The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.
{"title":"Logics for Temporal Information Systems in Rough Set Theory","authors":"Md. Aquil Khan, M. Banerjee, Sibsankar Panda","doi":"10.1145/3549075","DOIUrl":"https://doi.org/10.1145/3549075","url":null,"abstract":"The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":" ","pages":"1 - 29"},"PeriodicalIF":0.5,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49318217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, H. Wehrheim
In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.
{"title":"Unifying Operational Weak Memory Verification: An Axiomatic Approach","authors":"Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, H. Wehrheim","doi":"10.1145/3545117","DOIUrl":"https://doi.org/10.1145/3545117","url":null,"abstract":"In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 39"},"PeriodicalIF":0.5,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41288180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.
{"title":"A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types","authors":"Jason Z. S. Hu, B. Pientka, Ulrich Schöpp","doi":"10.1145/3545115","DOIUrl":"https://doi.org/10.1145/3545115","url":null,"abstract":"We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 36"},"PeriodicalIF":0.5,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45776127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.
{"title":"A Generalized Realizability and Intuitionistic Logic","authors":"A. Y. Konovalov","doi":"10.1145/3565367","DOIUrl":"https://doi.org/10.1145/3565367","url":null,"abstract":"Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 15"},"PeriodicalIF":0.5,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48224824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z1,..., zk have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connk the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + connk + 1 is strictly more expressive than FO + connk for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) xi and yi for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPk that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.
{"title":"First-order Logic with Connectivity Operators","authors":"Nicole Schirrmacher, S. Siebertz, Alexandre Vigny","doi":"10.1145/3595922","DOIUrl":"https://doi.org/10.1145/3595922","url":null,"abstract":"First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z1,..., zk have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connk the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + connk + 1 is strictly more expressive than FO + connk for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) xi and yi for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPk that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 23"},"PeriodicalIF":0.5,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48028297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Carvalho, Florent R. Madelaine, B. Martin, Dmitriy Zhuk
Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.
{"title":"The Complexity of Quantified Constraints: Collapsibility, Switchability, and the Algebraic Formulation","authors":"C. Carvalho, Florent R. Madelaine, B. Martin, Dmitriy Zhuk","doi":"10.1145/3568397","DOIUrl":"https://doi.org/10.1145/3568397","url":null,"abstract":"Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 26"},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model...
{"title":"Expressiveness and Nash Equilibrium in Iterated Boolean Games","authors":"GutiérrezJulián, HarrensteinPaul, PerelliGiuseppe, WooldridgeMichael","doi":"10.1145/3439900","DOIUrl":"https://doi.org/10.1145/3439900","url":null,"abstract":"We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model...","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"22 1","pages":"1-38"},"PeriodicalIF":0.5,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3439900","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44371734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.
{"title":"A Subatomic Proof System for Decision Trees","authors":"Chris Barrett, Alessio Guglielmi","doi":"10.1145/3545116","DOIUrl":"https://doi.org/10.1145/3545116","url":null,"abstract":"We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 25"},"PeriodicalIF":0.5,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45735718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dylan Bellier, M. Benerecetti, Dario Della Monica, F. Mogavero
An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. Inspired by the work on logics of dependence and independence, we provide a new compositional semantics for QPTL that allows for expressing such functional dependencies among variables. The fragment where only restricted quantifications are considered, called behavioral quantifications, allows for linear-time properties that are satisfiable if and only if they are realisable in the Pnueli-Rosner sense. This fragment can be decided, for both model checking and satisfiability, in 2Exp Time and is expressively equivalent to QPTL, though significantly less succinct.
{"title":"Good-for-Game QPTL: An Alternating Hodges Semantics","authors":"Dylan Bellier, M. Benerecetti, Dario Della Monica, F. Mogavero","doi":"10.1145/3565365","DOIUrl":"https://doi.org/10.1145/3565365","url":null,"abstract":"An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. Inspired by the work on logics of dependence and independence, we provide a new compositional semantics for QPTL that allows for expressing such functional dependencies among variables. The fragment where only restricted quantifications are considered, called behavioral quantifications, allows for linear-time properties that are satisfiable if and only if they are realisable in the Pnueli-Rosner sense. This fragment can be decided, for both model checking and satisfiability, in 2Exp Time and is expressively equivalent to QPTL, though significantly less succinct.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 57"},"PeriodicalIF":0.5,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46679373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (random with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.
{"title":"The Intersection of Algorithmically Random Closed Sets and Effective Dimension","authors":"Adam Case, Christopher P. Porter","doi":"10.1145/3545114","DOIUrl":"https://doi.org/10.1145/3545114","url":null,"abstract":"In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (random with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 19"},"PeriodicalIF":0.5,"publicationDate":"2021-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41998782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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