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On Quantitative Algebraic Higher-Order Theories 关于定量代数高阶理论
International Conference on Formal Structures for Computation and Deduction Pub Date : 2022-04-28 DOI: 10.48550/arXiv.2204.13654
Ugo Dal Lago, F. Honsell, Marina Lenisa, Paolo Pistone
{"title":"On Quantitative Algebraic Higher-Order Theories","authors":"Ugo Dal Lago, F. Honsell, Marina Lenisa, Paolo Pistone","doi":"10.48550/arXiv.2204.13654","DOIUrl":"https://doi.org/10.48550/arXiv.2204.13654","url":null,"abstract":"We explore the possibility of extending Mardare et al quantitative algebras to the structures which naturally emerge from Combinatory Logic and the λ -calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results which clearly delineate to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123330052","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: 4
Strategies for Asymptotic Normalization 渐近归一化策略
International Conference on Formal Structures for Computation and Deduction Pub Date : 2022-04-19 DOI: 10.48550/arXiv.2204.08772
C. Faggian, Giulio Guerrieri
{"title":"Strategies for Asymptotic Normalization","authors":"C. Faggian, Giulio Guerrieri","doi":"10.48550/arXiv.2204.08772","DOIUrl":"https://doi.org/10.48550/arXiv.2204.08772","url":null,"abstract":"We present a technique to study normalizing strategies when termination is asymptotic, that is, it appears as a limit, as opposite to reaching a normal form in a finite number of steps. Asymptotic termination occurs in several settings, such as effectful, and in particular probabilistic computation -- where the limits are distributions over the possible outputs -- or infinitary lambda-calculi -- where the limits are infinitary normal forms such as Boehm trees. As a concrete application, we obtain a result which is of independent interest: a normalization theorem for Call-by-Value (and -- in a uniform way -- for Call-by-Name) probabilistic lambda-calculus.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127379679","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
mwp-Analysis Improvement and Implementation: Realizing Implicit Computational Complexity mwp分析改进与实现:实现隐式计算复杂性
International Conference on Formal Structures for Computation and Deduction Pub Date : 2022-03-08 DOI: 10.48550/arXiv.2203.03943
Clément Aubert, Thomas Rubiano, Neea Rusch, T. Seiller
{"title":"mwp-Analysis Improvement and Implementation: Realizing Implicit Computational Complexity","authors":"Clément Aubert, Thomas Rubiano, Neea Rusch, T. Seiller","doi":"10.48550/arXiv.2203.03943","DOIUrl":"https://doi.org/10.48550/arXiv.2203.03943","url":null,"abstract":"Implicit Computational Complexity (ICC) drives better understanding of complexity classes, but it also guides the development of resources-aware languages and static source code analyzers. Among the methods developed, the mwp-flow analysis certifies polynomial bounds on the size of the values manipulated by an imperative program. This result is obtained by bounding the transitions between states instead of focusing on states in isolation, as most static analyzers do, and is not concerned with termination or tight bounds on values. Those differences, along with its built-in compositionality, make the mwp-flow analysis a good target for determining how ICC-inspired techniques diverge compared with more traditional static analysis methods. This paper's contributions are threefold: we fine-tune the internal machinery of the original analysis to make it tractable in practice; we extend the analysis to function calls and leverage its machinery to compute the result of the analysis efficiently; and we implement the resulting analysis as a lightweight tool to automatically perform data-size analysis of C programs. This documented effort prepares and enables the development of certified complexity analysis, by transforming a costly analysis into a tractable program, that furthermore decorrelates the problem of deciding if a bound exist with the problem of computing it.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125209147","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: 4
Addition and Differentiation of ZX-Diagrams zx图的加法与微分
International Conference on Formal Structures for Computation and Deduction Pub Date : 2022-02-23 DOI: 10.4230/LIPIcs.FSCD.2022.13
E. Jeandel, S. Perdrix, Margarita Veshchezerova
{"title":"Addition and Differentiation of ZX-Diagrams","authors":"E. Jeandel, S. Perdrix, Margarita Veshchezerova","doi":"10.4230/LIPIcs.FSCD.2022.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.13","url":null,"abstract":"The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127953744","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
Galois connecting call-by-value and call-by-name 伽罗瓦连接按值调用和按名称调用
International Conference on Formal Structures for Computation and Deduction Pub Date : 2022-02-16 DOI: 10.4230/LIPIcs.FSCD.2022.32
Dylan McDermott, A. Mycroft
{"title":"Galois connecting call-by-value and call-by-name","authors":"Dylan McDermott, A. Mycroft","doi":"10.4230/LIPIcs.FSCD.2022.32","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.32","url":null,"abstract":"We establish a general framework for reasoning about the relationship between call-by-value and call-by-name. In languages with side-effects, call-by-value and call-by-name executions of programs often have different, but related, observable behaviours. For example, if a program might diverge but otherwise has no side-effects, then whenever it terminates under call-by-value, it terminates with the same result under call-by-name. We propose a technique for stating and proving these properties. The key ingredient is Levy's call-by-push-value calculus, which we use as a framework for reasoning about evaluation orders. We show that the call-by-value and call-by-name translations of expressions into call-by-push-value have related observable behaviour under certain conditions on side-effects, which we identify. We then use this fact to construct maps between the call-by-value and call-by-name interpretations of types, and identify further properties of side-effects that imply these maps form a Galois connection. These properties hold for some side-effects (such as divergence), but not others (such as mutable state). This gives rise to a general reasoning principle that relates call-by-value and call-by-name. We apply the reasoning principle to example side-effects including divergence and nondeterminism.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125269200","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
Concurrent Realizability on Conjunctive Structures 连接结构的并发可实现性
International Conference on Formal Structures for Computation and Deduction Pub Date : 2021-12-29 DOI: 10.4230/LIPIcs.FSCD.2023.28
E. Beffara, F'elix Castro, Mauricio Guillermo
{"title":"Concurrent Realizability on Conjunctive Structures","authors":"E. Beffara, F'elix Castro, Mauricio Guillermo","doi":"10.4230/LIPIcs.FSCD.2023.28","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.28","url":null,"abstract":"The point of this work is to explore axiomatisations of concurrent computation using the technology of proof theory and realizability. To deal with this problem, we redefine the Concurrent Realizability of Beffara using as realizers a $pi$-calculus with global fusions. We define a variant of the Conjunctive Structures of 'E Miquey as a general structure where belong realizers and truth values from realizability. As for Secuential Realizability, we encode the realizers into the algebraic structure by means of a combinatory presentation, following the work of Honda&Yoshida. In this first work we restricted to work with the $pi$-calculus without replication and its corresponding type system is the multiplicative linear logic (MLL).","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122254318","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
Type-Based Termination for Futures 基于类型的期货终止
International Conference on Formal Structures for Computation and Deduction Pub Date : 2021-05-13 DOI: 10.4230/LIPIcs.FSCD.2022.12
Siva Somayyajula, F. Pfenning
{"title":"Type-Based Termination for Futures","authors":"Siva Somayyajula, F. Pfenning","doi":"10.4230/LIPIcs.FSCD.2022.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.12","url":null,"abstract":"In sequential functional languages, sized types enable termination checking of programs with complex patterns of recursion in the presence of mixed inductive-coinductive types. In this paper, we adapt sized types and their metatheory to the concurrent setting. We extend the semi-axiomatic sequent calculus, a subsuming paradigm for futures-based functional concurrency, and its underlying operational semantics with recursion and arithmetic refinements. The latter enables a new and highly general sized type scheme we call sized type refinements. As a widely applicable technical device, we type recursive programs with infinitely deep typing derivations that unfold all recursive calls. Then, we observe that certain such derivations can be made infinitely wide but finitely deep. The resulting trees serve as the induction target of our termination result, which we develop via a novel logical relations argument.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134583705","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
Recursion and Sequentiality in Categories of Sheaves 捆类的递归与序性
International Conference on Formal Structures for Computation and Deduction Pub Date : 2021-05-05 DOI: 10.4230/LIPIcs.FSCD.2021.25
C. Matache, Sean K. Moss, S. Staton
{"title":"Recursion and Sequentiality in Categories of Sheaves","authors":"C. Matache, Sean K. Moss, S. Staton","doi":"10.4230/LIPIcs.FSCD.2021.25","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.25","url":null,"abstract":"We present a fully abstract model of a call-by-value language with higher-order functions, recursion and natural numbers, as an exponential ideal in a topos. Our model is inspired by the fully abstract models of O'Hearn, Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based on cpo's, we treat recursion as just one feature in a model built by combining a choice of modular components.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"268 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125824228","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: 4
What's Decidable about (Atomic) Polymorphism 关于(原子)多态，什么是可确定的
International Conference on Formal Structures for Computation and Deduction Pub Date : 2021-05-03 DOI: 10.4230/LIPIcs.FSCD.2021.27
Paolo Pistone, L. Tranchini
{"title":"What's Decidable about (Atomic) Polymorphism","authors":"Paolo Pistone, L. Tranchini","doi":"10.4230/LIPIcs.FSCD.2021.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.27","url":null,"abstract":"Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most well-known being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127008479","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
Tuple Interpretations for Higher-Order Complexity 高阶复杂性的元组解释
International Conference on Formal Structures for Computation and Deduction Pub Date : 2021-05-03 DOI: 10.4230/LIPIcs.FSCD.2021.31
Cynthia Kop, Deivid Vale
{"title":"Tuple Interpretations for Higher-Order Complexity","authors":"Cynthia Kop, Deivid Vale","doi":"10.4230/LIPIcs.FSCD.2021.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2021.31","url":null,"abstract":"We develop a class of algebraic interpretations for many-sorted and higher-order term rewriting systems that takes type information into account. Specifically, base-type terms are mapped to emph{tuples} of natural numbers and higher-order terms to functions between those tuples. Tuples may carry information relevant to the type; for instance, a term of type $mathsf{nat}$ may be associated to a pair $(mathsf{cost}, mathsf{size})$ representing its evaluation cost and size. This class of interpretations results in a more fine-grained notion of complexity than runtime or derivational complexity, which makes it particularly useful to obtain complexity bounds for higher-order rewriting systems. We show that rewriting systems compatible with tuple interpretations admit finite bounds on derivation height. Furthermore, we demonstrate how to mechanically construct tuple interpretations and how to orient $beta$ and $eta$ reductions within our technique. Finally, we relate our method to runtime complexity and prove that specific interpretation shapes imply certain runtime complexity bounds.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123215641","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: 8
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