A complete unification algorithm is presented for the combination of two arbitrary equational theories E in T(F,X) and E/sup 1/ in T(F',X), where F and F' denote two disjoint sets of function symbols. The method adapts to unification of infinite trees. It is applied to two well-known open problems, when E is the theory of Boolean rings or the theory of Abelian groups, and E is the free theory. The interest to Boolean rings originates in VSLI verification.<>
给出了两个任意方程理论E in T(F,X)和E/sup 1/ in T(F'，X)组合的完全统一算法，其中F和F'表示两个不相交的函数符号集。该方法适用于无限树的统一。它应用于两个著名的开放问题，当E是布尔环理论或阿贝尔群理论，E是自由理论。对布尔环的兴趣起源于VSLI验证。
{"title":"Unification in free extensions of Boolean rings and Abelian groups","authors":"Alexandre Boudet, J. Jouannaud, M. Schmidt-Schauß","doi":"10.1109/LICS.1988.5110","DOIUrl":"https://doi.org/10.1109/LICS.1988.5110","url":null,"abstract":"A complete unification algorithm is presented for the combination of two arbitrary equational theories E in T(F,X) and E/sup 1/ in T(F',X), where F and F' denote two disjoint sets of function symbols. The method adapts to unification of infinite trees. It is applied to two well-known open problems, when E is the theory of Boolean rings or the theory of Abelian groups, and E is the free theory. The interest to Boolean rings originates in VSLI verification.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123521382","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}
The author develops an operational model of an impredicative version of explicit polymorphism, namely, an extension of the second-order lambda-calculus, including a fixed-point combinator and a multisorted first-order algebra. He shows that the typical properties of the lambda -calculus are preserved, and he investigates novel aspects that arise from second-order type structure as well as the relationships with well-known, simply typed, PCF-like languages. A suitable theory of observably equivalent terms is defined, and its complexity is explored. A denotional model suggesting interesting extensions of a language is studied.<>
{"title":"A fixed point of the second order lambda-calculus: observable equivalences and models","authors":"R. Amadio","doi":"10.1109/LICS.1988.5100","DOIUrl":"https://doi.org/10.1109/LICS.1988.5100","url":null,"abstract":"The author develops an operational model of an impredicative version of explicit polymorphism, namely, an extension of the second-order lambda-calculus, including a fixed-point combinator and a multisorted first-order algebra. He shows that the typical properties of the lambda -calculus are preserved, and he investigates novel aspects that arise from second-order type structure as well as the relationships with well-known, simply typed, PCF-like languages. A suitable theory of observably equivalent terms is defined, and its complexity is explored. A denotional model suggesting interesting extensions of a language is studied.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132709572","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}
The composition and extraction mechanisms of Petri nets are at present inadequate. This problem is solved by viewing place/transition Petri nets as ordinary, directed graphs equipped with two algebraic operations corresponding to parallel and sequential composition of transitions. A distributive law between the two operations captures a basic fact about concurrency. Novel morphisms are defined, mapping single, atomic transitions into whole computations, thus relating system descriptions at different levels of abstraction. Categories equipped with products and coproducts (corresponding to parallel and nondeterministic compositions) are introduced for Petri nets with and without initial markings. It is briefly indicated how the approach yields function spaces and novel interpretations of duality and invariants. The results provide a formal basis for expressing the semantics of concurrent languages in terms of Petri nets and an understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models and contribute to the conceptual unification of concurrency.<>
{"title":"Petri nets are monoids: a new algebraic foundation for net theory","authors":"J. Meseguer, U. Montanari","doi":"10.1109/LICS.1988.5114","DOIUrl":"https://doi.org/10.1109/LICS.1988.5114","url":null,"abstract":"The composition and extraction mechanisms of Petri nets are at present inadequate. This problem is solved by viewing place/transition Petri nets as ordinary, directed graphs equipped with two algebraic operations corresponding to parallel and sequential composition of transitions. A distributive law between the two operations captures a basic fact about concurrency. Novel morphisms are defined, mapping single, atomic transitions into whole computations, thus relating system descriptions at different levels of abstraction. Categories equipped with products and coproducts (corresponding to parallel and nondeterministic compositions) are introduced for Petri nets with and without initial markings. It is briefly indicated how the approach yields function spaces and novel interpretations of duality and invariants. The results provide a formal basis for expressing the semantics of concurrent languages in terms of Petri nets and an understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models and contribute to the conceptual unification of concurrency.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"195 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114249235","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}
ML/sup +/ is an extension of the functional language ML that allows the actual parameters of recursively called functions to have types that are generic instances of the (derived) types of corresponding formal parameters. It is shown that the polymorphism allowed by the original ML can be eliminated without loss of computational power, specifically, it is shown that its computational power (in all interpretations) is the same as that of finitely typed functional programs. It is proved that the polymorphism of ML/sup +/ cannot be eliminated, in that its computational power far exceeds that of finitely typed functional programs and therefore that of the original ML too.<>
{"title":"On the computational power of universally polymorphic recursion","authors":"A. Kfoury, J. Tiuryn, P. Urzyczyn","doi":"10.1109/LICS.1988.5102","DOIUrl":"https://doi.org/10.1109/LICS.1988.5102","url":null,"abstract":"ML/sup +/ is an extension of the functional language ML that allows the actual parameters of recursively called functions to have types that are generic instances of the (derived) types of corresponding formal parameters. It is shown that the polymorphism allowed by the original ML can be eliminated without loss of computational power, specifically, it is shown that its computational power (in all interpretations) is the same as that of finitely typed functional programs. It is proved that the polymorphism of ML/sup +/ cannot be eliminated, in that its computational power far exceeds that of finitely typed functional programs and therefore that of the original ML too.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126049248","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}
An abstract framework is proposed for the description of the type dependency semantics. It is claimed that the notion of fibration introduced by A. Grothendieck in the 1960s is perfectly adapted to this goal and provides the greatest simplicity and generality. This semantics is extended to higher order, and an explanation is given of what a general definition for the semantics of the theory of constructions could be.<>
{"title":"A categorical semantics of constructions","authors":"T. Ehrhard","doi":"10.1109/LICS.1988.5125","DOIUrl":"https://doi.org/10.1109/LICS.1988.5125","url":null,"abstract":"An abstract framework is proposed for the description of the type dependency semantics. It is claimed that the notion of fibration introduced by A. Grothendieck in the 1960s is perfectly adapted to this goal and provides the greatest simplicity and generality. This semantics is extended to higher order, and an explanation is given of what a general definition for the semantics of the theory of constructions could be.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126648024","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}
A bridge is presented between term-rewriting theory in computer science and proof theory in logic. It is shown that proof-theoretic tools are very useful for analyzing two basic attributes of term rewriting systems, the termination property and the Church-Rosser property. A counterexample is given to show that Knuth's critical pair lemma does not hold for conditional rewrite systems. Two restrictions on conditional systems under which the critical pair lemma holds are presented. One is considered a generalization of Bergstra-Klop's former result; the other is concerned with a generalization of Kaplan's and Jouannaud-Waldmann's systems.<>
{"title":"Proof-theoretic techniques for term rewriting theory","authors":"N. Dershowitz, M. Okada","doi":"10.1109/LICS.1988.5108","DOIUrl":"https://doi.org/10.1109/LICS.1988.5108","url":null,"abstract":"A bridge is presented between term-rewriting theory in computer science and proof theory in logic. It is shown that proof-theoretic tools are very useful for analyzing two basic attributes of term rewriting systems, the termination property and the Church-Rosser property. A counterexample is given to show that Knuth's critical pair lemma does not hold for conditional rewrite systems. Two restrictions on conditional systems under which the critical pair lemma holds are presented. One is considered a generalization of Bergstra-Klop's former result; the other is concerned with a generalization of Kaplan's and Jouannaud-Waldmann's systems.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131178720","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}
Complete axiomizations for the algebras of infinite trees and infinite trees are presented. The axiomizations are parameterized by the alphabet of function symbols for both the finite trees and infinite trees. There are two main cases, depending on whether the number of function symbols is finite or infinite. In the former case an extra axiom is necessary to obtain completeness. The method of proof is an elimination of quantifiers. Although a full elimination of quantifiers is not possible, the method forms the basis of decision procedures for the theories of the corresponding algebras. As a corollary to the results in infinite trees, the elementary equivalence of the algebra of rational trees and the algebra of infinite trees is obtained.<>
{"title":"Complete axiomatizations of the algebras of finite, rational and infinite trees","authors":"Michael J. Maher","doi":"10.1109/LICS.1988.5132","DOIUrl":"https://doi.org/10.1109/LICS.1988.5132","url":null,"abstract":"Complete axiomizations for the algebras of infinite trees and infinite trees are presented. The axiomizations are parameterized by the alphabet of function symbols for both the finite trees and infinite trees. There are two main cases, depending on whether the number of function symbols is finite or infinite. In the former case an extra axiom is necessary to obtain completeness. The method of proof is an elimination of quantifiers. Although a full elimination of quantifiers is not possible, the method forms the basis of decision procedures for the theories of the corresponding algebras. As a corollary to the results in infinite trees, the elementary equivalence of the algebra of rational trees and the algebra of infinite trees is obtained.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114775265","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}
Every proof system for (partial) correctness yields an enumeration procedure for correctness assertions. Other researchers have proved results on the existence of (sound and complete) enumeration procedures for assertions about programs from an acceptable programming language where the assertion language is first-order logic. It is shown that some of the assumptions are stronger than necessary, whereas others must not be weakened. Two novel procedures are given that work for more interpretations with a smaller oracle than those known up to now.<>
{"title":"On the existence of effective Hoare logics","authors":"M. Grabowski, H. Hungar","doi":"10.1109/LICS.1988.5140","DOIUrl":"https://doi.org/10.1109/LICS.1988.5140","url":null,"abstract":"Every proof system for (partial) correctness yields an enumeration procedure for correctness assertions. Other researchers have proved results on the existence of (sound and complete) enumeration procedures for assertions about programs from an acceptable programming language where the assertion language is first-order logic. It is shown that some of the assumptions are stronger than necessary, whereas others must not be weakened. Two novel procedures are given that work for more interpretations with a smaller oracle than those known up to now.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120983650","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}
A precise notion of a formal framework, meant to capture the intuition of an open-ended range of deductive interpreted languages, is proposed. A particular framework called the logical theory of constructions (LTC) is developed as an example. A series of languages in the LTC framework is defined, demonstrating how a language can be thought of as gradually evolving.<>
{"title":"The notion of a framework and a framework for LTC","authors":"P. Mendler, P. Aczel","doi":"10.1109/LICS.1988.5136","DOIUrl":"https://doi.org/10.1109/LICS.1988.5136","url":null,"abstract":"A precise notion of a formal framework, meant to capture the intuition of an open-ended range of deductive interpreted languages, is proposed. A particular framework called the logical theory of constructions (LTC) is developed as an example. A series of languages in the LTC framework is defined, demonstrating how a language can be thought of as gradually evolving.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131990822","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}
Denotational semantics is founded on a theory of higher order computation called domain theory, which formalizes a computation as a potentially infinite enumeration of finite elements that approximate the answer with progressively higher accuracy. Although existing formulations of domain theory provide an elegant framework for defining the abstract meaning of programs, these definitions are not effective because they fail to specify when computations terminate. A formulation of domain theory is presented that gives a natural topological characterization of termination: the evaluation of a program expression should terminate if and only if the expression denotes an element that is finite and maximal.<>
{"title":"The topology of program termination","authors":"Robert Cartwright, A. Demers","doi":"10.1109/LICS.1988.5128","DOIUrl":"https://doi.org/10.1109/LICS.1988.5128","url":null,"abstract":"Denotational semantics is founded on a theory of higher order computation called domain theory, which formalizes a computation as a potentially infinite enumeration of finite elements that approximate the answer with progressively higher accuracy. Although existing formulations of domain theory provide an elegant framework for defining the abstract meaning of programs, these definitions are not effective because they fail to specify when computations terminate. A formulation of domain theory is presented that gives a natural topological characterization of termination: the evaluation of a program expression should terminate if and only if the expression denotes an element that is finite and maximal.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132335415","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}
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