Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. Several algorithms that solve unification in EL have previously been presented. In this paper, we summarize recent extensions of these algorithms that can deal with general concept inclusion axioms (GCIs), role hierarchies (H), and transitive roles (R). For the algorithms to be complete, however, the ontology consisting of the GCIs and role axioms needs to satisfy a certain cycle restriction.
{"title":"Recent Advances in Unification for the EL Family","authors":"F. Baader, Stefan Borgwardt, Barbara Morawska","doi":"10.29007/q5px","DOIUrl":"https://doi.org/10.29007/q5px","url":null,"abstract":"Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. Several algorithms that solve unification in EL have previously been presented. In this paper, we summarize recent extensions of these algorithms that can deal with general concept inclusion axioms (GCIs), role hierarchies (H), and transitive roles (R). For the algorithms to be complete, however, the ontology consisting of the GCIs and role axioms needs to satisfy a certain cycle restriction.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129380593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss the use of type systems in a non-strict sense when designing uni cation algorithms. We rst give a new (rule-based) algorithm for an equational theory which represents a property of El-Gamal signature schemes and show how a type system can be used to prove termination of the algorithm. Lastly, we reproduce a termination result for theory of partial exponentiation given earlier.
{"title":"The use of types in designing unification algorithms: two case studies","authors":"Serdar Erbatur, Santiago Escobar, P. Narendran","doi":"10.29007/lbk5","DOIUrl":"https://doi.org/10.29007/lbk5","url":null,"abstract":"We discuss the use of type systems in a non-strict sense when designing uni cation algorithms. We rst give a new (rule-based) algorithm for an equational theory which represents a property of El-Gamal signature schemes and show how a type system can be used to prove termination of the algorithm. Lastly, we reproduce a termination result for theory of partial exponentiation given earlier.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126057648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a procedure for the bounded unication of higher-order terms [24]. The procedure extends G. P. Huet’s pre-unication procedure [11] with rules for the generation and folding of regular terms. The concise form of the procedure allows the reuse of the pre-unication correctness proof. Furthermore, the regular terms can be restricted in order to get a decidable uniability problem. Finally, the procedure avoids re-computation of terms in a non-deterministic search which leads to a better performance in practice when compared to other bounded unication algorithms.
我们提出了一个高阶项的有界统一过程[24]。该过程扩展了G. P. Huet的预统一过程[11]，其中包含正则项的生成和折叠规则。该程序的简洁形式允许重用统一前的正确性证明。此外，还可以对正则项进行限制，从而得到一个可判定的不确定性问题。最后，该方法避免了在不确定搜索中重新计算项，与其他有界通信算法相比，在实践中具有更好的性能。
{"title":"Bounded Higher-order Unification using Regular Terms","authors":"Tomer Libal","doi":"10.29007/zhpc","DOIUrl":"https://doi.org/10.29007/zhpc","url":null,"abstract":"We present a procedure for the bounded unication of higher-order terms [24]. The procedure extends G. P. Huet’s pre-unication procedure [11] with rules for the generation and folding of regular terms. The concise form of the procedure allows the reuse of the pre-unication correctness proof. Furthermore, the regular terms can be restricted in order to get a decidable uniability problem. Finally, the procedure avoids re-computation of terms in a non-deterministic search which leads to a better performance in practice when compared to other bounded unication algorithms.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133222131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kimberly A. Gero, Christopher Bouchard, P. Narendran
Unification modulo convergent term rewrite systems is an important research area with many applications. In their seminal paper Lynch and Morawska gave three conditions on rewrite systems that guarantee that unifiability can be checked in polynomial time (P). We show that these conditions are tight, in the sense that relaxing any one of them will “upset the applecart,” giving rise to unification problems that are not in P (unless P = NP), and in doing so address an open problem posed by Lynch and Morawska. We also investigate a related decision problem: we show the undecidability of subterm-collapse for the restricted term rewriting systems that we are considering.
{"title":"Some Notes on Basic Syntactic Mutation","authors":"Kimberly A. Gero, Christopher Bouchard, P. Narendran","doi":"10.29007/sdp1","DOIUrl":"https://doi.org/10.29007/sdp1","url":null,"abstract":"Unification modulo convergent term rewrite systems is an important research area with many applications. In their seminal paper Lynch and Morawska gave three conditions on rewrite systems that guarantee that unifiability can be checked in polynomial time (P). We show that these conditions are tight, in the sense that relaxing any one of them will “upset the applecart,” giving rise to unification problems that are not in P (unless P = NP), and in doing so address an open problem posed by Lynch and Morawska. We also investigate a related decision problem: we show the undecidability of subterm-collapse for the restricted term rewriting systems that we are considering.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115949598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-06-09DOI: 10.1007/978-3-642-38574-2_16
Serdar Erbatur, Santiago Escobar, D. Kapur, Zhiqiang Liu, C. Lynch, C. Meadows, J. Meseguer, P. Narendran, Sonia Santiago, R. Sasse
{"title":"Asymmetric Unification: A New Unification Paradigm for Cryptographic Protocol Analysis","authors":"Serdar Erbatur, Santiago Escobar, D. Kapur, Zhiqiang Liu, C. Lynch, C. Meadows, J. Meseguer, P. Narendran, Sonia Santiago, R. Sasse","doi":"10.1007/978-3-642-38574-2_16","DOIUrl":"https://doi.org/10.1007/978-3-642-38574-2_16","url":null,"abstract":"","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"293 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123113677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the Tiden and Arnborg algorithm for equational unification modulo one-sided distributivity is not polynomial time bounded as previously thought. A set of counterexamples is developed that demonstrates that the algorithm goes through exponentially many steps.
{"title":"On the Complexity of the Tiden-Arnborg Algorithm for Unification modulo One-Sided Distributivity","authors":"P. Narendran, Andrew M. Marshall, B. Mahapatra","doi":"10.4204/EPTCS.42.5","DOIUrl":"https://doi.org/10.4204/EPTCS.42.5","url":null,"abstract":"We prove that the Tiden and Arnborg algorithm for equational unification modulo one-sided distributivity is not polynomial time bounded as previously thought. A set of counterexamples is developed that demonstrates that the algorithm goes through exponentially many steps.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"151 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133305888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present formalized proofs verifying that the first-order unification algorithm defined over lists of satisfiable constraints generates a most general unifier (MGU), which also happens to be idempotent. All of our proofs have been formalized in the Coq theorem prover. Our proofs show that finite maps produced by the unification algorithm provide a model of the axioms characterizing idempotent MGUs of lists of constraints. The axioms that serve as the basis for our verification are derived from a standard set by extending them to lists of constraints. For us, constraints are equalities between terms in the language of simple types. Substitutions are formally modeled as finite maps using the Coq library Coq.FSets.FMapInterface. Coq's method of functional induction is the main proof technique used in proving many of the axioms.
{"title":"A Machine Checked Model of Idempotent MGU Axioms For Lists of Equational Constraints","authors":"Sunil Kothari, J. Caldwell","doi":"10.4204/EPTCS.42.3","DOIUrl":"https://doi.org/10.4204/EPTCS.42.3","url":null,"abstract":"We present formalized proofs verifying that the first-order unification algorithm defined over lists of satisfiable constraints generates a most general unifier (MGU), which also happens to be idempotent. All of our proofs have been formalized in the Coq theorem prover. Our proofs show that finite maps produced by the unification algorithm provide a model of the axioms characterizing idempotent MGUs of lists of constraints. The axioms that serve as the basis for our verification are derived from a standard set by extending them to lists of constraints. For us, constraints are equalities between terms in the language of simple types. Substitutions are formally modeled as finite maps using the Coq library Coq.FSets.FMapInterface. Coq's method of functional induction is the main proof technique used in proving many of the axioms.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125640908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.
{"title":"Towards Correctness of Program Transformations Through Unification and Critical Pair Computation","authors":"Conrad Rau, M. Schmidt-Schauß","doi":"10.4204/EPTCS.42.4","DOIUrl":"https://doi.org/10.4204/EPTCS.42.4","url":null,"abstract":"Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124826436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement protocols. In our earlier work, we gave many decidability as well as undecidability results for multiple equational theories, involving various properties of modular exponentiation. Here, we consider a partial subtheory focussing only on exponentiation and multiplication operators. Two main results are proved. The first result is positive, namely, that the unification problem for the above theory (in which no additional property is assumed of the multiplication operators) is decidable. The second result is negative: if we assume that the two multiplication operators belong to two different abelian groups, then the unification problem becomes undecidable.
{"title":"Unification modulo a partial theory of exponentiation","authors":"D. Kapur, Andrew M. Marshall, P. Narendran","doi":"10.4204/EPTCS.42.2","DOIUrl":"https://doi.org/10.4204/EPTCS.42.2","url":null,"abstract":"Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement protocols. In our earlier work, we gave many decidability as well as undecidability results for multiple equational theories, involving various properties of modular exponentiation. Here, we consider a partial subtheory focussing only on exponentiation and multiplication operators. Two main results are proved. The first result is positive, namely, that the unification problem for the above theory (in which no additional property is assumed of the multiplication operators) is decidable. The second result is negative: if we assume that the two multiplication operators belong to two different abelian groups, then the unification problem becomes undecidable.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134516744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nominal unification calculates substitutions that make terms involving binders equal modulo alpha-equivalence. Although nominal unification can be seen as equivalent to Miller's higher-order pattern unification, it has properties, such as the use of first-order terms with names (as opposed to alpha-equivalence classes) and that no new names need to be generated during unification, which set it clearly apart from higher-order pattern unification. The purpose of this paper is to simplify a clunky proof from the original paper on nominal unification and to give an overview over some results about nominal unification.
{"title":"Nominal Unification Revisited","authors":"Christian Urban","doi":"10.4204/EPTCS.42.1","DOIUrl":"https://doi.org/10.4204/EPTCS.42.1","url":null,"abstract":"Nominal unification calculates substitutions that make terms involving binders equal modulo alpha-equivalence. Although nominal unification can be seen as equivalent to Miller's higher-order pattern unification, it has properties, such as the use of first-order terms with names (as opposed to alpha-equivalence classes) and that no new names need to be generated during unification, which set it clearly apart from higher-order pattern unification. The purpose of this paper is to simplify a clunky proof from the original paper on nominal unification and to give an overview over some results about nominal unification.","PeriodicalId":164988,"journal":{"name":"UNIF","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129520037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：