Pub Date : 2021-04-30DOI: 10.4230/LIPIcs.FSCD.2021.21
Joseph W. N. Paulus, D. Nantes-Sobrinho, Jorge A. Pérez
We study encodings of the lambda-calculus into the pi-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider lambdafail, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider spi, a pi-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of lambdafail into spi and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in lambdafail via typed processes in spi. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interaction protocols.
{"title":"Non-Deterministic Functions as Non-Deterministic Processes (Extended Version)","authors":"Joseph W. N. Paulus, D. Nantes-Sobrinho, Jorge A. Pérez","doi":"10.4230/LIPIcs.FSCD.2021.21","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.21","url":null,"abstract":"We study encodings of the lambda-calculus into the pi-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider lambdafail, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider spi, a pi-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of lambdafail into spi and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in lambdafail via typed processes in spi. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interaction protocols.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"2 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127004655","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 : 2021-02-25DOI: 10.4230/LIPIcs.FSCD.2021.28
Hans-Peter Deifel, Stefan Milius, Thorsten Wißmann
Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a fully fledged minimization algorithm and tool by integrating two new aspects: (1) the computation of the transition structure on the minimized state set, and (2) the computation of the reachable part of the given system. In our generic coalgebraic setting these two aspects turn out to be surprisingly non-trivial requiring us to extend the previous theory. In particular, we identify a sufficient condition on encodings of coalgebras, and we show how to augment the existing interface, which encapsulates computations that are specific for the coalgebraic type functor, to make the above extensions possible. Both extensions have linear run time.
{"title":"Coalgebra Encoding for Efficient Minimization","authors":"Hans-Peter Deifel, Stefan Milius, Thorsten Wißmann","doi":"10.4230/LIPIcs.FSCD.2021.28","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.28","url":null,"abstract":"Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a fully fledged minimization algorithm and tool by integrating two new aspects: (1) the computation of the transition structure on the minimized state set, and (2) the computation of the reachable part of the given system. In our generic coalgebraic setting these two aspects turn out to be surprisingly non-trivial requiring us to extend the previous theory. In particular, we identify a sufficient condition on encodings of coalgebras, and we show how to augment the existing interface, which encapsulates computations that are specific for the coalgebraic type functor, to make the above extensions possible. Both extensions have linear run time.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125800147","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 : 2021-02-22DOI: 10.4230/LIPIcs.FSCD.2021.26
Pieter J. W. Hofstra, Jason Parker, P. Scott
We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply this characterization to prove that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore, we obtain an explicit description of the covariant isotropy group of a presheaf category.
{"title":"Polymorphic Automorphisms and the Picard Group","authors":"Pieter J. W. Hofstra, Jason Parker, P. Scott","doi":"10.4230/LIPIcs.FSCD.2021.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.26","url":null,"abstract":"We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply this characterization to prove that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore, we obtain an explicit description of the covariant isotropy group of a presheaf category.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124501993","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 : 2020-11-18DOI: 10.46298/lmcs-17(4:18)2021
P. Vukmirović, A. Bentkamp, V. Nummelin
We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers. We identify a new such fragment and describe a procedure for computing its unifiers. Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.
{"title":"Efficient Full Higher-Order Unification","authors":"P. Vukmirović, A. Bentkamp, V. Nummelin","doi":"10.46298/lmcs-17(4:18)2021","DOIUrl":"https://doi.org/10.46298/lmcs-17(4:18)2021","url":null,"abstract":"We developed a procedure to enumerate complete sets of higher-order unifiers\u0000based on work by Jensen and Pietrzykowski. Our procedure removes many redundant\u0000unifiers by carefully restricting the search space and tightly integrating\u0000decision procedures for fragments that admit a finite complete set of unifiers.\u0000We identify a new such fragment and describe a procedure for computing its\u0000unifiers. Our unification procedure, together with new higher-order term\u0000indexing data structures, is implemented in the Zipperposition theorem prover.\u0000Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's\u0000procedure.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131092536","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 : 2020-10-30DOI: 10.4230/LIPIcs.FSCD.2020.13
F. Blanqui
The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. In this paper, we give a new method to check that property using Knuth-Bendix completion.
{"title":"Type Safety of Rewrite Rules in Dependent Types","authors":"F. Blanqui","doi":"10.4230/LIPIcs.FSCD.2020.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.13","url":null,"abstract":"The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. In this paper, we give a new method to check that property using Knuth-Bendix completion.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127476844","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 : 2020-10-30DOI: 10.4230/LIPIcs.FSCD.2020.35
Gabriel Hondet, F. Blanqui
Dedukti is a type-checker for the $lambda$$Pi$-calculus modulo rewriting, an extension of Edinburgh's logicalframework LF where functions and type symbols can be defined by rewrite rules. It thereforecontains an engine for rewriting LF terms and types according to the rewrite rules given by the user.A key component of this engine is the matching algorithm to find which rules can be fired. In thispaper, we describe the class of rewrite rules supported by Dedukti and the new implementation ofthe matching algorithm. Dedukti supports non-linear rewrite rules on terms with binders usinghigher-order pattern-matching as in Combinatory Reduction Systems (CRS). The new matchingalgorithm extends the technique of decision trees introduced by Luc Maranget in the OCamlcompiler to this more general context.
{"title":"The New Rewriting Engine of Dedukti (System Description)","authors":"Gabriel Hondet, F. Blanqui","doi":"10.4230/LIPIcs.FSCD.2020.35","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.35","url":null,"abstract":"Dedukti is a type-checker for the $lambda$$Pi$-calculus modulo rewriting, an extension of Edinburgh's logicalframework LF where functions and type symbols can be defined by rewrite rules. It thereforecontains an engine for rewriting LF terms and types according to the rewrite rules given by the user.A key component of this engine is the matching algorithm to find which rules can be fired. In thispaper, we describe the class of rewrite rules supported by Dedukti and the new implementation ofthe matching algorithm. Dedukti supports non-linear rewrite rules on terms with binders usinghigher-order pattern-matching as in Combinatory Reduction Systems (CRS). The new matchingalgorithm extends the technique of decision trees introduced by Luc Maranget in the OCamlcompiler to this more general context.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122171585","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 : 2020-05-06DOI: 10.4230/LIPIcs.FSCD.2020.15
Andrej Ivaskovic, A. Mycroft, Dominic A. Orchard
In static analysis, two frameworks have been studied extensively: monotone data-flow analysis and type-and-effect systems. Whilst both are seen as general analysis frameworks, their relationship has remained unclear. Here we show that monotone data-flow analyses can be encoded as effect systems in a uniform way, via algebras of transfer functions. This helps to answer questions about the most appropriate structure for general effect algebras, especially with regards capturing control-flow precisely. Via the perspective of capturing data-flow analyses, we show the recent suggestion of using effect quantales is not general enough as it excludes non-distributive analyses e.g., constant propagation. By rephrasing the McCarthy transformation, we then model monotone data-flow effects via graded monads. This provides a model of data-flow analyses that can be used to reason about analysis correctness at the semantic level, and to embed data-flow analyses into type systems. 2012 ACM Subject Classification Theory of computation → Type theory
{"title":"Data-Flow Analyses as Effects and Graded Monads","authors":"Andrej Ivaskovic, A. Mycroft, Dominic A. Orchard","doi":"10.4230/LIPIcs.FSCD.2020.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.15","url":null,"abstract":"In static analysis, two frameworks have been studied extensively: monotone data-flow analysis and type-and-effect systems. Whilst both are seen as general analysis frameworks, their relationship has remained unclear. Here we show that monotone data-flow analyses can be encoded as effect systems in a uniform way, via algebras of transfer functions. This helps to answer questions about the most appropriate structure for general effect algebras, especially with regards capturing control-flow precisely. Via the perspective of capturing data-flow analyses, we show the recent suggestion of using effect quantales is not general enough as it excludes non-distributive analyses e.g., constant propagation. By rephrasing the McCarthy transformation, we then model monotone data-flow effects via graded monads. This provides a model of data-flow analyses that can be used to reason about analysis correctness at the semantic level, and to embed data-flow analyses into type systems. 2012 ACM Subject Classification Theory of computation → Type theory","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122415270","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 : 2020-05-06DOI: 10.4230/LIPIcs.FSCD.2020.34
C. Cohen, Kazuhiko Sakaguchi, Enrico Tassi
It is nowadays customary to organize libraries of machine checked proofs around hierarchies of algebraic structures [2, 6, 8, 16, 18, 23, 27]. One influential example is the Mathematical Components library on top of which the long and intricate proof of the Odd Order Theorem could be fully formalized [14]. Still, building algebraic hierarchies in a proof assistant such as Coq [9] requires a lot of manual labor and often a deep expertise in the internals of the prover [13, 17]. Moreover, according to our experience [26], making a hierarchy evolve without causing breakage in client code is equally tricky: even a simple refactoring such as splitting a structure into two simpler ones is hard to get right. In this paper we describe HB, a high level language to build hierarchies of algebraic structures and to make these hierarchies evolve without breaking user code. The key concepts are the ones of factory, builder and abbreviation that let the hierarchy developer describe an actual interface for their library. Behind that interface the developer can provide appropriate code to ensure retro compatibility. We implement the HB language in the hierarchy-builder addon for the Coq system using the Elpi [11, 28] extension language.
{"title":"Hierarchy Builder: Algebraic hierarchies Made Easy in Coq with Elpi (System Description)","authors":"C. Cohen, Kazuhiko Sakaguchi, Enrico Tassi","doi":"10.4230/LIPIcs.FSCD.2020.34","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.34","url":null,"abstract":"It is nowadays customary to organize libraries of machine checked proofs around hierarchies of algebraic structures [2, 6, 8, 16, 18, 23, 27]. One influential example is the Mathematical Components library on top of which the long and intricate proof of the Odd Order Theorem could be fully formalized [14]. Still, building algebraic hierarchies in a proof assistant such as Coq [9] requires a lot of manual labor and often a deep expertise in the internals of the prover [13, 17]. Moreover, according to our experience [26], making a hierarchy evolve without causing breakage in client code is equally tricky: even a simple refactoring such as splitting a structure into two simpler ones is hard to get right. In this paper we describe HB, a high level language to build hierarchies of algebraic structures and to make these hierarchies evolve without breaking user code. The key concepts are the ones of factory, builder and abbreviation that let the hierarchy developer describe an actual interface for their library. Behind that interface the developer can provide appropriate code to ensure retro compatibility. We implement the HB language in the hierarchy-builder addon for the Coq system using the Elpi [11, 28] extension language.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115620398","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 : 2020-04-24DOI: 10.4230/LIPICS.FSCD.2020.7
Dariusz Biernacki, Sergueï Lenglet, Piotr Polesiuk
Reactive systems a la Leifer and Milner, an abstract categorical framework for rewriting, provide a suitable framework for deriving bisimulation congruences. This is done by synthesizing interactions with the environment in order to obtain a compositional semantics. We enrich the notion of reactive systems by conditions on two levels: first, as in earlier work, we consider rules enriched with application conditions and second, we investigate the notion of conditional bisimilarity. Conditional bisimilarity allows us to say that two system states are bisimilar provided that the environment satisfies a given condition. We present several equivalent definitions of conditional bisimilarity, including one that is useful for concrete proofs and that employs an up-to-context technique, and we compare with related behavioural equivalences. We instantiate reactive systems in order to obtain DPO graph rewriting and consider a case study in this setting.
{"title":"A Complete Normal-Form Bisimilarity for Algebraic Effects and Handlers","authors":"Dariusz Biernacki, Sergueï Lenglet, Piotr Polesiuk","doi":"10.4230/LIPICS.FSCD.2020.7","DOIUrl":"https://doi.org/10.4230/LIPICS.FSCD.2020.7","url":null,"abstract":"Reactive systems a la Leifer and Milner, an abstract categorical framework for rewriting, provide a suitable framework for deriving bisimulation congruences. This is done by synthesizing interactions with the environment in order to obtain a compositional semantics. We enrich the notion of reactive systems by conditions on two levels: first, as in earlier work, we consider rules enriched with application conditions and second, we investigate the notion of conditional bisimilarity. Conditional bisimilarity allows us to say that two system states are bisimilar provided that the environment satisfies a given condition. We present several equivalent definitions of conditional bisimilarity, including one that is useful for concrete proofs and that employs an up-to-context technique, and we compare with related behavioural equivalences. We instantiate reactive systems in order to obtain DPO graph rewriting and consider a case study in this setting.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114202241","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 : 2020-03-10DOI: 10.4230/LIPIcs.FSCD.2019.27
Dominique Larchey-Wendling, Y. Forster
We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $mu$-recursive functions and give a certified compiler from $mu$-recursive functions to Minsky machines.
{"title":"Hilbert's Tenth Problem in Coq","authors":"Dominique Larchey-Wendling, Y. Forster","doi":"10.4230/LIPIcs.FSCD.2019.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2019.27","url":null,"abstract":"We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $mu$-recursive functions and give a certified compiler from $mu$-recursive functions to Minsky machines.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124029574","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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