaune.  ...  Rewriting with rules R modulo axioms E is a widely used technique in both rule-based programming languages and in automated deduction.  ...  Durán and S. Lucas were partially supported by Spanish MEC grants TIN 2008-03107 and TIN 2007-68093-C02, respectively. J. Meseguer was partially supported by NSF grants CNS 07-16638 and CNS 08-31064.  ... 
doi:10.1007/978-3-642-04222-5_15 fatcat:wqihagyaazf2vfof6vpgw3ukmi

Building and Combining Matching Algorithms [chapter]

Christophe Ringeissen
2019 Lecture Notes in Computer Science  
The concept of matching is ubiquitous in declarative programming and in automated reasoning.  ...  For instance, it is a key mechanism to run rule-based programs and to simplify clauses generated by theorem provers.  ...  For the class of finite syntactic theories, we present a rule-based matching algorithm along the lines of the classical rule-based syntactic unification algorithm.  ... 
doi:10.1007/978-3-030-22102-7_24 fatcat:6nx7ollvnzchrcuacrpa522f6q

Variant-Based Satisfiability in Initial Algebras [chapter]

José Meseguer
2016 Communications in Computer and Information Science  
for the initial algebras of its user-specified input theories when such theories satisfy Comon-Delaune's finite variant property (FVP) and some extra conditions.  ...  in such combinations.  ...  I thank Andrew Cholewa, Steven Eker, Santiago Escobar, Ralf Sasse, and Carolyn Talcott for their contributions to the development of the theory and Maude implementation of folding variant narrowing.  ... 
doi:10.1007/978-3-319-29510-7_1 fatcat:d67mbh76efafllryccq4drbon4

Functional Logic Programming in Maude [chapter]

Santiago Escobar
2014 Lecture Notes in Computer Science  
Functional logic programming languages combine the most important features of functional programming languages and logic programming languages.  ...  As a practical application we use the Missionaries and Cannibals equational logic program given by Goguen and Meseguer for Eqlog in the eighties.  ...  When the theory also satisfies the finite variant property [19] , a finitary and complete unification algorithm based on a narrowing strategy called folding variant narrowing is provided in [19] .  ... 
doi:10.1007/978-3-642-54624-2_16 fatcat:5xvsjyk56net3azx4mb7rjzpqq
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