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The Big-O Problem 大o问题
Log. Methods Comput. Sci. Pub Date : 2020-07-15 DOI: 10.46298/lmcs-18(1:40)2022
D. Chistikov, S. Kiefer, A. Murawski, David Purser
{"title":"The Big-O Problem","authors":"D. Chistikov, S. Kiefer, A. Murawski, David Purser","doi":"10.46298/lmcs-18(1:40)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(1:40)2022","url":null,"abstract":"Given two weighted automata, we consider the problem of whether one is big-O\u0000of the other, i.e., if the weight of every finite word in the first is not\u0000greater than some constant multiple of the weight in the second.\u0000 We show that the problem is undecidable, even for the instantiation of\u0000weighted automata as labelled Markov chains. Moreover, even when it is known\u0000that one weighted automaton is big-O of another, the problem of finding or\u0000approximating the associated constant is also undecidable.\u0000 Our positive results show that the big-O problem is polynomial-time solvable\u0000for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e.,\u0000when the alphabet is a single character) and decidable, subject to Schanuel's\u0000conjecture, when the language is bounded (i.e., a subset of $w_1^*dots w_m^*$\u0000for some finite words $w_1,dots,w_m$) or when the automaton has finite\u0000ambiguity.\u0000 On labelled Markov chains, the problem can be restated as a ratio total\u0000variation distance, which, instead of finding the maximum difference between\u0000the probabilities of any two events, finds the maximum ratio between the\u0000probabilities of any two events. The problem is related to\u0000$varepsilon$-differential privacy, for which the optimal constant of the big-O\u0000notation is exactly $exp(varepsilon)$.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130992675","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
Foundations of Online Structure Theory II: The Operator Approach 在线结构理论基础II:算子方法
Log. Methods Comput. Sci. Pub Date : 2020-07-14 DOI: 10.46298/lmcs-17(3:6)2021
R. Downey, A. Melnikov, K. Ng
{"title":"Foundations of Online Structure Theory II: The Operator Approach","authors":"R. Downey, A. Melnikov, K. Ng","doi":"10.46298/lmcs-17(3:6)2021","DOIUrl":"https://doi.org/10.46298/lmcs-17(3:6)2021","url":null,"abstract":"We introduce a framework for online structure theory. Our approach\u0000generalises notions arising independently in several areas of computability\u0000theory and complexity theory. We suggest a unifying approach using operators\u0000where we allow the input to be a countable object of an arbitrary complexity.\u0000We give a new framework which (i) ties online algorithms with computable\u0000analysis, (ii) shows how to use modifications of notions from computable\u0000analysis, such as Weihrauch reducibility, to analyse finite but uniform\u0000combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine\u0000structure of finite analogs of infinite combinatorial problems, and (iv) see\u0000how similar ideas can be amalgamated from areas such as EX-learning, computable\u0000analysis, distributed computing and the like. One of the key ideas is that\u0000online algorithms can be viewed as a sub-area of computable analysis.\u0000Conversely, we also get an enrichment of computable analysis from classical\u0000online algorithms.\u0000","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130382555","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: 13
Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions 基于模糊Lax扩展的行为半度量的特征逻辑
Log. Methods Comput. Sci. Pub Date : 2020-07-02 DOI: 10.46298/lmcs-18(2:19)2022
P. Wild, Lutz Schröder
{"title":"Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions","authors":"P. Wild, Lutz Schröder","doi":"10.46298/lmcs-18(2:19)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(2:19)2022","url":null,"abstract":"In systems involving quantitative data, such as probabilistic, fuzzy, or\u0000metric systems, behavioural distances provide a more fine-grained comparison of\u0000states than two-valued notions of behavioural equivalence or behaviour\u0000inclusion. Like in the two-valued case, the wide variation found in system\u0000types creates a need for generic methods that apply to many system types at\u0000once. Approaches of this kind are emerging within the paradigm of universal\u0000coalgebra, based either on lifting pseudometrics along set functors or on\u0000lifting general real-valued (fuzzy) relations along functors by means of fuzzy\u0000lax extensions. An immediate benefit of the latter is that they allow bounding\u0000behavioural distance by means of fuzzy (bi-)simulations that need not\u0000themselves be hemi- or pseudometrics; this is analogous to classical\u0000simulations and bisimulations, which need not be preorders or equivalence\u0000relations, respectively. The known generic pseudometric liftings, specifically\u0000the generic Kantorovich and Wasserstein liftings, both can be extended to yield\u0000fuzzy lax extensions, using the fact that both are effectively given by a\u0000choice of quantitative modalities. Our central result then shows that in fact\u0000all fuzzy lax extensions are Kantorovich extensions for a suitable set of\u0000quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy\u0000lax extensions, this allows for the extraction of quantitative modal logics\u0000that characterize behavioural distance, i.e. satisfy a quantitative version of\u0000the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a\u0000quantitative version of Moss' coalgebraic logic. All our results explicitly\u0000hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative\u0000simulation.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116953849","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: 13
Algebraic coherent confluence and higher globular Kleene algebras 代数相干合流与高球状Kleene代数
Log. Methods Comput. Sci. Pub Date : 2020-06-29 DOI: 10.46298/lmcs-18(4:9)2022
Cameron Calk, É. Goubault, P. Malbos, G. Struth
{"title":"Algebraic coherent confluence and higher globular Kleene algebras","authors":"Cameron Calk, É. Goubault, P. Malbos, G. Struth","doi":"10.46298/lmcs-18(4:9)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(4:9)2022","url":null,"abstract":"We extend the formalisation of confluence results in Kleene algebras to a\u0000formalisation of coherent confluence proofs. For this, we introduce the\u0000structure of higher globular Kleene algebra, a higher-dimensional\u0000generalisation of modal and concurrent Kleene algebra. We calculate a coherent\u0000Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras\u0000by equational reasoning. We instantiate these results in the context of higher\u0000rewriting systems modelled by polygraphs.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132796391","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
Predicative theories of continuous lattices 连续格的谓词理论
Log. Methods Comput. Sci. Pub Date : 2020-06-10 DOI: 10.23638/LMCS-17(2:22)2021
Tatsuji Kawai
{"title":"Predicative theories of continuous lattices","authors":"Tatsuji Kawai","doi":"10.23638/LMCS-17(2:22)2021","DOIUrl":"https://doi.org/10.23638/LMCS-17(2:22)2021","url":null,"abstract":"The paper introduces strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices. Strong proximity join-semilattices can be characterised by the coalgebras of the lower powerlocale on the wider category of proximity posets (known as abstract bases and R-structure). Moreover, locally compact locales can be characterised in terms of strong proximity join-semilattices by the coalgebras of the double powerlocale on the category of proximity posets. The paper also introduces more logical characterisation of a strong proximity join-semilattice, called a strong continuous finitary cover, which uses an entailment relation to present the underlying join-semilattice. It is shown that this structure naturally corresponds to the notion of continuous lattice in the predicative pointfree topology. The result makes the predicative and finitary aspect of the notion of continuous lattice in point-free topology more explicit.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121028211","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
A Complete Axiomatisation for Quantifier-Free Separation Logic 无量词分离逻辑的完全公理化
Log. Methods Comput. Sci. Pub Date : 2020-06-09 DOI: 10.46298/lmcs-17(3:17)2021
Stephane Demri, É. Lozes, Alessio Mansutti
{"title":"A Complete Axiomatisation for Quantifier-Free Separation Logic","authors":"Stephane Demri, É. Lozes, Alessio Mansutti","doi":"10.46298/lmcs-17(3:17)2021","DOIUrl":"https://doi.org/10.46298/lmcs-17(3:17)2021","url":null,"abstract":"We present the first complete axiomatisation for quantifier-free separation\u0000logic. The logic is equipped with the standard concrete heaplet semantics and\u0000the proof system has no external feature such as nominals/labels. It is not\u0000possible to rely completely on proof systems for Boolean BI as the concrete\u0000semantics needs to be taken into account. Therefore, we present the first\u0000internal Hilbert-style axiomatisation for quantifier-free separation logic. The\u0000calculus is divided in three parts: the axiomatisation of core formulae where\u0000Boolean combinations of core formulae capture the expressivity of the whole\u0000logic, axioms and inference rules to simulate a bottom-up elimination of\u0000separating connectives, and finally structural axioms and inference rules from\u0000propositional calculus and Boolean BI with the magic wand.\u0000","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116758158","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
Satisfiability and Model Checking for the Logic of Sub-Intervals under the Homogeneity Assumption 齐次假设下子区间逻辑的可满足性及模型检验
Log. Methods Comput. Sci. Pub Date : 2020-06-08 DOI: 10.46298/lmcs-18(1:24)2022
L. Bozzelli, A. Molinari, A. Montanari, A. Peron, P. Sala
{"title":"Satisfiability and Model Checking for the Logic of Sub-Intervals under the Homogeneity Assumption","authors":"L. Bozzelli, A. Molinari, A. Montanari, A. Peron, P. Sala","doi":"10.46298/lmcs-18(1:24)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(1:24)2022","url":null,"abstract":"The expressive power of interval temporal logics (ITLs) makes them one of the\u0000most natural choices in a number of application domains, ranging from the\u0000specification and verification of complex reactive systems to automated\u0000planning. However, for a long time, because of their high computational\u0000complexity, they were considered not suitable for practical purposes. The\u0000recent discovery of several computationally well-behaved ITLs has finally\u0000changed the scenario.\u0000 In this paper, we investigate the finite satisfiability and model checking\u0000problems for the ITL D, that has a single modality for the sub-interval\u0000relation, under the homogeneity assumption (that constrains a proposition\u0000letter to hold over an interval if and only if it holds over all its points).\u0000We first prove that the satisfiability problem for D, over finite linear\u0000orders, is PSPACE-complete, and then we show that the same holds for its model\u0000checking problem, over finite Kripke structures. In such a way, we enrich the\u0000set of tractable interval temporal logics with a new meaningful representative.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116683809","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: 14
A program for the full axiom of choice 一个程序为充分的公理选择
Log. Methods Comput. Sci. Pub Date : 2020-06-08 DOI: 10.46298/lmcs-17(3:21)2021
J. Krivine
{"title":"A program for the full axiom of choice","authors":"J. Krivine","doi":"10.46298/lmcs-17(3:21)2021","DOIUrl":"https://doi.org/10.46298/lmcs-17(3:21)2021","url":null,"abstract":"The theory of classical realizability is a framework for the Curry-Howard\u0000correspondence which enables to associate a program with each proof in\u0000Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in\u0000physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent\u0000choice (DC) or even the (full) axiom of choice (AC). It is therefore important\u0000to find explicit programs for these axioms. Various solutions have been found\u0000for DC, for instance the lambda-term called \"bar recursion\" or the instruction\u0000\"quote\" of LISP. We present here the first program for AC.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125577636","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: 5
Foundations of regular coinduction 正则共归纳的基础
Log. Methods Comput. Sci. Pub Date : 2020-06-04 DOI: 10.46298/lmcs-17(4:2)2021
Francesco Dagnino
{"title":"Foundations of regular coinduction","authors":"Francesco Dagnino","doi":"10.46298/lmcs-17(4:2)2021","DOIUrl":"https://doi.org/10.46298/lmcs-17(4:2)2021","url":null,"abstract":"Inference systems are a widespread framework used to define possibly\u0000recursive predicates by means of inference rules. They allow both inductive and\u0000coinductive interpretations that are fairly well-studied. In this paper, we\u0000consider a middle way interpretation, called regular, which combines advantages\u0000of both approaches: it allows non-well-founded reasoning while being finite. We\u0000show that the natural proof-theoretic definition of the regular interpretation,\u0000based on regular trees, coincides with a rational fixed point. Then, we provide\u0000an equivalent inductive characterization, which leads to an algorithm which\u0000looks for a regular derivation of a judgment. Relying on these results, we\u0000define proof techniques for regular reasoning: the regular coinduction\u0000principle, to prove completeness, and an inductive technique to prove\u0000soundness, based on the inductive characterization of the regular\u0000interpretation. Finally, we show the regular approach can be smoothly extended\u0000to inference systems with corules, a recently introduced, generalised\u0000framework, which allows one to refine the coinductive interpretation, proving\u0000that also this flexible regular interpretation admits an equivalent inductive\u0000characterisation.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"223 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126814124","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: 9
Regular matching problems for infinite trees 无限树的正则匹配问题
Log. Methods Comput. Sci. Pub Date : 2020-04-21 DOI: 10.46298/lmcs-18(1:25)2022
Carlos Camino, V. Diekert, B. Dundua, M. Marin, G'eraud S'enizergues
{"title":"Regular matching problems for infinite trees","authors":"Carlos Camino, V. Diekert, B. Dundua, M. Marin, G'eraud S'enizergues","doi":"10.46298/lmcs-18(1:25)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(1:25)2022","url":null,"abstract":"We study the matching problem of regular tree languages, that is, \"$exists\u0000sigma:sigma(L)subseteq R$?\" where $L,R$ are regular tree languages over the\u0000union of finite ranked alphabets $Sigma$ and $mathcal{X}$ where $mathcal{X}$\u0000is an alphabet of variables and $sigma$ is a substitution such that\u0000$sigma(x)$ is a set of trees in $T(Sigmacup H)setminus H$ for all $xin\u0000mathcal{X}$. Here, $H$ denotes a set of \"holes\" which are used to define a\u0000\"sorted\" concatenation of trees. Conway studied this problem in the special\u0000case for languages of finite words in his classical textbook \"Regular algebra\u0000and finite machines\" published in 1971. He showed that if $L$ and $R$ are\u0000regular, then the problem \"$exists sigma forall xin mathcal{X}:\u0000sigma(x)neq emptysetwedge sigma(L)subseteq R$?\" is decidable. Moreover,\u0000there are only finitely many maximal solutions, the maximal solutions are\u0000regular substitutions, and they are effectively computable. We extend Conway's\u0000results when $L,R$ are regular languages of finite and infinite trees, and\u0000language substitution is applied inside-out, in the sense of Engelfriet and\u0000Schmidt (1977/78). More precisely, we show that if $Lsubseteq\u0000T(Sigmacupmathcal{X})$ and $Rsubseteq T(Sigma)$ are regular tree languages\u0000over finite or infinite trees, then the problem \"$exists sigma forall xin\u0000mathcal{X}: sigma(x)neq emptysetwedge sigma_{mathrm{io}}(L)subseteq\u0000R$?\" is decidable. Here, the subscript \"$mathrm{io}$\" in\u0000$sigma_{mathrm{io}}(L)$ refers to \"inside-out\". Moreover, there are only\u0000finitely many maximal solutions $sigma$, the maximal solutions are regular\u0000substitutions and effectively computable. The corresponding question for the\u0000outside-in extension $sigma_{mathrm{oi}}$ remains open, even in the\u0000restricted setting of finite trees.","PeriodicalId":314387,"journal":{"name":"Log. Methods Comput. Sci.","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124005064","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
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