Pub Date : 1900-01-01DOI: 10.4230/LIPIcs.FSCD.2022.20
Anupam Das, A. De, A. Saurin
Linear logic is an important logic for modelling resources and decomposing computational interpretations of proofs. Decision problems for fragments of linear logic exhibiting “infinitary” behaviour (such as exponentials) are notoriously complicated. In this work, we address the decision problems for variations of linear logic with fixed points ( µ MALL ), in particular, recent systems based on “circular” and “non-wellfounded” reasoning. In this paper, we show that µ MALL is undecidable. More explicitly, we show that the general non-wellfounded system is Π 01 -hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ 01 ). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ 01 -complete via a reduction to the reachability problem of alternating vector addition systems with states. This implies that both the circular system and the finitary system (with explicit (co)induction) are Σ 01 -complete.
{"title":"Decision Problems for Linear Logic with Least and Greatest Fixed Points","authors":"Anupam Das, A. De, A. Saurin","doi":"10.4230/LIPIcs.FSCD.2022.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.20","url":null,"abstract":"Linear logic is an important logic for modelling resources and decomposing computational interpretations of proofs. Decision problems for fragments of linear logic exhibiting “infinitary” behaviour (such as exponentials) are notoriously complicated. In this work, we address the decision problems for variations of linear logic with fixed points ( µ MALL ), in particular, recent systems based on “circular” and “non-wellfounded” reasoning. In this paper, we show that µ MALL is undecidable. More explicitly, we show that the general non-wellfounded system is Π 01 -hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ 01 ). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ 01 -complete via a reduction to the reachability problem of alternating vector addition systems with states. This implies that both the circular system and the finitary system (with explicit (co)induction) are Σ 01 -complete.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"5 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":"114996702","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: 10.4230/LIPIcs.FSCD.2020.22
Kazuyuki Asada, N. Kobayashi
Higher-order grammars have recently been studied actively in the context of automated verification of higher-order programs. Asada and Kobayashi have previously shown that, for any order-(n+ 1) word grammar, there exists an order-n grammar whose frontier language coincides with the language generated by the word grammar. Their translation, however, blows up the size of the grammar, which inhibited complexity-preserving reductions from decision problems on word grammars to those on tree grammars. In this paper, we present a new translation from order-(n+ 1) word grammars to order-n tree grammars that is size-preserving in the sense that the size of the output tree grammar is polynomial in the size of an input tree grammar. The new translation and its correctness proof are arguably much simpler than the previous translation and proof. 2012 ACM Subject Classification Theory of computation→ Formal languages and automata theory
{"title":"Size-Preserving Translations from Order-(n+1) Word Grammars to Order-n Tree Grammars","authors":"Kazuyuki Asada, N. Kobayashi","doi":"10.4230/LIPIcs.FSCD.2020.22","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.22","url":null,"abstract":"Higher-order grammars have recently been studied actively in the context of automated verification of higher-order programs. Asada and Kobayashi have previously shown that, for any order-(n+ 1) word grammar, there exists an order-n grammar whose frontier language coincides with the language generated by the word grammar. Their translation, however, blows up the size of the grammar, which inhibited complexity-preserving reductions from decision problems on word grammars to those on tree grammars. In this paper, we present a new translation from order-(n+ 1) word grammars to order-n tree grammars that is size-preserving in the sense that the size of the output tree grammar is polynomial in the size of an input tree grammar. The new translation and its correctness proof are arguably much simpler than the previous translation and proof. 2012 ACM Subject Classification Theory of computation→ Formal languages and automata theory","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"9 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":"116803198","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: 10.4230/LIPIcs.FSCD.2017.6
Y. Akama
For the lambda-calculus with surjective pairing and terminal type, Curien and Di Cosmo, inspired by Knuth-Bendix completion, introduced a confluent rewriting system of the naive rewriting system. Their system is a confluent (CR) rewriting system stable under contexts. They left the strong normalization (SN) of their rewriting system open. By Girard's reducibility method with restricting reducibility theorem, we prove SN of their rewriting, and SN of the extensions by polymorphism and (terminal types caused by parametric polymorphism). We extend their system by sum types and eta-like reductions, and prove the SN. We compare their system to type-directed expansions.
{"title":"The Confluent Terminating Context-Free Substitutive Rewriting System for the lambda-Calculus with Surjective Pairing and Terminal Type","authors":"Y. Akama","doi":"10.4230/LIPIcs.FSCD.2017.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.6","url":null,"abstract":"For the lambda-calculus with surjective pairing and terminal type, Curien and Di Cosmo, inspired by Knuth-Bendix completion, introduced a confluent rewriting system of the naive rewriting system. Their system is a confluent (CR) rewriting system stable under contexts. They left the strong normalization (SN) of their rewriting system open. By Girard's reducibility method with restricting reducibility theorem, we prove SN of their rewriting, and SN of the extensions by polymorphism and (terminal types caused by parametric polymorphism). We extend their system by sum types and eta-like reductions, and prove the SN. We compare their system to type-directed expansions.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"21 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":"125760442","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: 10.4230/LIPIcs.FSCD.2022.24
F. Blanqui
The encoding of proof systems and type theories in logical frameworks is key to allow the translation of proofs from one system to the other. The λ Π-calculus modulo rewriting is a powerful logical framework in which various systems have already been encoded, including type systems with an infinite hierarchy of type universes equipped with a unary successor operator and a binary max operator: Matita, Coq, Agda and Lean. However, to decide the word problem in this max-successor algebra, all the encodings proposed so far use rewriting with matching modulo associativity and commutativity (AC), which is of high complexity and difficult to integrate in usual algorithms for β -reduction and type-checking. In this paper, we show that we do not need matching modulo AC by enforcing terms to be in some special canonical form wrt associativity and commutativity, and by using rewriting rules taking advantage of this canonical form. This work has been implemented in the proof assistant Lambdapi. paper, Gaspard Férey for his remarks on a first version of this paper, as well as the anonymous reviewers for their suggestions.
{"title":"Encoding Type Universes Without Using Matching Modulo Associativity and Commutativity","authors":"F. Blanqui","doi":"10.4230/LIPIcs.FSCD.2022.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.24","url":null,"abstract":"The encoding of proof systems and type theories in logical frameworks is key to allow the translation of proofs from one system to the other. The λ Π-calculus modulo rewriting is a powerful logical framework in which various systems have already been encoded, including type systems with an infinite hierarchy of type universes equipped with a unary successor operator and a binary max operator: Matita, Coq, Agda and Lean. However, to decide the word problem in this max-successor algebra, all the encodings proposed so far use rewriting with matching modulo associativity and commutativity (AC), which is of high complexity and difficult to integrate in usual algorithms for β -reduction and type-checking. In this paper, we show that we do not need matching modulo AC by enforcing terms to be in some special canonical form wrt associativity and commutativity, and by using rewriting rules taking advantage of this canonical form. This work has been implemented in the proof assistant Lambdapi. paper, Gaspard Férey for his remarks on a first version of this paper, as well as the anonymous reviewers for their suggestions.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"59 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":"129461124","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: 10.4230/LIPIcs.FSCD.2023.24
Thorsten Altenkirch, A. Kaposi, Artjoms Šinkarovs, Tamás Végh
It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with
{"title":"Combinatory Logic and Lambda Calculus Are Equal, Algebraically","authors":"Thorsten Altenkirch, A. Kaposi, Artjoms Šinkarovs, Tamás Végh","doi":"10.4230/LIPIcs.FSCD.2023.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.24","url":null,"abstract":"It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"50 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":"127551387","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: 10.4230/LIPIcs.FSCD.2018.30
S. Winkler, A. Middeldorp
We propose an abstract completion procedure for logically constrained term rewrite systems (LCTRSs). This procedure can be instantiated to both standard Knuth-Bendix completion and ordered completion for LCTRSs, and we present a succinct and uniform correctness proof. A prototype implementation illustrates the viability of the new completion approach. 2012 ACM Subject Classification Theory of computation → Rewrite systems, Theory of computation → Equational logic and rewriting, Theory of computation → Automated reasoning
{"title":"Completion for Logically Constrained Rewriting","authors":"S. Winkler, A. Middeldorp","doi":"10.4230/LIPIcs.FSCD.2018.30","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2018.30","url":null,"abstract":"We propose an abstract completion procedure for logically constrained term rewrite systems (LCTRSs). This procedure can be instantiated to both standard Knuth-Bendix completion and ordered completion for LCTRSs, and we present a succinct and uniform correctness proof. A prototype implementation illustrates the viability of the new completion approach. 2012 ACM Subject Classification Theory of computation → Rewrite systems, Theory of computation → Equational logic and rewriting, Theory of computation → Automated reasoning","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"101 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":"130631633","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: 10.4230/LIPIcs.FSCD.2016.35
Stéphane Gimenez, David Obwaller
We introduce interaction automata as a topological model of computation and present the conceptual plane interpreter ia2d. Interaction automata form a refinement of both interaction nets and cellular automata models that combine data deployment, memory management and structured computation mechanisms. Their local structure is inspired from pointer machines and allows an asynchronous spatial distribution of the computation. Our tool can be considered as a proof-of-concept piece of abstract hardware on which functional programs can be run in parallel.
{"title":"Interaction Automata and the ia2d Interpreter","authors":"Stéphane Gimenez, David Obwaller","doi":"10.4230/LIPIcs.FSCD.2016.35","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2016.35","url":null,"abstract":"We introduce interaction automata as a topological model of \u0000computation and present the conceptual plane interpreter ia2d. \u0000Interaction automata form a refinement of both interaction nets and \u0000cellular automata models that combine data deployment, memory \u0000management and structured computation mechanisms. Their local \u0000structure is inspired from pointer machines and allows an asynchronous \u0000spatial distribution of the computation. Our tool can be considered \u0000as a proof-of-concept piece of abstract hardware on which functional \u0000programs can be run in parallel.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"17 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":"130732130","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: 10.4230/LIPIcs.FSCD.2023.31
D. Galmiche, D. Méry
In this paper, we define the logic of Linear Temporal Bunched Implications ( LTBI ), a temporal extension of the Bunched Implications logic BI that deals with resource evolution over time, by combining the BI separation connectives and the LTL temporal connectives. We first present the syntax and semantics of LTBI and illustrate its expressiveness with a significant example. Then we introduce a tableau calculus with labels and constraints, called T LTBI , and prove its soundness w.r.t. the Kripke-style semantics of LTBI . Finally we discuss and analyze the issues that make the completeness of the calculus not trivial in the general case of unbounded timelines and explain how to solve the issues in the more restricted case of bounded timelines.
{"title":"Labelled Tableaux for Linear Time Bunched Implication Logic","authors":"D. Galmiche, D. Méry","doi":"10.4230/LIPIcs.FSCD.2023.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.31","url":null,"abstract":"In this paper, we define the logic of Linear Temporal Bunched Implications ( LTBI ), a temporal extension of the Bunched Implications logic BI that deals with resource evolution over time, by combining the BI separation connectives and the LTL temporal connectives. We first present the syntax and semantics of LTBI and illustrate its expressiveness with a significant example. Then we introduce a tableau calculus with labels and constraints, called T LTBI , and prove its soundness w.r.t. the Kripke-style semantics of LTBI . Finally we discuss and analyze the issues that make the completeness of the calculus not trivial in the general case of unbounded timelines and explain how to solve the issues in the more restricted case of bounded timelines.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"80 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":"132391462","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: 10.4230/LIPIcs.FSCD.2016.3
G. Huet
We describe experiments in teaching fundamental informatics notions around mathematical structures for formal concepts, and effective algorithms to manipulate them. The major themes of lambda-calculus and type theory served as guides for the effective implementation of functional programming languages and higher-order proof assistants, appropriate for reflecting the theoretical material into effective tools to represent constructively the concepts and formally certify the proofs of their properties. Progressively, a literate programming and proving style replaced informal mathematics in the presentation of the material as executable course notes. The talk will evoke the various stages of (in)completion of the corresponding set of notes along the years, and tell how their elaboration proved to be essential to the discovery of fundamental results.
{"title":"Teaching Foundations of Computation and Deduction Through Literate Functional Programming and Type Theory Formalization","authors":"G. Huet","doi":"10.4230/LIPIcs.FSCD.2016.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2016.3","url":null,"abstract":"We describe experiments in teaching fundamental informatics notions around mathematical structures for formal concepts, and effective algorithms to manipulate them. The major themes of lambda-calculus and type theory served as guides for the effective implementation of functional programming languages and higher-order proof assistants, appropriate for reflecting the theoretical material into effective tools to represent constructively the concepts and formally certify the proofs of their properties. Progressively, a literate programming and proving style replaced informal mathematics in the presentation of the material as executable course notes. The talk will evoke the various stages of (in)completion of the corresponding set of notes along the years, and tell how their elaboration proved to be essential to the discovery of fundamental results.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 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":"131232170","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: 10.4230/LIPIcs.FSCD.2021.14
Anupam Das, A. Rice
A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems. Systems of linear logic and, later, deep inference are founded upon two particular linear inferences, switch : x ∧ ( y ∨ z ) → ( x ∧ y ) ∨ z , and medial : ( w ∧ x ) ∨ ( y ∧ z ) → ( w ∨ y ) ∧ ( x ∨ z ). It is well-known that these two are not enough to derive all linear inferences (even modulo all valid linear equations), but beyond this little more is known about the structure of linear inferences in general. In particular despite recurring attention in the literature, the smallest linear inference not derivable under switch and medial (‘switch-medial-independent’) was not previously known. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find two ‘minimal’ 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. One of these new inferences derives some previously found independent linear inferences. The other exhibits structure seemingly beyond the scope of previous approaches we are aware of; in particular, its existence contradicts a conjecture of Das and Strassburger.
{"title":"New Minimal Linear Inferences in Boolean Logic Independent of Switch and Medial","authors":"Anupam Das, A. Rice","doi":"10.4230/LIPIcs.FSCD.2021.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.14","url":null,"abstract":"A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems. Systems of linear logic and, later, deep inference are founded upon two particular linear inferences, switch : x ∧ ( y ∨ z ) → ( x ∧ y ) ∨ z , and medial : ( w ∧ x ) ∨ ( y ∧ z ) → ( w ∨ y ) ∧ ( x ∨ z ). It is well-known that these two are not enough to derive all linear inferences (even modulo all valid linear equations), but beyond this little more is known about the structure of linear inferences in general. In particular despite recurring attention in the literature, the smallest linear inference not derivable under switch and medial (‘switch-medial-independent’) was not previously known. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find two ‘minimal’ 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. One of these new inferences derives some previously found independent linear inferences. The other exhibits structure seemingly beyond the scope of previous approaches we are aware of; in particular, its existence contradicts a conjecture of Das and Strassburger.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"13 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":"120947514","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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