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Fixed Points in Computer Science最新文献

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Non-monotonic Pre-fixed Points and Learning 非单调前不动点与学习
Fixed Points in Computer Science Pub Date : 2013-09-03 DOI: 10.4204/EPTCS.126.1
S. Berardi, Ugo de'Liguoro
{"title":"Non-monotonic Pre-fixed Points and Learning","authors":"S. Berardi, Ugo de'Liguoro","doi":"10.4204/EPTCS.126.1","DOIUrl":"https://doi.org/10.4204/EPTCS.126.1","url":null,"abstract":"We consider the problem of finding pre-fixed points of interactive realizers over arbitrary knowledge spaces, obtaining a relative recursive procedure. Knowledge spaces and interactive realizers are an abstract setting to represent learning processes, that can interpret non-constructive proofs. Atomic pieces of information of a knowledge space are stratified into levels, and evaluated into truth values depending on knowledge states. Realizers are then used to define operators that extend a given state by adding and possibly removing atoms: in a learning process states of knowledge change nonmonotonically. Existence of a pre-fixed point of a realizer is equivalent to the termination of the learning process with some state of knowledge which is free of patent contradictions and such that there is nothing to add. In this paper we generalize our previous results in the case of level 2 knowledge spaces and deterministic operators to the case of omega-level knowledge spaces and of non-deterministic operators.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129659574","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
A Coinductive Approach to Proof Search 证明搜索的一种共归纳方法
Fixed Points in Computer Science Pub Date : 2013-08-28 DOI: 10.4204/EPTCS.126.3
J. E. Santo, R. Matthes, L. Pinto
{"title":"A Coinductive Approach to Proof Search","authors":"J. E. Santo, R. Matthes, L. Pinto","doi":"10.4204/EPTCS.126.3","DOIUrl":"https://doi.org/10.4204/EPTCS.126.3","url":null,"abstract":"We propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin's LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the \"solution spaces\" (called Bohm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent). As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study \"co-contraction\" of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"79 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120850478","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
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion 保护你的匕首和痕迹:关于保护(Co-)递归的等式性质
Fixed Points in Computer Science Pub Date : 2013-08-28 DOI: 10.4204/EPTCS.126.6
Stefan Milius, Tadeusz Litak
{"title":"Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion","authors":"Stefan Milius, Tadeusz Litak","doi":"10.4204/EPTCS.126.6","DOIUrl":"https://doi.org/10.4204/EPTCS.126.6","url":null,"abstract":"Motivated by the recent interest in models of guarded (co-)recursion we study its equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading to the description of classifying theories for guarded recursion and hence completeness results involving our axioms of guarded fixpoint operators in future work.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"272 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116118580","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
The Fixed-Point Theory of Strictly Contracting Functions on Generalized Ultrametric Semilattices 广义超度量半格上严格收缩函数的不动点理论
Fixed Points in Computer Science Pub Date : 2013-08-28 DOI: 10.4204/EPTCS.126.5
E. Matsikoudis, Edward A. Lee
{"title":"The Fixed-Point Theory of Strictly Contracting Functions on Generalized Ultrametric Semilattices","authors":"E. Matsikoudis, Edward A. Lee","doi":"10.4204/EPTCS.126.5","DOIUrl":"https://doi.org/10.4204/EPTCS.126.5","url":null,"abstract":"We introduce a new class of abstract structures, which we call generalized ultrametric semilattices, and in which the meet operation of the semilattice coexists with a generalized distance function in a tightly coordinated way. We prove a constructive fixed-point theorem for strictly contracting functions on directed-complete generalized ultrametric semilattices, and introduce a corresponding induction principle. We cite examples of application in the semantics of logic programming and timed computation, where, until now, the only tool available has been the non-constructive fixed-point theorem of Priess-Crampe and Ribenboim for strictly contracting functions on spherically complete generalized ultrametric semilattices.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123500891","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: 3
The Equivalence of Game and Denotational Semantics for the Probabilistic mu-Calculus 概率mu-微积分的博弈等价性和指称语义
Fixed Points in Computer Science Pub Date : 2012-05-01 DOI: 10.2168/LMCS-8(2:7)2012
M. Mio
{"title":"The Equivalence of Game and Denotational Semantics for the Probabilistic mu-Calculus","authors":"M. Mio","doi":"10.2168/LMCS-8(2:7)2012","DOIUrl":"https://doi.org/10.2168/LMCS-8(2:7)2012","url":null,"abstract":"The probabilistic (or quantitative) modal mu-calculus is a fixed-point logic de- signed for expressing properties of probabilistic labeled transition systems (PLTS). Two semantics have been studied for this logic, both assigning to every process state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS's, but the equivalence of the two semantics on arbitrary models has been open in literature. In this paper we prove that the equivalence indeed holds for arbitrary infinite models, and thus our result strengthens the fruitful connection between denotational and game semantics. Our proof adapts the unraveling or unfolding method, a general proof technique for proving result of parity games by induction on their complexity.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"139 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120998691","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: 16
Characteristic Formulae for Relations with Nested Fixed Points 嵌套不动点关系的特征公式
Fixed Points in Computer Science Pub Date : 2012-02-15 DOI: 10.4204/EPTCS.77.3
L. Aceto, A. Ingólfsdóttir
{"title":"Characteristic Formulae for Relations with Nested Fixed Points","authors":"L. Aceto, A. Ingólfsdóttir","doi":"10.4204/EPTCS.77.3","DOIUrl":"https://doi.org/10.4204/EPTCS.77.3","url":null,"abstract":"A general framework for the connection between characteristic formulae and behavioral semantics is described in [2]. This approach does not suitably cover semantics defined by nested fixed points, such as the n-nested simulation semantics for n greater than 2. In this study we address this deficiency and give a description of nested fixed points that extends the approach for single fixed points in an intuitive and comprehensive way.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121463569","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
Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types 基于类型的终止、膨胀不动点和混合诱导-共归纳类型
Fixed Points in Computer Science Pub Date : 2012-02-15 DOI: 10.4204/EPTCS.77.1
Andreas Abel
{"title":"Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types","authors":"Andreas Abel","doi":"10.4204/EPTCS.77.1","DOIUrl":"https://doi.org/10.4204/EPTCS.77.1","url":null,"abstract":"Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130346408","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: 33
Structured general corecursion and coinductive graphs [extended abstract] 结构化一般共递和协归纳图[扩展摘要]
Fixed Points in Computer Science Pub Date : 2012-02-15 DOI: 10.4204/EPTCS.77.8
Tarmo Uustalu
{"title":"Structured general corecursion and coinductive graphs [extended abstract]","authors":"Tarmo Uustalu","doi":"10.4204/EPTCS.77.8","DOIUrl":"https://doi.org/10.4204/EPTCS.77.8","url":null,"abstract":"Bove and Capretta’s popular method for justifying function definitions by general recursive equations is based on the observation that any structured general recursion equation defines an inductive subset of the intended domain (the “domain of definedness”) for which the equation has a unique solution. To accept the definition, it is hence enough to prove that this subset contains the whole intended domain. This approach works very well for “terminating” definitions. But it fails to account for “productive” definitions, such as typical definitions of stream-valued functions. We argue that such definitions can be treated in a similar spirit, proceeding from a different unique solvability criterion. Any structured recursive equation defines a coinductive relation between the intended domain and intended codomain (the “coinductive graph”). This relation in turn determines a subset of the intended domain and a quotient of the intended codomain with the property that the equation is uniquely solved for the subset and quotient. The equation is therefore guaranteed to have a unique solution for the intended domain and intended codomain whenever the subset is the full set and the quotient is by equality.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125292311","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
Cut-elimination for the mu-calculus with one variable 一元微积分的消元法
Fixed Points in Computer Science Pub Date : 2012-02-15 DOI: 10.4204/EPTCS.77.7
G. Mints, T. Studer
{"title":"Cut-elimination for the mu-calculus with one variable","authors":"G. Mints, T. Studer","doi":"10.4204/EPTCS.77.7","DOIUrl":"https://doi.org/10.4204/EPTCS.77.7","url":null,"abstract":"We establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Our method is based on a recent cut-elimination technique by Mints that makes use of Buchholz’ W-rule.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128903623","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: 7
Initial Semantics for Strengthened Signatures 增强签名的初始语义
Fixed Points in Computer Science Pub Date : 2012-02-15 DOI: 10.4204/EPTCS.77.5
A. Hirschowitz, M. Maggesi
{"title":"Initial Semantics for Strengthened Signatures","authors":"A. Hirschowitz, M. Maggesi","doi":"10.4204/EPTCS.77.5","DOIUrl":"https://doi.org/10.4204/EPTCS.77.5","url":null,"abstract":"We give a new general definition of arity, yielding the companion notions of signature and associated syntax. This setting is modular in the sense requested by Ghani and Uustalu: merging two extensions of syntax corresponds to building an amalgamated sum. These signatures are too general in the sense that we are not able to prove the existence of an associated syntax in this general context. So we have to select arities and signatures for which there exists the desired initial monad. For this, we follow a track opened by Matthes and Uustalu: we introduce a notion of strengthened arity and prove that the corresponding signatures have initial semantics (i.e. associated syntax). Our strengthened arities admit colimits, which allows the treatment of the lambda-calculus with explicit substitution.","PeriodicalId":119563,"journal":{"name":"Fixed Points in Computer Science","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134641641","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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