Carlos Areces, Raul Fervari, Andrés R Saravia, Fernando R Velázquez-Quesada
Abstract We introduce a novel semantics for a multi-agent epistemic operator of knowing how, based on an indistinguishability relation between plans. Our proposal is, arguably, closer to the standard presentation of knowing that modalities in classical epistemic logic. We study the relationship between this new semantics and previous approaches, showing that our setting is general enough to capture them. We also study the logical properties of the new semantics. First, we define a sound and complete axiomatization. Second, we define a suitable notion of bisimulation and prove correspondence theorems. Finally, we investigate the computational complexity of the model checking and satisfiability problems for the new logic.
{"title":"Uncertainty-based knowing how logic","authors":"Carlos Areces, Raul Fervari, Andrés R Saravia, Fernando R Velázquez-Quesada","doi":"10.1093/logcom/exad056","DOIUrl":"https://doi.org/10.1093/logcom/exad056","url":null,"abstract":"Abstract We introduce a novel semantics for a multi-agent epistemic operator of knowing how, based on an indistinguishability relation between plans. Our proposal is, arguably, closer to the standard presentation of knowing that modalities in classical epistemic logic. We study the relationship between this new semantics and previous approaches, showing that our setting is general enough to capture them. We also study the logical properties of the new semantics. First, we define a sound and complete axiomatization. Second, we define a suitable notion of bisimulation and prove correspondence theorems. Finally, we investigate the computational complexity of the model checking and satisfiability problems for the new logic.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135146156","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}
Abstract Unification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $textbf{K5}$ are exactly the extensions of $textbf{K45}$. This resolves the open question of whether $textbf{K5}$ is projective.
{"title":"Projective relative unification through duality","authors":"Philippe Balbiani, Quentin Gougeon","doi":"10.1093/logcom/exad058","DOIUrl":"https://doi.org/10.1093/logcom/exad058","url":null,"abstract":"Abstract Unification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $textbf{K5}$ are exactly the extensions of $textbf{K45}$. This resolves the open question of whether $textbf{K5}$ is projective.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894747","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}
Francesco Olivieri, Guido Governatori, Matteo Cristani, Antonino Rotolo, Abdul Sattar
Abstract The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al. (2021, Computing defeasible meta-logic. In JELIA 2021, LNCS, vol. 12678, pp. 69–84. Springer.). This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic to model meta-rules in the application area we just alluded to has been investigated. Within this line of research, the study mentioned above was not focusing on the general computational properties of meta-rules. This study fills this gap with two major contributions. First, we introduce and formalise two variants of Defeasible Deontic Logic (DDL) with meta-rules to represent (i) defeasible meta-theories with deontic modalities and (ii) two different types of conflicts among rules: Simple Conflict DDL and Cautious Conflict DDL. Second, we advance efficient algorithms to compute the extensions for both variants.
{"title":"Deontic meta-rules","authors":"Francesco Olivieri, Guido Governatori, Matteo Cristani, Antonino Rotolo, Abdul Sattar","doi":"10.1093/logcom/exac081","DOIUrl":"https://doi.org/10.1093/logcom/exac081","url":null,"abstract":"Abstract The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al. (2021, Computing defeasible meta-logic. In JELIA 2021, LNCS, vol. 12678, pp. 69–84. Springer.). This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic to model meta-rules in the application area we just alluded to has been investigated. Within this line of research, the study mentioned above was not focusing on the general computational properties of meta-rules. This study fills this gap with two major contributions. First, we introduce and formalise two variants of Defeasible Deontic Logic (DDL) with meta-rules to represent (i) defeasible meta-theories with deontic modalities and (ii) two different types of conflicts among rules: Simple Conflict DDL and Cautious Conflict DDL. Second, we advance efficient algorithms to compute the extensions for both variants.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135719693","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}
Abstract The fundamental relations in private law are claims and duties. These legal relations can be changed by agents with the appropriate legal powers. We use propositional dynamic logic and ideas about propositional control from the agency literature to formalize these changes in legal relations. Our models are sets of states with functions specifying atomic facts, agents’ abilities to change atomic facts, legal relations between agents concerning changing atomic facts and agents’ powers. We present a formal language that allows us to describe models and changes of models caused by two kinds of actions: actions that change atomic facts and actions that change legal relations. Next, we present a sound and complete calculus for this language. The paper demonstrates that the perspective on actions borrowed from computer science can be used to shed interesting light on the dynamics of legal relations.
{"title":"Modeling dynamics of legal relations with dynamic logic","authors":"Jan van Eijck, Fengkui Ju, Tianwen Xu","doi":"10.1093/logcom/exac055","DOIUrl":"https://doi.org/10.1093/logcom/exac055","url":null,"abstract":"Abstract The fundamental relations in private law are claims and duties. These legal relations can be changed by agents with the appropriate legal powers. We use propositional dynamic logic and ideas about propositional control from the agency literature to formalize these changes in legal relations. Our models are sets of states with functions specifying atomic facts, agents’ abilities to change atomic facts, legal relations between agents concerning changing atomic facts and agents’ powers. We present a formal language that allows us to describe models and changes of models caused by two kinds of actions: actions that change atomic facts and actions that change legal relations. Next, we present a sound and complete calculus for this language. The paper demonstrates that the perspective on actions borrowed from computer science can be used to shed interesting light on the dynamics of legal relations.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135770640","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}
Abstract We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczyński’s pure logic of necessitation $textsf{N}$. Moreover, we introduce three extensions $textsf{N4}$, $textsf{NR}$ and $textsf{NR4}$ of $textsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $textsf{N4}$, $textsf{NR}$ and $textsf{NR4}$ are the provability logics of all provability predicates satisfying the third condition $textbf{D3}$ of the derivability conditions, all Rosser provability predicates and all Rosser provability predicates satisfying $textbf{D3}$, respectively.
{"title":"The provability logic of all provability predicates","authors":"Taishi Kurahashi","doi":"10.1093/logcom/exad060","DOIUrl":"https://doi.org/10.1093/logcom/exad060","url":null,"abstract":"Abstract We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczyński’s pure logic of necessitation $textsf{N}$. Moreover, we introduce three extensions $textsf{N4}$, $textsf{NR}$ and $textsf{NR4}$ of $textsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $textsf{N4}$, $textsf{NR}$ and $textsf{NR4}$ are the provability logics of all provability predicates satisfying the third condition $textbf{D3}$ of the derivability conditions, all Rosser provability predicates and all Rosser provability predicates satisfying $textbf{D3}$, respectively.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135864677","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}
Abstract Reasoning about desire plays a significant role in logic, artificial intelligence and philosophy, etc. In this paper, we propose an interpretation of desire that takes into account causal inference. To formalize this idea, we build a desire-causality model by combing the betterness model in preference logic and the causal model in the logic for causal reasoning. We then develop a logic for desire based on this semantics, and an axiomatization for our formal system is given.
{"title":"A Logic for Desire Based on Causal Inference","authors":"Kaibo Xie, Jialiang Yan","doi":"10.1093/logcom/exac060","DOIUrl":"https://doi.org/10.1093/logcom/exac060","url":null,"abstract":"Abstract Reasoning about desire plays a significant role in logic, artificial intelligence and philosophy, etc. In this paper, we propose an interpretation of desire that takes into account causal inference. To formalize this idea, we build a desire-causality model by combing the betterness model in preference logic and the causal model in the logic for causal reasoning. We then develop a logic for desire based on this semantics, and an axiomatization for our formal system is given.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135864674","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}
Abstract In this paper, we provide a semantics for a range of positive substructural logics, including both logics with and logics without modal connectives. The semantics is novel insofar as it is meant to explicitly capture the computational flavor of these logics, and to do so in a way that builds in both nondeterministic and nonconcurrent computational processes.
{"title":"Nondeterministic and nonconcurrent computational semantics for BB+ and related logics","authors":"Shay Allen Logan","doi":"10.1093/logcom/exad057","DOIUrl":"https://doi.org/10.1093/logcom/exad057","url":null,"abstract":"Abstract In this paper, we provide a semantics for a range of positive substructural logics, including both logics with and logics without modal connectives. The semantics is novel insofar as it is meant to explicitly capture the computational flavor of these logics, and to do so in a way that builds in both nondeterministic and nonconcurrent computational processes.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"235 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135307184","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}
Abstract We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class $ {mathcal{C}}$ of Kripke frames from our semantics, we identify a condition such that $ {mathcal{C}}$ will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property.
{"title":"Dual counterpart intuitionistic logic","authors":"Anthony Cantor, Aaron Stump","doi":"10.1093/logcom/exad019","DOIUrl":"https://doi.org/10.1093/logcom/exad019","url":null,"abstract":"Abstract We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class $ {mathcal{C}}$ of Kripke frames from our semantics, we identify a condition such that $ {mathcal{C}}$ will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136024421","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}
Abstract Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but it didn’t turn out that way. There is one application domain, relational database management, that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.
{"title":"The umbilical cord of finite model theory","authors":"Yuri Gurevich","doi":"10.1093/logcom/exad055","DOIUrl":"https://doi.org/10.1093/logcom/exad055","url":null,"abstract":"Abstract Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but it didn’t turn out that way. There is one application domain, relational database management, that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135098959","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}
Abstract The field of belief revision in logic is still in evolution and holds a variety of disparate approaches; a consequence of theoretical conjecture. As a probabilistic model of supra-classical, non-monotonic (SCNM) logic, the Boltzmann machine, offers an experimental gateway into the field. How does the Boltzmann network adapt to new information? Catastrophic forgetting is the default response to retraining in any neural network. We have moderated this irrational non-monotonicity by alterations in the Boltzmann learning algorithm. The spectrum of experimental belief change is limited by the availability of ‘new’ information, a pragmatic realization co-related to the property of Rational Monotonicity in the domain of SCNM logic. Recognizing this upper boundary of defeasible belief simplifies the task of experimentally exploring machine adaptation. A minority of belief revisions involve new, but unsurprising information, that is at least partially consistent with the previous learned beliefs. In these circumstances, the Boltzmann network incrementally adjusts the priority of model state exemplars in accordance with preference; the traditional approach in SCNM logic. However, in the majority of situations the new information will be surprisingly inconsistent with the previous beliefs. In these circumstances, the pre-order on model states stratified by preference, will not have sufficient granularity to represent the conflicting requirements of ranking based on compositional atomic typicality. This novel experimental finding has not previously been considered in the logical conjecture on Belief Revision.
{"title":"Modelling supra-classical logic in a Boltzmann neural network: III adaptation","authors":"Glenn Blanchette, Anthony Robins","doi":"10.1093/logcom/exad052","DOIUrl":"https://doi.org/10.1093/logcom/exad052","url":null,"abstract":"Abstract The field of belief revision in logic is still in evolution and holds a variety of disparate approaches; a consequence of theoretical conjecture. As a probabilistic model of supra-classical, non-monotonic (SCNM) logic, the Boltzmann machine, offers an experimental gateway into the field. How does the Boltzmann network adapt to new information? Catastrophic forgetting is the default response to retraining in any neural network. We have moderated this irrational non-monotonicity by alterations in the Boltzmann learning algorithm. The spectrum of experimental belief change is limited by the availability of ‘new’ information, a pragmatic realization co-related to the property of Rational Monotonicity in the domain of SCNM logic. Recognizing this upper boundary of defeasible belief simplifies the task of experimentally exploring machine adaptation. A minority of belief revisions involve new, but unsurprising information, that is at least partially consistent with the previous learned beliefs. In these circumstances, the Boltzmann network incrementally adjusts the priority of model state exemplars in accordance with preference; the traditional approach in SCNM logic. However, in the majority of situations the new information will be surprisingly inconsistent with the previous beliefs. In these circumstances, the pre-order on model states stratified by preference, will not have sufficient granularity to represent the conflicting requirements of ranking based on compositional atomic typicality. This novel experimental finding has not previously been considered in the logical conjecture on Belief Revision.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135830467","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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