Pub Date : 1993-06-19DOI: 10.1109/LICS.1993.287598
L. Bachmair, H. Ganzinger, Uwe Waldmann
The authors investigate the relationship between set constraints and the monadic class of first-order formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence, they infer that the satisfiability problem for set constraints is complete for NEXPTIME. More precisely, it is proved that this problem has a lower bound of NTIME(c/sup n/log n/), for some c>0. The relationship between set constraints and the monadic class also gives decidability and complexity results for certain practically useful extensions of set constraints, in particular "negative" projections and subterm equality tests.<>
{"title":"Set constraints are the monadic class","authors":"L. Bachmair, H. Ganzinger, Uwe Waldmann","doi":"10.1109/LICS.1993.287598","DOIUrl":"https://doi.org/10.1109/LICS.1993.287598","url":null,"abstract":"The authors investigate the relationship between set constraints and the monadic class of first-order formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence, they infer that the satisfiability problem for set constraints is complete for NEXPTIME. More precisely, it is proved that this problem has a lower bound of NTIME(c/sup n/log n/), for some c>0. The relationship between set constraints and the monadic class also gives decidability and complexity results for certain practically useful extensions of set constraints, in particular \"negative\" projections and subterm equality tests.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85359977","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 : 1993-06-19DOI: 10.1109/LICS.1993.287578
V. Danos, L. Regnier
The authors build a confluent, local, asynchronous reduction on lambda -terms, using infinite objects (partial injections of Girard's (1988) algebra L*), which is simple (only one move), intelligible (semantic setting of the reduction), and general (based on a large-scale decomposition of beta ), and may be mechanized.<>
{"title":"Local and asynchronous beta-reduction (an analysis of Girard's execution formula)","authors":"V. Danos, L. Regnier","doi":"10.1109/LICS.1993.287578","DOIUrl":"https://doi.org/10.1109/LICS.1993.287578","url":null,"abstract":"The authors build a confluent, local, asynchronous reduction on lambda -terms, using infinite objects (partial injections of Girard's (1988) algebra L*), which is simple (only one move), intelligible (semantic setting of the reduction), and general (based on a large-scale decomposition of beta ), and may be mechanized.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72723934","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 : 1993-06-19DOI: 10.1109/LICS.1993.287576
Anil Seth
We consider functionals of type 2 as transformers between functions of type 1. An intuitively feasible functional must preserve the complexity of the input function in some broad sense. We show that the well quasi-order functional, which has been proposed by S.A. Cook (1990) as being intuitively feasible, fails to preserve the class of Kalmar elementary functions. For the basic feasible functionals (BFF), we show that there are arbitrarily large complexity classes of type 1 functions, under the classical definition of a complexity class, which contain polynomial-time functions and are closed under composition but are not preserved by the BFF. However, for a more natural definition of a complexity class of type 1 functions, BFF is shown to preserve all such complexity classes. BFF is the largest known class with this property. We prove BFF to be the largest class of type 2 functionals which satisfies Cook's conditions and the Ritchie-Cobham property, and preserves all classes of type 1 computable functions that contain polynomial-time functions and are closed under composition and limited recursion on notation. These results give some evidence that basic feasible functionals may be the right notion of type 2 feasibility.<>
{"title":"Some desirable conditions for feasible functionals of type 2","authors":"Anil Seth","doi":"10.1109/LICS.1993.287576","DOIUrl":"https://doi.org/10.1109/LICS.1993.287576","url":null,"abstract":"We consider functionals of type 2 as transformers between functions of type 1. An intuitively feasible functional must preserve the complexity of the input function in some broad sense. We show that the well quasi-order functional, which has been proposed by S.A. Cook (1990) as being intuitively feasible, fails to preserve the class of Kalmar elementary functions. For the basic feasible functionals (BFF), we show that there are arbitrarily large complexity classes of type 1 functions, under the classical definition of a complexity class, which contain polynomial-time functions and are closed under composition but are not preserved by the BFF. However, for a more natural definition of a complexity class of type 1 functions, BFF is shown to preserve all such complexity classes. BFF is the largest known class with this property. We prove BFF to be the largest class of type 2 functionals which satisfies Cook's conditions and the Ritchie-Cobham property, and preserves all classes of type 1 computable functions that contain polynomial-time functions and are closed under composition and limited recursion on notation. These results give some evidence that basic feasible functionals may be the right notion of type 2 feasibility.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75078715","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 : 1993-06-19DOI: 10.1109/LICS.1993.287584
A. Felty
The author presents an encoding of the calculus of constructions (CC) in a higher-order intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, she demonstrates a direct correspondence between proofs in these two systems. The logic I is an extension of hereditary Harrop formulas (hh), which serve as the logical foundation of the logic programming language lambda Prolog. Like hh, I has the uniform proof property, which allows a complete nondeterministic search procedure to be described in a straightforward manner. Via the encoding, this search procedure provides a goal directed description of proof checking and proof search in CC.<>
{"title":"Encoding the calculus of constructions in a higher-order logic","authors":"A. Felty","doi":"10.1109/LICS.1993.287584","DOIUrl":"https://doi.org/10.1109/LICS.1993.287584","url":null,"abstract":"The author presents an encoding of the calculus of constructions (CC) in a higher-order intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, she demonstrates a direct correspondence between proofs in these two systems. The logic I is an extension of hereditary Harrop formulas (hh), which serve as the logical foundation of the logic programming language lambda Prolog. Like hh, I has the uniform proof property, which allows a complete nondeterministic search procedure to be described in a straightforward manner. Via the encoding, this search procedure provides a goal directed description of proof checking and proof search in CC.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73600442","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 : 1993-06-19DOI: 10.1109/LICS.1993.287567
L. Jagadeesan, A. Meyer
Introduces a unary "self-synchronization" operation on concurrent processes that synchronizes concurrent transitions within a process. Standard parallel synchronization and communicating action refinement operations can be reduced to simple combinations of self-synchronization and unsynchronized noncommunicating operations. Modifying familiar fully abstract process semantics, so that actions are replaced by action multisets (steps), typically yields semantics that are fully abstract for processes with self-synchronization.<>
{"title":"Self-synchronization of concurrent processes","authors":"L. Jagadeesan, A. Meyer","doi":"10.1109/LICS.1993.287567","DOIUrl":"https://doi.org/10.1109/LICS.1993.287567","url":null,"abstract":"Introduces a unary \"self-synchronization\" operation on concurrent processes that synchronizes concurrent transitions within a process. Standard parallel synchronization and communicating action refinement operations can be reduced to simple combinations of self-synchronization and unsynchronized noncommunicating operations. Modifying familiar fully abstract process semantics, so that actions are replaced by action multisets (steps), typically yields semantics that are fully abstract for processes with self-synchronization.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73495979","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 : 1993-06-19DOI: 10.1109/LICS.1993.287602
M. Parigot
The strong normalization theorem for second-order classical natural deduction is proved. The method used is an adaptation of the one of reducibility candidates introduced in a thesis by J.Y. Girard (Univ. Paris 7, 1972) for second-order intuitionistic natural deduction. The extension to the classical case requires, in particular, a simplification of the notion of reducibility candidates.<>
证明了二阶经典自然演绎的强归一化定理。所使用的方法是J.Y. Girard (Univ. Paris 7, 1972)在二阶直觉自然演绎的论文中引入的可约性候选方法之一的改编。对经典情况的推广，特别需要简化可约候选者的概念。
{"title":"Strong normalization for second order classical natural deduction","authors":"M. Parigot","doi":"10.1109/LICS.1993.287602","DOIUrl":"https://doi.org/10.1109/LICS.1993.287602","url":null,"abstract":"The strong normalization theorem for second-order classical natural deduction is proved. The method used is an adaptation of the one of reducibility candidates introduced in a thesis by J.Y. Girard (Univ. Paris 7, 1972) for second-order intuitionistic natural deduction. The extension to the classical case requires, in particular, a simplification of the notion of reducibility candidates.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73552126","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 axiomatization of the isomorphisms that hold in all Cartesian closed categories (CCCs), discovered independently by S.V. Soloviev (1983) and by K.B. Bruce and G. Longo (1985), leads to seven equalities. It is shown that the unification problem for this theory is undecidable, thus setting an open question. It is also shown that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by M. Rittri (1990, 1991). It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages.<>
{"title":"On the unification problem for Cartesian closed categories","authors":"P. Narendran, F. Pfenning, R. Statman","doi":"10.2307/2275552","DOIUrl":"https://doi.org/10.2307/2275552","url":null,"abstract":"An axiomatization of the isomorphisms that hold in all Cartesian closed categories (CCCs), discovered independently by S.V. Soloviev (1983) and by K.B. Bruce and G. Longo (1985), leads to seven equalities. It is shown that the unification problem for this theory is undecidable, thus setting an open question. It is also shown that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by M. Rittri (1990, 1991). It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85950159","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 : 1993-06-19DOI: 10.1109/LICS.1993.287595
M. Fiore
The concept of bisimulation from concurrency theory is used to reason about recursively defined data types. From two strong-extensionality theorems stating that the equality (resp. inequality) relation is maximal among all bisimulations, a proof principle for the final coalgebra of an endofunctor on a category of data types (resp. domains) is obtained. As an application of the theory developed, an internal full abstraction result for the canonical model of the untyped call-by-value lambda -calculus is proved. The operations notion of bisimulation and the denotational notion of final semantics are related by means of conditions under which both coincide.<>
{"title":"A coinduction principle for recursive data types based on bisimulation","authors":"M. Fiore","doi":"10.1109/LICS.1993.287595","DOIUrl":"https://doi.org/10.1109/LICS.1993.287595","url":null,"abstract":"The concept of bisimulation from concurrency theory is used to reason about recursively defined data types. From two strong-extensionality theorems stating that the equality (resp. inequality) relation is maximal among all bisimulations, a proof principle for the final coalgebra of an endofunctor on a category of data types (resp. domains) is obtained. As an application of the theory developed, an internal full abstraction result for the canonical model of the untyped call-by-value lambda -calculus is proved. The operations notion of bisimulation and the denotational notion of final semantics are related by means of conditions under which both coincide.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89510354","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}
Total well-founded orderings on monadic terms satisfying the replacement and full invariance properties are considered. It is shown that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega , omega /sup 2/, or omega /sup omega /. It is further shown that a familiar construction gives rise to continuum many such orderings of order type omega . A new family of such orderings of order type omega is constructed, and it is shown that there are only four such orderings of order type omega /sup omega /, the two familiar recursive path orderings and two closely related orderings. It is shown that any total well-founded ordering on N/sup n/ that is preserved under vector addition must have order type omega /sup lambda / for some 1>
{"title":"The order types of termination orderings on monadic terms, strings and multisets","authors":"U. Martin, E. Scott","doi":"10.2307/2275551","DOIUrl":"https://doi.org/10.2307/2275551","url":null,"abstract":"Total well-founded orderings on monadic terms satisfying the replacement and full invariance properties are considered. It is shown that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega , omega /sup 2/, or omega /sup omega /. It is further shown that a familiar construction gives rise to continuum many such orderings of order type omega . A new family of such orderings of order type omega is constructed, and it is shown that there are only four such orderings of order type omega /sup omega /, the two familiar recursive path orderings and two closely related orderings. It is shown that any total well-founded ordering on N/sup n/ that is preserved under vector addition must have order type omega /sup lambda / for some 1<or=k<or=n; if k<n, there are continuum many such orderings, and if k=n, there are only n-factorial, namely the n-factorial lexicographic orderings.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72764024","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 : 1993-06-19DOI: 10.1109/LICS.1993.287577
P. Beame, T. Pitassi
The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size bounded-depth Frege proofs. M. Ajtai (1990) previously showed that the matching principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential, and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes. In particular, oracle separations between the complexity classes PPA and PPAD and between PPA and PPP follow from our techniques.<>
{"title":"An exponential separation between the matching principle and the pigeonhole principle","authors":"P. Beame, T. Pitassi","doi":"10.1109/LICS.1993.287577","DOIUrl":"https://doi.org/10.1109/LICS.1993.287577","url":null,"abstract":"The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size bounded-depth Frege proofs. M. Ajtai (1990) previously showed that the matching principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential, and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes. In particular, oracle separations between the complexity classes PPA and PPAD and between PPA and PPP follow from our techniques.<<ETX>>","PeriodicalId":6322,"journal":{"name":"[1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75284394","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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