In the course of developing a mathematical theory or proof it is a common practice to introduce new notation to represent notation that is previously understood. The author presents a formal account that is intended to model the practice of introducing and using notational (abbreviate) definitions. The aim of this work is a pragmatic one: to provide a framework useful in the design and implementation of secure proof system interfaces which accommodate, as much as possible, conventional mathematical practice. A typed lambda -calculus is used to represent expressions of a given object language. A Delta -equation is introduced to model conventional definition equations.<>
{"title":"Notational definition-a formal account","authors":"T. Griffin","doi":"10.1109/LICS.1988.5134","DOIUrl":"https://doi.org/10.1109/LICS.1988.5134","url":null,"abstract":"In the course of developing a mathematical theory or proof it is a common practice to introduce new notation to represent notation that is previously understood. The author presents a formal account that is intended to model the practice of introducing and using notational (abbreviate) definitions. The aim of this work is a pragmatic one: to provide a framework useful in the design and implementation of secure proof system interfaces which accommodate, as much as possible, conventional mathematical practice. A typed lambda -calculus is used to represent expressions of a given object language. A Delta -equation is introduced to model conventional definition equations.<<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":"115639459","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}
Polymorphic type discipline for lambda -calculus is an extension of H.B. Curry's (1969) classical functionality theory, in which types can be universally quantified. An algorithm that, given a term M, builds a set of constraints, is satisfied. Moreover, all the typings for M (if any) are built from the set of constraints by substitutions. Using the set of constraints, some properties of polymorphic type discipline are proved.<>
{"title":"Characterization of typings in polymorphic type discipline","authors":"P. Giannini, S. D. Rocca","doi":"10.1109/LICS.1988.5101","DOIUrl":"https://doi.org/10.1109/LICS.1988.5101","url":null,"abstract":"Polymorphic type discipline for lambda -calculus is an extension of H.B. Curry's (1969) classical functionality theory, in which types can be universally quantified. An algorithm that, given a term M, builds a set of constraints, is satisfied. Moreover, all the typings for M (if any) are built from the set of constraints by substitutions. Using the set of constraints, some properties of polymorphic type discipline are proved.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"64 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":"126857761","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 authors show that the exact formulation of the rules of type theory is important for rules of subset type. It turns out that there are propositions involving subsets that are trivially true in naive set theory, but which cannot be proved in type theory. They examine the probability of a type proposition that is important when modularizing program derivations.<>
{"title":"The strength of the subset type in Martin-Lof's type theory","authors":"A. Salvesen, Jan M. Smith","doi":"10.1109/LICS.1988.5135","DOIUrl":"https://doi.org/10.1109/LICS.1988.5135","url":null,"abstract":"The authors show that the exact formulation of the rules of type theory is important for rules of subset type. It turns out that there are propositions involving subsets that are trivially true in naive set theory, but which cannot be proved in type theory. They examine the probability of a type proposition that is important when modularizing program derivations.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"18 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":"124465293","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 semantic condition necessary for the parametricity of polymorphic functions is considered. One of its instances is the stability condition for elements of variable type in the coherent domains semantics. A larger setting is presented that does not use retract pairs and keeps intact a basic feature of a certain function-type constructor. Polymorphic lambda terms are semantically parametric because of normalization.<>
{"title":"Semantic parametricity in polymorphic lambda calculus","authors":"P. Freyd, J. Girard, A. Scedrov, P. Scott","doi":"10.1109/LICS.1988.5126","DOIUrl":"https://doi.org/10.1109/LICS.1988.5126","url":null,"abstract":"A semantic condition necessary for the parametricity of polymorphic functions is considered. One of its instances is the stability condition for elements of variable type in the coherent domains semantics. A larger setting is presented that does not use retract pairs and keeps intact a basic feature of a certain function-type constructor. Polymorphic lambda terms are semantically parametric because of normalization.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"29 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":"121535525","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 method is proven for proving that equations are valid in the initial model of an equational variety. This proof by consistency procedure can be applied to equational theories that are represented as ground convergent rewrite systems. In contrast with so-called inductive completion procedures, the method requires no specific ordering on terms and can handle unorientable equations. The method is linear and refutationally complete, in that it refutes any equation which is not an inductive theorem.<>
{"title":"Proof by consistency in equational theories","authors":"L. Bachmair","doi":"10.1109/LICS.1988.5122","DOIUrl":"https://doi.org/10.1109/LICS.1988.5122","url":null,"abstract":"A method is proven for proving that equations are valid in the initial model of an equational variety. This proof by consistency procedure can be applied to equational theories that are represented as ground convergent rewrite systems. In contrast with so-called inductive completion procedures, the method requires no specific ordering on terms and can handle unorientable equations. The method is linear and refutationally complete, in that it refutes any equation which is not an inductive theorem.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"12 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":"122456463","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 problem of the relationship between truly concurrent operational and denotational semantics is tackled by mapping syntactic terms on similar semantic domains in both approaches. Occurrence nets are associated to terms through structural operational semantics based on a set of rewriting rules; event structures are defined as denotations for terms, without resorting to categorical constructions. The proof of the equivalence of the two semantics relies on the direct correspondence between occurrence nets and event structures. R. Milner's (1980) calculus of communicating systems is used as a test case; truly concurrent denotional and operational semantics are given for it and proved consistent. This equivalence is established for the first time in true concurrency approach. It is proved that G. Winskel's (1982) categorical denotational semantics is equivalent to that given here.<>
{"title":"On the consistency of 'truly concurrent' operational and denotational semantics","authors":"P. Degano, R. Nicola, U. Montanari","doi":"10.1109/LICS.1988.5112","DOIUrl":"https://doi.org/10.1109/LICS.1988.5112","url":null,"abstract":"The problem of the relationship between truly concurrent operational and denotational semantics is tackled by mapping syntactic terms on similar semantic domains in both approaches. Occurrence nets are associated to terms through structural operational semantics based on a set of rewriting rules; event structures are defined as denotations for terms, without resorting to categorical constructions. The proof of the equivalence of the two semantics relies on the direct correspondence between occurrence nets and event structures. R. Milner's (1980) calculus of communicating systems is used as a test case; truly concurrent denotional and operational semantics are given for it and proved consistent. This equivalence is established for the first time in true concurrency approach. It is proved that G. Winskel's (1982) categorical denotational semantics is equivalent to that given here.<<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":"122347520","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 novel logic is introduced for the introduction of nondeterministic and concurrent processes expressed in a process algebra. For a process algebra to be useful as a process language, it must possess compositionality, i.e. it should be possible to decompose the problem of correctness for a combined system with respect to a given specification of similar and simpler correctness problems for the components of the system. The logic presented allows such specifications to be expressed. It is an extension of process algebra in the sense that process constructs are included as connectives in the logic. Moreover, the formulas of the logic are given an operational interpretation based on which a refinement ordering between formulas is defined.<>
{"title":"A modal process logic","authors":"K. Larsen, Bent Thomsen","doi":"10.1109/LICS.1988.5119","DOIUrl":"https://doi.org/10.1109/LICS.1988.5119","url":null,"abstract":"A novel logic is introduced for the introduction of nondeterministic and concurrent processes expressed in a process algebra. For a process algebra to be useful as a process language, it must possess compositionality, i.e. it should be possible to decompose the problem of correctness for a combined system with respect to a given specification of similar and simpler correctness problems for the components of the system. The logic presented allows such specifications to be expressed. It is an extension of process algebra in the sense that process constructs are included as connectives in the logic. Moreover, the formulas of the logic are given an operational interpretation based on which a refinement ordering between formulas is defined.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"27 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":"115828726","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}
Behavioral preorders based on the idea of bisimulation equivalence and providing explicit treatments of the phenomenon of divergence in communicating systems are investigated. Particular study is made of those preorders in the context of R. Milner's (1980) calculus of communicating systems. The precongruence relations generated by the preorders are characterized and axiomated on the classes of finite closed terms and sequential terms. A compositional verification technique based on the theory of the preorders is introduced and illustrated.<>
{"title":"Bisimulations and divergence","authors":"D. Walker","doi":"10.1109/LICS.1988.5117","DOIUrl":"https://doi.org/10.1109/LICS.1988.5117","url":null,"abstract":"Behavioral preorders based on the idea of bisimulation equivalence and providing explicit treatments of the phenomenon of divergence in communicating systems are investigated. Particular study is made of those preorders in the context of R. Milner's (1980) calculus of communicating systems. The precongruence relations generated by the preorders are characterized and axiomated on the classes of finite closed terms and sequential terms. A compositional verification technique based on the theory of the preorders is introduced and illustrated.<<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":"128288945","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 algebra and a logic characterizing the complexity class NC/sup 1/, which consists of functions computed by uniform sequences of polynomial-size, log depth circuits, are presented. In both characterizations, NC/sup 1/ functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization, upward and downward tree recursion are applied to a class of simple functions. In the logical characterization, first-order logic is augmented by an operator for defining relations by primitive recursion. It is assumed that every structure has an underlying relation giving the binary representations of integers.<>
{"title":"An algebra and a logic for NC/sup 1/","authors":"K. Compton, C. Laflamme","doi":"10.1109/LICS.1988.5096","DOIUrl":"https://doi.org/10.1109/LICS.1988.5096","url":null,"abstract":"An algebra and a logic characterizing the complexity class NC/sup 1/, which consists of functions computed by uniform sequences of polynomial-size, log depth circuits, are presented. In both characterizations, NC/sup 1/ functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization, upward and downward tree recursion are applied to a class of simple functions. In the logical characterization, first-order logic is augmented by an operator for defining relations by primitive recursion. It is assumed that every structure has an underlying relation giving the binary representations of integers.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"4 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":"121863909","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}
To help understand the reason for continuity in denotational semantics, the author offers some global comments on goodness-to-fit criteria between semantic domains and symbolic evaluators. The appendices provide the key parts of a proof that Scott domains give a computationally adequate and fully abstract semantics for lambda calculus with simple recursive types.<>
{"title":"Semantical paradigms: notes for an invited lecture","authors":"A. Meyer","doi":"10.1109/LICS.1988.5123","DOIUrl":"https://doi.org/10.1109/LICS.1988.5123","url":null,"abstract":"To help understand the reason for continuity in denotational semantics, the author offers some global comments on goodness-to-fit criteria between semantic domains and symbolic evaluators. The appendices provide the key parts of a proof that Scott domains give a computationally adequate and fully abstract semantics for lambda calculus with simple recursive types.<<ETX>>","PeriodicalId":425186,"journal":{"name":"[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science","volume":"62 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":"120992923","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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