Pub Date : 2020-02-27DOI: 10.46298/lmcs-18(3:36)2022
Mirai Ikebuchi
It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier's homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs.
{"title":"A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods","authors":"Mirai Ikebuchi","doi":"10.46298/lmcs-18(3:36)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(3:36)2022","url":null,"abstract":"It is well-known that some equational theories such as groups or boolean\u0000algebras can be defined by fewer equational axioms than the original axioms.\u0000However, it is not easy to determine if a given set of axioms is the smallest\u0000or not. Malbos and Mimram investigated a general method to find a lower bound\u0000of the cardinality of the set of equational axioms (or rewrite rules) that is\u0000equivalent to a given equational theory (or term rewriting systems), using\u0000homological algebra. Their method is an analog of Squier's homology theory on\u0000string rewriting systems. In this paper, we develop the homology theory for\u0000term rewriting systems more and provide a better lower bound under a stronger\u0000notion of equivalence than their equivalence. The author also implemented a\u0000program to compute the lower bounds, and experimented with 64 complete TRSs.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123823640","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 : 2020-02-21DOI: 10.4230/LIPIcs.FSCD.2020.27
Yu-Yang Lin, N. Tzevelekos
We present a framework for symbolically executing and model checking higher-order programs with external (open) methods. We focus on the client-library paradigm and in particular we aim to check libraries with respect to any definable client. We combine traditional symbolic execution techniques with operational game semantics to build a symbolic execution semantics that captures arbitrary external behaviour. We prove the symbolic semantics to be sound and complete. This yields a bounded technique by imposing bounds on the depth of recursion and callbacks. We provide an implementation of our technique in the K framework and showcase its performance on a custom benchmark based on higher-order coding errors such as reentrancy bugs.
{"title":"Symbolic Execution Game Semantics","authors":"Yu-Yang Lin, N. Tzevelekos","doi":"10.4230/LIPIcs.FSCD.2020.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.27","url":null,"abstract":"We present a framework for symbolically executing and model checking higher-order programs with external (open) methods. We focus on the client-library paradigm and in particular we aim to check libraries with respect to any definable client. We combine traditional symbolic execution techniques with operational game semantics to build a symbolic execution semantics that captures arbitrary external behaviour. We prove the symbolic semantics to be sound and complete. This yields a bounded technique by imposing bounds on the depth of recursion and callbacks. We provide an implementation of our technique in the K framework and showcase its performance on a custom benchmark based on higher-order coding errors such as reentrancy bugs.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115487085","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 : 2020-02-01DOI: 10.4230/LIPIcs.FSCD.2020.17
G. Boisseau
Optics are a data representation for compositional data access, with lenses as a popular special case. Hedges has presented a diagrammatic calculus for lenses, but in a way that does not generalize to other classes of optic. We present a calculus that works for all optics, not just lenses; this is done by embedding optics into their presheaf category, which naturally features string diagrams. We apply our calculus to the common case of lenses, extend it to effectful lenses, and explore how the laws of optics manifest in this setting.
{"title":"String Diagrams for Optics","authors":"G. Boisseau","doi":"10.4230/LIPIcs.FSCD.2020.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.17","url":null,"abstract":"Optics are a data representation for compositional data access, with lenses as a popular special case. Hedges has presented a diagrammatic calculus for lenses, but in a way that does not generalize to other classes of optic. We present a calculus that works for all optics, not just lenses; this is done by embedding optics into their presheaf category, which naturally features string diagrams. We apply our calculus to the common case of lenses, extend it to effectful lenses, and explore how the laws of optics manifest in this setting.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116588654","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 translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are proven equivalent to the usual partial/primitive recursive functions. Complexity-wise, this framework captures in a streamlined way most of the relevant sub-polynomial classes. Pointer recursion with the safe/normal tiering discipline of Bellantoni and Cook corresponds to polylogtime computation. We introduce a new, non-size increasing tiering discipline, called tropical tiering. Tropical tiering and pointer recursion, used with some of the most common recursion schemes, capture the classes logspace, logspace/polylogtime, ptime, and NC. Finally, in a fashion reminiscent of the safe recursive functions, tropical tiering is expressed directly in the syntax of the function algebras, yielding the tropical recursive function algebras.
{"title":"Pointers in Recursion: Exploring the Tropics","authors":"Paulin Jacobé de Naurois","doi":"10.4204/EPTCS.298.3","DOIUrl":"https://doi.org/10.4204/EPTCS.298.3","url":null,"abstract":"We translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are proven equivalent to the usual partial/primitive recursive functions. Complexity-wise, this framework captures in a streamlined way most of the relevant sub-polynomial classes. Pointer recursion with the safe/normal tiering discipline of Bellantoni and Cook corresponds to polylogtime computation. We introduce a new, non-size increasing tiering discipline, called tropical tiering. Tropical tiering and pointer recursion, used with some of the most common recursion schemes, capture the classes logspace, logspace/polylogtime, ptime, and NC. Finally, in a fashion reminiscent of the safe recursive functions, tropical tiering is expressed directly in the syntax of the function algebras, yielding the tropical recursive function algebras.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116769959","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 : 2019-06-24DOI: 10.4230/LIPIcs.FSCD.2019.9
F. Blanqui, G. Genestier, O. Hermant
Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewriting systems. In this paper, we introduce an extension of this technique for a large class of dependently-typed higher-order rewriting systems. This extends previous resultsby Wahlstedt on the one hand and the first author on the other hand to strong normalization and non-orthogonal rewriting systems. This new criterion is implemented in the type-checker Dedukti.
{"title":"Dependency Pairs Termination in Dependent Type Theory Modulo Rewriting","authors":"F. Blanqui, G. Genestier, O. Hermant","doi":"10.4230/LIPIcs.FSCD.2019.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.9","url":null,"abstract":"Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewriting systems. In this paper, we introduce an extension of this technique for a large class of dependently-typed higher-order rewriting systems. This extends previous resultsby Wahlstedt on the one hand and the first author on the other hand to strong normalization and non-orthogonal rewriting systems. This new criterion is implemented in the type-checker Dedukti.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122945483","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 : 2019-06-24DOI: 10.4230/LIPIcs.FSCD.2019.19
C. Faggian
While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and normalizing strategies (is there a strategy to find a result with *greatest probability* ?). The goal is to have tools to analyze the operational properties of probabilistic calculi such as probabilistic lambda-calculi, whose evaluation is also non-deterministic, where non-determinism arises from a choice between several redexes. We investigate how the asymptotic behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, propose a robust analogue of the notion of "unique normal form", and we develop methods to study and compare strategies. Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific nature of the objects.
{"title":"Probabilistic Rewriting: Normalization, Termination, and Unique Normal Forms","authors":"C. Faggian","doi":"10.4230/LIPIcs.FSCD.2019.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.19","url":null,"abstract":"While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and normalizing strategies (is there a strategy to find a result with *greatest probability* ?). The goal is to have tools to analyze the operational properties of probabilistic calculi such as probabilistic lambda-calculi, whose evaluation is also non-deterministic, where non-determinism arises from a choice between several redexes. \u0000We investigate how the asymptotic behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, propose a robust analogue of the notion of \"unique normal form\", and we develop methods to study and compare strategies. Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific nature of the objects.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127725090","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 : 2019-06-18DOI: 10.4230/LIPICS.FSCD.2019.22
W. Heijltjes, Dominic J. D. Hughes, Lutz Straßburger
We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation.
{"title":"Proof Nets for First-Order Additive Linear Logic","authors":"W. Heijltjes, Dominic J. D. Hughes, Lutz Straßburger","doi":"10.4230/LIPICS.FSCD.2019.22","DOIUrl":"https://doi.org/10.4230/LIPICS.FSCD.2019.22","url":null,"abstract":"We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. \u0000 \u0000We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. \u0000 \u0000Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. \u0000 \u0000Both capture sequent calculus cut-elimination as a one-step global composition operation.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125309636","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 : 2019-04-21DOI: 10.4230/LIPIcs.FSCD.2019.13
Ugo Dal Lago, Thomas Leventis
We generalise Ehrhard and Regnier’s Taylor expansion from pure to probabilistic λ -terms through notions of probabilistic resource terms and explicit Taylor expansion. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic λ terms, and that there is a precise correspondence with probabilistic Böhm trees, as introduced by the second author.
{"title":"On the Taylor Expansion of Probabilistic lambda-terms","authors":"Ugo Dal Lago, Thomas Leventis","doi":"10.4230/LIPIcs.FSCD.2019.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.13","url":null,"abstract":"We generalise Ehrhard and Regnier’s Taylor expansion from pure to probabilistic λ -terms through notions of probabilistic resource terms and explicit Taylor expansion. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic λ terms, and that there is a precise correspondence with probabilistic Böhm trees, as introduced by the second author.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133927991","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 : 2019-04-18DOI: 10.4230/LIPIcs.FSCD.2019.31
Jonathan Sterling, C. Angiuli, Daniel Gratzer
We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-Lof's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel cubical extension (independently proposed by Awodey) of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.
{"title":"Cubical Syntax for Reflection-Free Extensional Equality","authors":"Jonathan Sterling, C. Angiuli, Daniel Gratzer","doi":"10.4230/LIPIcs.FSCD.2019.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.31","url":null,"abstract":"We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-Lof's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel cubical extension (independently proposed by Awodey) of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116250125","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 : 2019-03-03DOI: 10.4230/LIPIcs.FSCD.2019.6
B. Ahrens, A. Hirschowitz, Ambroise Lafont, M. Maggesi
In their work on second-order equational logic, Fiore and Hur have studied presentations of simply typed languages by generating binding constructions and equations among them. To each pair consisting of a binding signature and a set of equations, they associate a category of "models", and they give a monadicity result which implies that this category has an initial object, which is the language presented by the pair. In the present work, we propose, for the untyped setting, a variant of their approach where monads and modules over them are the central notions. More precisely, we study, for monads over sets, presentations by generating ("higher-order") operations and equations among them. We consider a notion of 2-signature which allows to specify a monad with a family of binding operations subject to a family of equations, as is the case for the paradigmatic example of the lambda calculus, specified by its two standard constructions (application and abstraction) subject to beta- and eta-equalities. Such a 2-signature is hence a pair (Sigma,E) of a binding signature Sigma and a family E of equations for Sigma. This notion of 2-signature has been introduced earlier by Ahrens in a slightly different context. We associate, to each 2-signature (Sigma,E), a category of "models of (Sigma,E)"; and we say that a 2-signature is "effective" if this category has an initial object; the monad underlying this (essentially unique) object is the "monad specified by the 2-signature". Not every 2-signature is effective; we identify a class of 2-signatures, which we call "algebraic", that are effective. Importantly, our 2-signatures together with their models enjoy "modularity": when we glue (algebraic) 2-signatures together, their initial models are glued accordingly. We provide a computer formalization for our main results.
{"title":"Modular specification of monads through higher-order presentations","authors":"B. Ahrens, A. Hirschowitz, Ambroise Lafont, M. Maggesi","doi":"10.4230/LIPIcs.FSCD.2019.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.6","url":null,"abstract":"In their work on second-order equational logic, Fiore and Hur have studied presentations of simply typed languages by generating binding constructions and equations among them. To each pair consisting of a binding signature and a set of equations, they associate a category of \"models\", and they give a monadicity result which implies that this category has an initial object, which is the language presented by the pair. \u0000In the present work, we propose, for the untyped setting, a variant of their approach where monads and modules over them are the central notions. More precisely, we study, for monads over sets, presentations by generating (\"higher-order\") operations and equations among them. We consider a notion of 2-signature which allows to specify a monad with a family of binding operations subject to a family of equations, as is the case for the paradigmatic example of the lambda calculus, specified by its two standard constructions (application and abstraction) subject to beta- and eta-equalities. Such a 2-signature is hence a pair (Sigma,E) of a binding signature Sigma and a family E of equations for Sigma. This notion of 2-signature has been introduced earlier by Ahrens in a slightly different context. \u0000We associate, to each 2-signature (Sigma,E), a category of \"models of (Sigma,E)\"; and we say that a 2-signature is \"effective\" if this category has an initial object; the monad underlying this (essentially unique) object is the \"monad specified by the 2-signature\". Not every 2-signature is effective; we identify a class of 2-signatures, which we call \"algebraic\", that are effective. \u0000Importantly, our 2-signatures together with their models enjoy \"modularity\": when we glue (algebraic) 2-signatures together, their initial models are glued accordingly. \u0000We provide a computer formalization for our main results.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125374576","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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