Pub Date : 2023-05-11DOI: 10.46298/lmcs-19(2:10)2023
Jan Friso Groote, Jan Martens, Erik. P. de Vink
We provide time lower bounds for sequential and parallel algorithms deciding bisimulation on labeled transition systems that use partition refinement. For sequential algorithms this is $Omega((m mkern1mu {+} mkern1mu n ) mkern-1mu log mkern-1mu n)$ and for parallel algorithms this is $Omega(n)$, where $n$ is the number of states and $m$ is the number of transitions. The lowerbounds are obtained by analysing families of deterministic transition systems, ultimately with two actions in the sequential case, and one action for parallel algorithms. For deterministic transition systems with one action, bisimilarity can be decided sequentially with fundamentally different techniques than partition refinement. In particular, Paige, Tarjan, and Bonic give a linear algorithm for this specific situation. We show, exploiting the concept of an oracle, that this approach is not of help to develop a faster generic algorithm for deciding bisimilarity. For parallel algorithms there is a similar situation where these techniques may be applied, too.
{"title":"Lowerbounds for Bisimulation by Partition Refinement","authors":"Jan Friso Groote, Jan Martens, Erik. P. de Vink","doi":"10.46298/lmcs-19(2:10)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:10)2023","url":null,"abstract":"We provide time lower bounds for sequential and parallel algorithms deciding bisimulation on labeled transition systems that use partition refinement. For sequential algorithms this is $Omega((m mkern1mu {+} mkern1mu n ) mkern-1mu log mkern-1mu n)$ and for parallel algorithms this is $Omega(n)$, where $n$ is the number of states and $m$ is the number of transitions. The lowerbounds are obtained by analysing families of deterministic transition systems, ultimately with two actions in the sequential case, and one action for parallel algorithms. For deterministic transition systems with one action, bisimilarity can be decided sequentially with fundamentally different techniques than partition refinement. In particular, Paige, Tarjan, and Bonic give a linear algorithm for this specific situation. We show, exploiting the concept of an oracle, that this approach is not of help to develop a faster generic algorithm for deciding bisimilarity. For parallel algorithms there is a similar situation where these techniques may be applied, too.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135526917","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}
Pub Date : 2023-05-08DOI: 10.46298/lmcs-19(2:9)2023
Jasmin Blanchette, Petar Vukmirović
We generalize several propositional preprocessing techniques to higher-order logic, building on existing first-order generalizations. These techniques eliminate literals, clauses, or predicate symbols from the problem, with the aim of making it more amenable to automatic proof search. We also introduce a new technique, which we call quasipure literal elimination, that strictly subsumes pure literal elimination. The new techniques are implemented in the Zipperposition theorem prover. Our evaluation shows that they sometimes help prove problems originating from Isabelle formalizations and the TPTP library.
{"title":"SAT-Inspired Higher-Order Eliminations","authors":"Jasmin Blanchette, Petar Vukmirović","doi":"10.46298/lmcs-19(2:9)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:9)2023","url":null,"abstract":"We generalize several propositional preprocessing techniques to higher-order logic, building on existing first-order generalizations. These techniques eliminate literals, clauses, or predicate symbols from the problem, with the aim of making it more amenable to automatic proof search. We also introduce a new technique, which we call quasipure literal elimination, that strictly subsumes pure literal elimination. The new techniques are implemented in the Zipperposition theorem prover. Our evaluation shows that they sometimes help prove problems originating from Isabelle formalizations and the TPTP library.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135806319","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}
Pub Date : 2023-04-25DOI: 10.46298/lmcs-19(2:7)2023
Bruno Dinis, Étienne Miquey
In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $cal{M}^{mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.
{"title":"Stateful Realizers for Nonstandard Analysis","authors":"Bruno Dinis, Étienne Miquey","doi":"10.46298/lmcs-19(2:7)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:7)2023","url":null,"abstract":"In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $cal{M}^{mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135066079","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}
Pub Date : 2023-04-20DOI: 10.46298/lmcs-19(2:3)2023
Martin Abadi, Gordon Plotkin
Describing systems in terms of choices and their resulting costs and rewards offers the promise of freeing algorithm designers and programmers from specifying how those choices should be made; in implementations, the choices can be realized by optimization techniques and, increasingly, by machine-learning methods. We study this approach from a programming-language perspective. We define two small languages that support decision-making abstractions: one with choices and rewards, and the other additionally with probabilities. We give both operational and denotational semantics. In the case of the second language we consider three denotational semantics, with varying degrees of correlation between possible program values and expected rewards. The operational semantics combine the usual semantics of standard constructs with optimization over spaces of possible execution strategies. The denotational semantics, which are compositional, rely on the selection monad, to handle choice, augmented with an auxiliary monad to handle other effects, such as rewards or probability. We establish adequacy theorems that the two semantics coincide in all cases. We also prove full abstraction at base types, with varying notions of observation in the probabilistic case corresponding to the various degrees of correlation. We present axioms for choice combined with rewards and probability, establishing completeness at base types for the case of rewards without probability.
{"title":"Smart Choices and the Selection Monad","authors":"Martin Abadi, Gordon Plotkin","doi":"10.46298/lmcs-19(2:3)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:3)2023","url":null,"abstract":"Describing systems in terms of choices and their resulting costs and rewards offers the promise of freeing algorithm designers and programmers from specifying how those choices should be made; in implementations, the choices can be realized by optimization techniques and, increasingly, by machine-learning methods. We study this approach from a programming-language perspective. We define two small languages that support decision-making abstractions: one with choices and rewards, and the other additionally with probabilities. We give both operational and denotational semantics. In the case of the second language we consider three denotational semantics, with varying degrees of correlation between possible program values and expected rewards. The operational semantics combine the usual semantics of standard constructs with optimization over spaces of possible execution strategies. The denotational semantics, which are compositional, rely on the selection monad, to handle choice, augmented with an auxiliary monad to handle other effects, such as rewards or probability. We establish adequacy theorems that the two semantics coincide in all cases. We also prove full abstraction at base types, with varying notions of observation in the probabilistic case corresponding to the various degrees of correlation. We present axioms for choice combined with rewards and probability, establishing completeness at base types for the case of rewards without probability.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135568819","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}
Pub Date : 2023-04-20DOI: 10.46298/lmcs-19(2:5)2023
Dana Fisman, Hadar Frenkel, Sandra Zilles
We study the learnability of symbolic finite state automata (SFA), a model shown useful in many applications in software verification. The state-of-the-art literature on this topic follows the query learning paradigm, and so far all obtained results are positive. We provide a necessary condition for efficient learnability of SFAs in this paradigm, from which we obtain the first negative result. The main focus of our work lies in the learnability of SFAs under the paradigm of identification in the limit using polynomial time and data, and its strengthening efficient identifiability, which are concerned with the existence of a systematic set of characteristic samples from which a learner can correctly infer the target language. We provide a necessary condition for identification of SFAs in the limit using polynomial time and data, and a sufficient condition for efficient learnability of SFAs. From these conditions we derive a positive and a negative result. The performance of a learning algorithm is typically bounded as a function of the size of the representation of the target language. Since SFAs, in general, do not have a canonical form, and there are trade-offs between the complexity of the predicates on the transitions and the number of transitions, we start by defining size measures for SFAs. We revisit the complexity of procedures on SFAs and analyze them according to these measures, paying attention to the special forms of SFAs: normalized SFAs and neat SFAs, as well as to SFAs over a monotonic effective Boolean algebra. This is an extended version of the paper with the same title published in CSL'22.
{"title":"Inferring Symbolic Automata","authors":"Dana Fisman, Hadar Frenkel, Sandra Zilles","doi":"10.46298/lmcs-19(2:5)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:5)2023","url":null,"abstract":"We study the learnability of symbolic finite state automata (SFA), a model shown useful in many applications in software verification. The state-of-the-art literature on this topic follows the query learning paradigm, and so far all obtained results are positive. We provide a necessary condition for efficient learnability of SFAs in this paradigm, from which we obtain the first negative result. The main focus of our work lies in the learnability of SFAs under the paradigm of identification in the limit using polynomial time and data, and its strengthening efficient identifiability, which are concerned with the existence of a systematic set of characteristic samples from which a learner can correctly infer the target language. We provide a necessary condition for identification of SFAs in the limit using polynomial time and data, and a sufficient condition for efficient learnability of SFAs. From these conditions we derive a positive and a negative result. The performance of a learning algorithm is typically bounded as a function of the size of the representation of the target language. Since SFAs, in general, do not have a canonical form, and there are trade-offs between the complexity of the predicates on the transitions and the number of transitions, we start by defining size measures for SFAs. We revisit the complexity of procedures on SFAs and analyze them according to these measures, paying attention to the special forms of SFAs: normalized SFAs and neat SFAs, as well as to SFAs over a monotonic effective Boolean algebra. This is an extended version of the paper with the same title published in CSL'22.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135568821","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}
Pub Date : 2023-04-14DOI: 10.46298/lmcs-19(2:2)2023
Olaf Beyersdorff, Benjamin Böhm
QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.
{"title":"Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution","authors":"Olaf Beyersdorff, Benjamin Böhm","doi":"10.46298/lmcs-19(2:2)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(2:2)2023","url":null,"abstract":"QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134998385","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}
Pub Date : 2023-02-15DOI: 10.46298/lmcs-18(1:3)2022
Karoliina Lehtinen, Martin Zimmermann
We introduce good-for-games $omega$-pushdown automata ($omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $omega$-GFG-PDA are more expressive than deterministic $omega$- pushdown automata and that solving infinite games with winning conditions specified by $omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $omega$-GFG- PDA and decidability of good-for-gameness of $omega$-pushdown automata and languages. Finally, we compare $omega$-GFG-PDA to $omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $omega$-GFG-PDA, and prove that the parity index hierarchy for $omega$-GFG-PDA is infinite. This is a corrected version of the paper arXiv:2001.04392v6 published originally on January 7, 2022.
{"title":"Good-for-games $omega$-Pushdown Automata","authors":"Karoliina Lehtinen, Martin Zimmermann","doi":"10.46298/lmcs-18(1:3)2022","DOIUrl":"https://doi.org/10.46298/lmcs-18(1:3)2022","url":null,"abstract":"We introduce good-for-games $omega$-pushdown automata ($omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $omega$-GFG-PDA are more expressive than deterministic $omega$- pushdown automata and that solving infinite games with winning conditions specified by $omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $omega$-GFG- PDA and decidability of good-for-gameness of $omega$-pushdown automata and languages. Finally, we compare $omega$-GFG-PDA to $omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $omega$-GFG-PDA, and prove that the parity index hierarchy for $omega$-GFG-PDA is infinite. This is a corrected version of the paper arXiv:2001.04392v6 published originally on January 7, 2022.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135538195","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}
Pub Date : 2021-05-18DOI: 10.46298/lmcs-19(3:8)2023
M. Loreti, M. Quadrini
Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes.
{"title":"A Spatial Logic for Simplicial Models","authors":"M. Loreti, M. Quadrini","doi":"10.46298/lmcs-19(3:8)2023","DOIUrl":"https://doi.org/10.46298/lmcs-19(3:8)2023","url":null,"abstract":"Collective Adaptive Systems often consist of many heterogeneous components\u0000typically organised in groups. These entities interact with each other by\u0000adapting their behaviour to pursue individual or collective goals. In these\u0000systems, the distribution of these entities determines a space that can be\u0000either physical or logical. The former is defined in terms of a physical\u0000relation among components. The latter depends on logical relations, such as\u0000being part of the same group. In this context, specification and verification\u0000of spatial properties play a fundamental role in supporting the design of\u0000systems and predicting their behaviour. For this reason, different tools and\u0000techniques have been proposed to specify and verify the properties of space,\u0000mainly described as graphs. Therefore, the approaches generally use model\u0000spatial relations to describe a form of proximity among pairs of entities.\u0000Unfortunately, these graph-based models do not permit considering relations\u0000among more than two entities that may arise when one is interested in\u0000describing aspects of space by involving interactions among groups of entities.\u0000In this work, we propose a spatial logic interpreted on simplicial complexes.\u0000These are topological objects, able to represent surfaces and volumes\u0000efficiently that generalise graphs with higher-order edges. We discuss how the\u0000satisfaction of logical formulas can be verified by a correct and complete\u0000model checking algorithm, which is linear to the dimension of the simplicial\u0000complex and logical formula. The expressiveness of the proposed logic is\u0000studied in terms of the spatial variants of classical bisimulation and\u0000branching bisimulation relations defined over simplicial complexes.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"19 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70491501","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-63777-4
R. Antonsen
{"title":"Logical Methods: The Art of Thinking Abstractly and Mathematically","authors":"R. Antonsen","doi":"10.1007/978-3-030-63777-4","DOIUrl":"https://doi.org/10.1007/978-3-030-63777-4","url":null,"abstract":"","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":"53 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87990214","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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