Pub Date : 1997-06-29DOI: 10.1109/LICS.1997.614966
J. Goubault-Larrecq
While unification in the simple theory of types (a.k.a. higher-order logic) is undecidable. we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem already at order 2. In impure ramified higher-order logics, expressive predicative second-order subsystems of arithmetic or of inductive theories have concise axiomatisations; because of this and our decidability result for the pure case, we argue that ramified systems are expressive higher-order frameworks in which automated proof search should be practical.
{"title":"Ramified higher-order unification","authors":"J. Goubault-Larrecq","doi":"10.1109/LICS.1997.614966","DOIUrl":"https://doi.org/10.1109/LICS.1997.614966","url":null,"abstract":"While unification in the simple theory of types (a.k.a. higher-order logic) is undecidable. we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem already at order 2. In impure ramified higher-order logics, expressive predicative second-order subsystems of arithmetic or of inductive theories have concise axiomatisations; because of this and our decidability result for the pure case, we argue that ramified systems are expressive higher-order frameworks in which automated proof search should be practical.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"40 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131858892","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 : 1997-06-29DOI: 10.1109/LICS.1997.614961
F. Henglein, J. Rehof
A subtyping /spl tau//spl les//spl tau/' is entailed by a set of subtyping constraints C, written C |=/spl tau//spl les//spl tau/', if every valuation (mapping of type variables to ground types) that satisfies C also satisfies /spl tau//spl les//spl tau/'. We study the complexity of subtype entailment for simple types over lattices of base types. We show that: deciding C |=/spl tau//spl les//spl tau/' is coNP-complete; deciding C |=/spl alpha//spl les//spl beta/ for consistent, atomic C and /spl alpha/, /spl beta/ atomic can be done in linear time. The structural lower (coNP-hardness) and upper (membership in coNP) bounds as well as the optimal algorithm for atomic entailment are new. The coNP-hardness result indicates that entailment is strictly harder than satisfiability, which is known to be in PTIME for lattices of base types. The proof of coNP-completeness gives an improved algorithm for deciding entailment and puts a precise complexity-theoretic marker on the intuitive "exponential explosion" in the algorithm. Central to our results is a novel characterization of C |=/spl alpha//spl les//spl beta/ for atomic, consistent C. This is the basis for correctness of the linear-time algorithm as well as a complete axiomatization of C |=/spl alpha//spl les//spl beta/ for atomic C by extending the usual proof rules for subtype inference. It also incorporates the fundamental insight for understanding the structural complexity bounds in the general case.
{"title":"The complexity of subtype entailment for simple types","authors":"F. Henglein, J. Rehof","doi":"10.1109/LICS.1997.614961","DOIUrl":"https://doi.org/10.1109/LICS.1997.614961","url":null,"abstract":"A subtyping /spl tau//spl les//spl tau/' is entailed by a set of subtyping constraints C, written C |=/spl tau//spl les//spl tau/', if every valuation (mapping of type variables to ground types) that satisfies C also satisfies /spl tau//spl les//spl tau/'. We study the complexity of subtype entailment for simple types over lattices of base types. We show that: deciding C |=/spl tau//spl les//spl tau/' is coNP-complete; deciding C |=/spl alpha//spl les//spl beta/ for consistent, atomic C and /spl alpha/, /spl beta/ atomic can be done in linear time. The structural lower (coNP-hardness) and upper (membership in coNP) bounds as well as the optimal algorithm for atomic entailment are new. The coNP-hardness result indicates that entailment is strictly harder than satisfiability, which is known to be in PTIME for lattices of base types. The proof of coNP-completeness gives an improved algorithm for deciding entailment and puts a precise complexity-theoretic marker on the intuitive \"exponential explosion\" in the algorithm. Central to our results is a novel characterization of C |=/spl alpha//spl les//spl beta/ for atomic, consistent C. This is the basis for correctness of the linear-time algorithm as well as a complete axiomatization of C |=/spl alpha//spl les//spl beta/ for atomic C by extending the usual proof rules for subtype inference. It also incorporates the fundamental insight for understanding the structural complexity bounds in the general case.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127931648","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 : 1997-06-29DOI: 10.1109/LICS.1997.614963
M. Escardó, T. Streicher
The partial real line is the continuous domain of compact real intervals ordered by reverse inclusion. The idea is that singleton intervals represent total real numbers, and that the remaining intervals represent partial real numbers. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allows us to verify that Real PCF programs meet their specification. The theory is based on a domain-equation-like presentation of the partial unit interval, which we refer to as a biquotient of a bifree algebra.
{"title":"Induction and recursion on the partial real line via biquotients of bifree algebras","authors":"M. Escardó, T. Streicher","doi":"10.1109/LICS.1997.614963","DOIUrl":"https://doi.org/10.1109/LICS.1997.614963","url":null,"abstract":"The partial real line is the continuous domain of compact real intervals ordered by reverse inclusion. The idea is that singleton intervals represent total real numbers, and that the remaining intervals represent partial real numbers. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allows us to verify that Real PCF programs meet their specification. The theory is based on a domain-equation-like presentation of the partial unit interval, which we refer to as a biquotient of a bifree algebra.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127476360","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 : 1997-06-11DOI: 10.1109/LICS.1997.614953
J. Riecke, Anders Sandholm
We construct a model for FPC, a purely functional, sequential, call-by-value language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.
{"title":"A relational account of call-by-value sequentiality","authors":"J. Riecke, Anders Sandholm","doi":"10.1109/LICS.1997.614953","DOIUrl":"https://doi.org/10.1109/LICS.1997.614953","url":null,"abstract":"We construct a model for FPC, a purely functional, sequential, call-by-value language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134415738","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 : 1997-01-04DOI: 10.1109/LICS.1997.614943
R. Blute, Josée Desharnais, A. Edalat, P. Panangaden
In this paper we introduce a new class of labelled transition systems-Labelled Markov Processes-and define bisimulation for them. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete, it could be the reals, for example. We assume that it is a Polish space (the underlying topological space for a complete separable metric space). The mathematical theory of such systems is completely new from the point of view of the extant literature on probabilistic process algebra; of course, it uses classical ideas from measure theory and Markov process theory. The notion of bisimulation builds on the ideas of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result that we prove is that a notion of bisimulation for Markov processes on Polish spaces, which extends the Larsen-Skou definition for discrete systems, is indeed an equivalence relation. This turns our to be a rather hard mathematical result which, as far as we know, embodies a new result in pure probability theory. This work heavily uses continuous mathematics which is becoming an important part of work on hybrid systems.
{"title":"Bisimulation for labelled Markov processes","authors":"R. Blute, Josée Desharnais, A. Edalat, P. Panangaden","doi":"10.1109/LICS.1997.614943","DOIUrl":"https://doi.org/10.1109/LICS.1997.614943","url":null,"abstract":"In this paper we introduce a new class of labelled transition systems-Labelled Markov Processes-and define bisimulation for them. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete, it could be the reals, for example. We assume that it is a Polish space (the underlying topological space for a complete separable metric space). The mathematical theory of such systems is completely new from the point of view of the extant literature on probabilistic process algebra; of course, it uses classical ideas from measure theory and Markov process theory. The notion of bisimulation builds on the ideas of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result that we prove is that a notion of bisimulation for Markov processes on Polish spaces, which extends the Larsen-Skou definition for discrete systems, is indeed an equivalence relation. This turns our to be a rather hard mathematical result which, as far as we know, embodies a new result in pure probability theory. This work heavily uses continuous mathematics which is becoming an important part of work on hybrid systems.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134191848","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 : 1996-12-02DOI: 10.1109/LICS.1997.614946
P. Thiagarajan, I. Walukiewicz
A basic result concerning LTL, the propositional temporal logic of linear time is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL-specifications. We show that LTrL, our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also provides a syntactic characterisation of the so called trace consistent (robust) LTL-specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.
{"title":"An expressively complete linear time temporal logic for Mazurkiewicz traces","authors":"P. Thiagarajan, I. Walukiewicz","doi":"10.1109/LICS.1997.614946","DOIUrl":"https://doi.org/10.1109/LICS.1997.614946","url":null,"abstract":"A basic result concerning LTL, the propositional temporal logic of linear time is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL-specifications. We show that LTrL, our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also provides a syntactic characterisation of the so called trace consistent (robust) LTL-specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131296191","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 : 1900-01-01DOI: 10.1109/LICS.1997.614947
D. Kozen
We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E/spl rarr/s=t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *-continuous Kleene algebra, if E contains only commutativity assumptions pq=qp, the problem is II/sub 1//sup 0/-complete; if E contains only monoid equations, the problem is II/sub 2//sup 0/-complete; for arbitrary equations E, the problem is II/sub 1//sup 1/-complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994).
{"title":"On the complexity of reasoning in Kleene algebra","authors":"D. Kozen","doi":"10.1109/LICS.1997.614947","DOIUrl":"https://doi.org/10.1109/LICS.1997.614947","url":null,"abstract":"We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E/spl rarr/s=t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *-continuous Kleene algebra, if E contains only commutativity assumptions pq=qp, the problem is II/sub 1//sup 0/-complete; if E contains only monoid equations, the problem is II/sub 2//sup 0/-complete; for arbitrary equations E, the problem is II/sub 1//sup 1/-complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994).","PeriodicalId":272903,"journal":{"name":"Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130950267","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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