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Types as Resources for Classical Natural Deduction 类型作为经典自然演绎的资源
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2017.24
D. Kesner, Pierre Vial
{"title":"Types as Resources for Classical Natural Deduction","authors":"D. Kesner, Pierre Vial","doi":"10.4230/LIPIcs.FSCD.2017.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.24","url":null,"abstract":"We define two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments –based on decreasing measures of type derivations– to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences. 1998 ACM Subject Classification F.4.1 Mathematical Logic","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"73 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":"115691268","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}
引用次数: 32
A Bicategorical Model for Finite Nondeterminism 有限不确定性的双范畴模型
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2021.10
Z. Galal
{"title":"A Bicategorical Model for Finite Nondeterminism","authors":"Z. Galal","doi":"10.4230/LIPIcs.FSCD.2021.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.10","url":null,"abstract":"Finiteness spaces were introduced by Ehrhard as a refinement of the relational model of linear logic. A finiteness space is a set equipped with a class of finitary subsets which can be thought of being subsets that behave like finite sets. A morphism between finiteness spaces is a relation that preserves the finitary structure. This model provided a semantics for finite non-determism and it gave a semantical motivation for differential linear logic and the syntactic notion of Taylor expansion. In this paper, we present a bicategorical extension of this construction where the relational model is replaced with the model of generalized species of structures introduced by Fiore et al. and the finiteness property now relies on finite presentability. 2012 ACM Subject Classification Theory of computation → Linear logic; Theory of computation → Categorical semantics","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":"123078472","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}
引用次数: 2
Beth Semantics and Labelled Deduction for Intuitionistic Sentential Calculus with Identity 具有恒等的直觉句演算的贝斯语义与标记演绎
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2021.13
D. Galmiche, Marta Gawek, D. Méry
{"title":"Beth Semantics and Labelled Deduction for Intuitionistic Sentential Calculus with Identity","authors":"D. Galmiche, Marta Gawek, D. Méry","doi":"10.4230/LIPIcs.FSCD.2021.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.13","url":null,"abstract":"In this paper we consider the intuitionistic sentential calculus with Suszko’s identity (ISCI). After recalling the basic concepts of the logic and its associated Hilbert proof system, we introduce a new sound and complete class of models for ISCI which can be viewed as algebraic counterparts (and extensions) of sheaf-theoretic topological models of intuitionistic logic. We use this new class of models, called Beth semantics for ISCI, to derive a first labelled sequent calculus and show its adequacy w.r.t. the standard Hilbert axiomatization of ISCI. This labelled proof system, like all other current proof systems for ISCI that we know of, does not enjoy the subformula property, which is problematic for achieving termination. We therefore introduce a second labelled sequent calculus in which the standard rules for identity are replaced with new special rules and show that this second calculus admits cut-elimination. Finally, using a key regularity property of the forcing relation in Beth models, we show that the eigenvariable condition can be dropped, thus leading to the termination and decidability results. 2012 ACM Subject Classification Theory of computation","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"29 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":"116700007","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}
引用次数: 0
Hydra Battles and AC Termination 九头蛇战斗和AC终止
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2023.12
Nao Hirokawa, A. Middeldorp
{"title":"Hydra Battles and AC Termination","authors":"Nao Hirokawa, A. Middeldorp","doi":"10.4230/LIPIcs.FSCD.2023.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.12","url":null,"abstract":"We present a new encoding of the Battle of Hercules and Hydra as a rewrite system with AC symbols. Unlike earlier term rewriting encodings, it faithfully models any strategy of Hercules to beat Hydra. To prove the termination of our encoding, we employ type introduction in connection with many-sorted semantic labeling for AC rewriting and AC-RPO. 2012 ACM Subject Classification Theory of computation → Equational logic and rewriting; Theory of computation → Rewrite systems; Theory of computation → Computability Keywords","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"47 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":"123581588","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}
引用次数: 1
Automata-Based Verification of Relational Properties of Functions over Algebraic Data Structures 代数数据结构上函数关系性质的自动机验证
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2023.7
Théo Losekoot, T. Genet, T. Jensen
{"title":"Automata-Based Verification of Relational Properties of Functions over Algebraic Data Structures","authors":"Théo Losekoot, T. Genet, T. Jensen","doi":"10.4230/LIPIcs.FSCD.2023.7","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.7","url":null,"abstract":"This paper is concerned with automatically proving properties about the input-output relation of functional programs operating over algebraic data types. Recent results show how to approximate the image of a functional program using a regular tree language. Though expressive, those techniques cannot prove properties relating the input and the output of a function, e.g., proving that the output of a function reversing a list has the same length as the input list. In this paper, we built upon those results and define a procedure to compute or over-approximate such a relation. Instead of representing the image of a function by a regular set of terms, we represent (an approximation of) the input-output relation by a regular set of tuples of terms. Regular languages of tuples of terms are recognized using a tree automaton recognizing convolutions of terms, where a convolution transforms a tuple of terms into a term built on tuples of symbols. Both the program and the properties are transformed into predicates and Constrained Horn clauses (CHCs). Then, using an Implication Counter Example procedure (ICE), we infer a model of the clauses, associating to each predicate a regular relation. In this ICE procedure, checking if a given model satisfies the clauses is undecidable in general. We overcome undecidability by proposing an incomplete but sound inference procedure for such relational regular properties. Though the procedure is incomplete, its implementation performs well on 120 examples. It efficiently proves non-trivial relational properties or finds counter-examples.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"49 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":"122800895","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}
引用次数: 0
Unital Anti-Unification: Type and Algorithms 统一反统一:类型和算法
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2020.26
David M. Cerna, Temur Kutsia
{"title":"Unital Anti-Unification: Type and Algorithms","authors":"David M. Cerna, Temur Kutsia","doi":"10.4230/LIPIcs.FSCD.2020.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2020.26","url":null,"abstract":"Unital equational theories are defined by axioms that assert the existence of the unit element for 9 some function symbols. We study anti-unification (AU) in unital theories and address the problems 10 of establishing generalization type and designing anti-unification algorithms. First, we prove that 11 when the term signature contains at least two unital functions, anti-unification is of the nullary 12 type by showing that there exists an AU problem, which does not have a minimal complete set of 13 generalizations. Next, we consider two special cases: the linear variant and the fragment with only 14 one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, 15 complete, and return tree grammars from which the set of generalizations can be constructed. 16 Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment 17 is extended to the unrestricted case. It terminates and returns a tree grammar which produces an 18 infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification 19 might affect the anti-unification problem in some combined theories, and list some open questions.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"31 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":"124178218","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}
引用次数: 6
Analysing Privacy-Type Properties in Cryptographic Protocols (Invited Talk) 分析加密协议中的隐私型属性(特邀演讲)
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2018.1
S. Delaune
{"title":"Analysing Privacy-Type Properties in Cryptographic Protocols (Invited Talk)","authors":"S. Delaune","doi":"10.4230/LIPIcs.FSCD.2018.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2018.1","url":null,"abstract":"","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":"114722162","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}
引用次数: 1
Categorical Coherence from Term Rewriting Systems 术语重写系统的范畴相干性
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2023.16
S. Mimram
{"title":"Categorical Coherence from Term Rewriting Systems","authors":"S. Mimram","doi":"10.4230/LIPIcs.FSCD.2023.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.16","url":null,"abstract":"The celebrated Squier theorem allows to prove coherence properties of algebraic structures, such as MacLane’s coherence theorem for monoidal categories, based on rewriting techniques. We are interested here in extending the theory and associated tools simultaneously in two directions. Firstly, we want to take in account situations where coherence is partial, in the sense that it only applies for a subset of structural morphisms (for instance, in the case of the coherence theorem for symmetric monoidal categories, we do not want to strictify the symmetry). Secondly, we are interested in structures where variables can be duplicated or erased. We develop theorems and rewriting techniques in order to achieve this, first in the setting of abstract rewriting systems, and then extend them to term rewriting systems, suitably generalized in order to take coherence in account. As an illustration of our results, we explain how to recover the coherence theorems for monoidal and symmetric monoidal categories.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"2013 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":"125678328","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}
引用次数: 0
Convolution Products on Double Categories and Categorification of Rule Algebras 双范畴上的卷积积与规则代数的范畴化
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2023.17
Nicolas Behr, Paul-André Melliès, N. Zeilberger
{"title":"Convolution Products on Double Categories and Categorification of Rule Algebras","authors":"Nicolas Behr, Paul-André Melliès, N. Zeilberger","doi":"10.4230/LIPIcs.FSCD.2023.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.17","url":null,"abstract":"Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"57 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":"131440241","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}
引用次数: 0
Duality in Action (Invited Talk) 双重性在行动(特邀演讲)
International Conference on Formal Structures for Computation and Deduction Pub Date : 1900-01-01 DOI: 10.4230/LIPIcs.FSCD.2021.1
P. Downen, Z. M. Ariola
{"title":"Duality in Action (Invited Talk)","authors":"P. Downen, Z. M. Ariola","doi":"10.4230/LIPIcs.FSCD.2021.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.1","url":null,"abstract":"","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"29 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":"128809618","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}
引用次数: 0
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