Abstract We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $textbf {KTB}$, $textbf {GL}$ and $textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.
{"title":"Complexity function and complexity of validity of modal and superintuitionistic propositional logics","authors":"Mikhail Rybakov, Dmitry Shkatov","doi":"10.1093/logcom/exac085","DOIUrl":"https://doi.org/10.1093/logcom/exac085","url":null,"abstract":"Abstract We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $textbf {KTB}$, $textbf {GL}$ and $textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"179 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135604416","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}
Incomplete Argumentation Frameworks (IAFs) have been defined to incorporate some qualitative uncertainty in abstract argumentation: information such as ‘I am not sure whether this argument exists’ or ‘I am not sure whether this argument attacks that one’ can be expressed. Reasoning with IAFs is classically based on a set of completions, i.e. standard argumentation frameworks (AFs) that represent the possible worlds encoded in the IAF. The number of these completions may be exponential with respect to the number of arguments in the IAF. This leads, in some cases, to an increase of the complexity of reasoning, compared to the complexity of standard AFs. In this paper, we follow an approach that was initiated for Partial Argumentation Frameworks (PAFs) (a subclass of IAFs), which consists in defining new forms of conflict-freeness and defense, the properties that underly the definition of Dung's semantics for AFs. We generalize these semantics from PAFs to IAFs. We show that, among three possible types of admissibility, only two of them satisfy some desirable properties. We use them to define two new families of extension-based semantics. We study the properties of these semantics, and in particular, we show that their complexity remains the same as in the case of Dung's AFs. Finally, we propose a logical encoding of these semantics, and we show experimentally that this encoding can be used efficiently to reason with IAFs, thanks to the power of modern SAT solvers.
{"title":"Extension-based semantics for incomplete argumentation frameworks: properties, complexity and algorithms","authors":"Jean-Guy Mailly","doi":"10.1093/logcom/exac099","DOIUrl":"https://doi.org/10.1093/logcom/exac099","url":null,"abstract":"Incomplete Argumentation Frameworks (IAFs) have been defined to incorporate some qualitative uncertainty in abstract argumentation: information such as ‘I am not sure whether this argument exists’ or ‘I am not sure whether this argument attacks that one’ can be expressed. Reasoning with IAFs is classically based on a set of completions, i.e. standard argumentation frameworks (AFs) that represent the possible worlds encoded in the IAF. The number of these completions may be exponential with respect to the number of arguments in the IAF. This leads, in some cases, to an increase of the complexity of reasoning, compared to the complexity of standard AFs. In this paper, we follow an approach that was initiated for Partial Argumentation Frameworks (PAFs) (a subclass of IAFs), which consists in defining new forms of conflict-freeness and defense, the properties that underly the definition of Dung's semantics for AFs. We generalize these semantics from PAFs to IAFs. We show that, among three possible types of admissibility, only two of them satisfy some desirable properties. We use them to define two new families of extension-based semantics. We study the properties of these semantics, and in particular, we show that their complexity remains the same as in the case of Dung's AFs. Finally, we propose a logical encoding of these semantics, and we show experimentally that this encoding can be used efficiently to reason with IAFs, thanks to the power of modern SAT solvers.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"406-435"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49931764","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}
A logic L has the ‘variable-sharing property’ (VSP) if in all L-theorems of the form $Arightarrow B$ , $A$ and $B$ share at least a propositional variable. A logic L has the ‘Ackermann property’ (AP) if in all L-theorems of the form $Arightarrow (Brightarrow C)$ , $A$ contains at least a conditional connective ( $rightarrow $ ). Anderson and Belnap consider the VSP a necessary property of any relevant logic, and both the VSP and the AP necessary properties of any (relevant) entailment logic. Now, among relevant logicians, Brady's logic BN4 is widely viewed as the adequate 4-valued implicative logic. But BN4 lacks the VSP and the AP. The aim of this paper is to define the logics BN4 $^{text {VSP}}$ and BN4 $^{text {AP}}$ . The former one has the VSP, whereas the latter one has the VSP and the AP. Moreover, both logics have some properties that do not support their consideration as mere artificial constructs.
{"title":"Two 4-valued implicative expansions of first-degree entailment logic: The relevant logic BN4VSP and the (relevant) entailment logic BN4AP","authors":"Gemma Robles","doi":"10.1093/logcom/exac101","DOIUrl":"https://doi.org/10.1093/logcom/exac101","url":null,"abstract":"A logic L has the ‘variable-sharing property’ (VSP) if in all L-theorems of the form \u0000<tex>$Arightarrow B$</tex>\u0000, \u0000<tex>$A$</tex>\u0000 and \u0000<tex>$B$</tex>\u0000 share at least a propositional variable. A logic L has the ‘Ackermann property’ (AP) if in all L-theorems of the form \u0000<tex>$Arightarrow (Brightarrow C)$</tex>\u0000, \u0000<tex>$A$</tex>\u0000 contains at least a conditional connective (\u0000<tex>$rightarrow $</tex>\u0000). Anderson and Belnap consider the VSP a necessary property of any relevant logic, and both the VSP and the AP necessary properties of any (relevant) entailment logic. Now, among relevant logicians, Brady's logic BN4 is widely viewed as the adequate 4-valued implicative logic. But BN4 lacks the VSP and the AP. The aim of this paper is to define the logics BN4\u0000<tex>$^{text {VSP}}$</tex>\u0000 and BN4\u0000<tex>$^{text {AP}}$</tex>\u0000. The former one has the VSP, whereas the latter one has the VSP and the AP. Moreover, both logics have some properties that do not support their consideration as mere artificial constructs.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"462-484"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016805/10068384/10068395.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49962378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we provide a formal framework for modeling the burden of persuasion in legal reasoning. The framework is based on abstract argumentation, a frequently studied method of non-monotonic reasoning, and can be applied to different argumentation semantics; it supports burdens of persuasion with arbitrary many levels, and allows for the placement of a burden of persuasion on any subset of an argumentation framework's arguments. Our framework can be considered an extension of related works that raise questions on how burdens of persuasion should be handled in some conflict scenarios that can be modeled with abstract argumentation. An open source software implementation of the introduced formal notions is available as an extension of an argumentation reasoning library. A theoretical analysis shows that our approach can be generalized to a novel method for the preference-based selection of extensions from argumentation frameworks.
{"title":"A comprehensive account of the burden of persuasion in abstract argumentation","authors":"Timotheus Kampik;Dov Gabbay;Giovanni Sartor","doi":"10.1093/logcom/exac094","DOIUrl":"https://doi.org/10.1093/logcom/exac094","url":null,"abstract":"In this paper, we provide a formal framework for modeling the burden of persuasion in legal reasoning. The framework is based on abstract argumentation, a frequently studied method of non-monotonic reasoning, and can be applied to different argumentation semantics; it supports burdens of persuasion with arbitrary many levels, and allows for the placement of a burden of persuasion on any subset of an argumentation framework's arguments. Our framework can be considered an extension of related works that raise questions on how burdens of persuasion should be handled in some conflict scenarios that can be modeled with abstract argumentation. An open source software implementation of the introduced formal notions is available as an extension of an argumentation reasoning library. A theoretical analysis shows that our approach can be generalized to a novel method for the preference-based selection of extensions from argumentation frameworks.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"257-288"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016805/10068384/10068388.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49931759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preface: Special Issue on Logic and Argumentation","authors":"Pietro Baroni;Christoph Benzmüller;Yì N Wáng","doi":"10.1093/logcom/exac091","DOIUrl":"https://doi.org/10.1093/logcom/exac091","url":null,"abstract":"","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"189-191"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016805/10068384/10068385.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49962379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The issue of how a semantics should deal with self-attacking arguments was always a subject of debate among argumentation scholars. A consensus exists for extension-based semantics because those arguments are always rejected (as soon as the semantics in question respects conflict-freeness). In case of gradual semantics, the question is more complex, since other criteria are taken into account. In this paper, we check the impact of those arguments by using a principle-based approach. Principles like self-contradiction and strong self-contradiction prescribe how to deal with self-attacking arguments. We show that they are incompatible with the well-known equivalence principle (which is satisfied by almost all the existing gradual semantics), as well as with some other principles (e.g. counting). This incompatibility was not studied until now and the class of semantics satisfying self-contradiction is under-explored. In the present paper, we explore that class of semantics. We show links and incompatibilities between several principles. We define a new general oriented argumentation semantics that satisfies (strong) self-contradiction and a maximal number of compatible principles. We introduce an iterative algorithm to calculate our semantics and prove that it always converges. We also provide a characterization of our semantics. Finally, we experimentally show that our semantics is computationally efficient.
{"title":"A Principle-based Account of Self-attacking Arguments in Gradual Semantics","authors":"Vivien Beuselinck;Jérôme Delobelle;Srdjan Vesic","doi":"10.1093/logcom/exac093","DOIUrl":"https://doi.org/10.1093/logcom/exac093","url":null,"abstract":"The issue of how a semantics should deal with self-attacking arguments was always a subject of debate among argumentation scholars. A consensus exists for extension-based semantics because those arguments are always rejected (as soon as the semantics in question respects conflict-freeness). In case of gradual semantics, the question is more complex, since other criteria are taken into account. In this paper, we check the impact of those arguments by using a principle-based approach. Principles like self-contradiction and strong self-contradiction prescribe how to deal with self-attacking arguments. We show that they are incompatible with the well-known equivalence principle (which is satisfied by almost all the existing gradual semantics), as well as with some other principles (e.g. counting). This incompatibility was not studied until now and the class of semantics satisfying self-contradiction is under-explored. In the present paper, we explore that class of semantics. We show links and incompatibilities between several principles. We define a new general oriented argumentation semantics that satisfies (strong) self-contradiction and a maximal number of compatible principles. We introduce an iterative algorithm to calculate our semantics and prove that it always converges. We also provide a characterization of our semantics. Finally, we experimentally show that our semantics is computationally efficient.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"230-256"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49931758","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}
{"title":"Game equivalence and expressive power of game description languages: a bisimulation approach","authors":"Guifei Jiang;Laurent Perrussel;Dongmo Zhang;Heng Zhang;Yuzhi Zhang","doi":"10.1093/logcom/exac034","DOIUrl":"https://doi.org/10.1093/logcom/exac034","url":null,"abstract":"","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 1","pages":"163-187"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49982125","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}
Collective argumentation is the process of reaching a collective decision that is acceptable to the group in a debate. We introduce the notion of topological restriction to enrich the study of collective argumentation. Topological restrictions are rational constraints assumed to be satisfied by individual agents. We assume that in a debate, for every pair of arguments under consideration, every agent indicates whether the first argument attacks the second, i.e. an agent's argumentative stance is characterized as an argumentation framework, and only argumentation frameworks that satisfy topological restrictions are allowed. The topological restrictions we consider in this paper include various topological properties in the literature, such as acyclicity, symmetry, coherence and determinedness, as well as three topological restrictions that generalize classic social-choice-theoretic domain conditions. We show that when the profile of the argumentation frameworks provided by the agents satisfies topological restrictions, impossibility results during aggregation can be avoided. Furthermore, if a profile is topologically restricted with respect to restrictions that generalize domain conditions, then the majority rule preserves several desirable properties during aggregation.
{"title":"Collective argumentation with topological restrictions: the case of aggregating abstract argumentation frameworks","authors":"Weiwei Chen","doi":"10.1093/logcom/exac096","DOIUrl":"https://doi.org/10.1093/logcom/exac096","url":null,"abstract":"Collective argumentation is the process of reaching a collective decision that is acceptable to the group in a debate. We introduce the notion of topological restriction to enrich the study of collective argumentation. Topological restrictions are rational constraints assumed to be satisfied by individual agents. We assume that in a debate, for every pair of arguments under consideration, every agent indicates whether the first argument attacks the second, i.e. an agent's argumentative stance is characterized as an argumentation framework, and only argumentation frameworks that satisfy topological restrictions are allowed. The topological restrictions we consider in this paper include various topological properties in the literature, such as acyclicity, symmetry, coherence and determinedness, as well as three topological restrictions that generalize classic social-choice-theoretic domain conditions. We show that when the profile of the argumentation frameworks provided by the agents satisfies topological restrictions, impossibility results during aggregation can be avoided. Furthermore, if a profile is topologically restricted with respect to restrictions that generalize domain conditions, then the majority rule preserves several desirable properties during aggregation.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 2","pages":"319-343"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49931761","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}
Nominal schemas have been proposed as an extension to Description Logics (DL), the knowledge representation paradigm underlying the Web Ontology Language (OWL). They provide for a very tight integration of DL and rules. Nominal schemas can be understood as syntactic sugar on top of OWL. However, this naive perspective leads to inefficient reasoning procedures. In order to develop an efficient reasoning procedure for the language ELV ++ , which results from extending the OWL profile language OWL EL with nominal schemas, we propose a transformation from ELV ++ ontologies into Datalog-like rule programs that can be used for satisfiability checking and assertion retrieval. The use of this transformation enables the use of powerful Datalog engines to solve reasoning tasks over ELV ++ ontologies. We implement and then evaluate our approach on several real-world, data-intensive ontologies, and find that it can outperform state-of-the-art reasoners such as Konclude and ELK. As a lesser side result we also provide a self-contained description of a rule-based algorithm for EL ++ which does not require a normal form transformation.
{"title":"An efficient algorithm for reasoning over OWL EL ontologies with nominal schemas","authors":"David Carral;Joseph Zalewski;Pascal Hitzler","doi":"10.1093/logcom/exac032","DOIUrl":"https://doi.org/10.1093/logcom/exac032","url":null,"abstract":"Nominal schemas have been proposed as an extension to Description Logics (DL), the knowledge representation paradigm underlying the Web Ontology Language (OWL). They provide for a very tight integration of DL and rules. Nominal schemas can be understood as syntactic sugar on top of OWL. However, this naive perspective leads to inefficient reasoning procedures. In order to develop an efficient reasoning procedure for the language ELV ++ , which results from extending the OWL profile language OWL EL with nominal schemas, we propose a transformation from ELV ++ ontologies into Datalog-like rule programs that can be used for satisfiability checking and assertion retrieval. The use of this transformation enables the use of powerful Datalog engines to solve reasoning tasks over ELV ++ ontologies. We implement and then evaluate our approach on several real-world, data-intensive ontologies, and find that it can outperform state-of-the-art reasoners such as Konclude and ELK. As a lesser side result we also provide a self-contained description of a rule-based algorithm for EL ++ which does not require a normal form transformation.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 1","pages":"136-162"},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49982124","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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