Matthias Baaz, Marcel Ertel, Reinhard Kahle, Thomas Piecha, Jan von Plato
This volume is one of two special issues collecting articles by invited speakers of the conference ‘Celebrating 90 Years of Gödel’s Incompleteness Theorems’ held in Nürtingen (Germany) in July 2021. The conference was organized by the Carl Friedrich von Weizsäcker Center at the University of Tübingen with support by the ERC-funded project Gödel Enigma: Rediscovering Kurt Gödel through his unpublished works at the University of Helsinki and by the Kurt Gödel Society in Vienna.
本卷是两期特刊之一，收录了2021年7月在德国纽廷根举行的 "庆祝哥德尔不完备性定理问世90周年 "会议特邀发言人的文章。此次会议由图宾根大学卡尔-弗里德里希-冯-魏茨泽克中心（Carl Friedrich von Weizsäcker）主办，并得到了欧洲研究理事会（ERC）资助的 "哥德尔之谜 "项目的支持：通过赫尔辛基大学库尔特-哥德尔未发表的作品重新发现库尔特-哥德尔 "项目和维也纳库尔特-哥德尔协会的支持下举办的。
{"title":"90 years of Gödel’s incompleteness theorems: Logic and computation","authors":"Matthias Baaz, Marcel Ertel, Reinhard Kahle, Thomas Piecha, Jan von Plato","doi":"10.1093/logcom/exae026","DOIUrl":"https://doi.org/10.1093/logcom/exae026","url":null,"abstract":"This volume is one of two special issues collecting articles by invited speakers of the conference ‘Celebrating 90 Years of Gödel’s Incompleteness Theorems’ held in Nürtingen (Germany) in July 2021. The conference was organized by the Carl Friedrich von Weizsäcker Center at the University of Tübingen with support by the ERC-funded project Gödel Enigma: Rediscovering Kurt Gödel through his unpublished works at the University of Helsinki and by the Kurt Gödel Society in Vienna.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194713","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 formalize the modal operators from the concurrent dynamic logics of Peleg, Nerode and Wijesekera in a multirelational algebraic language based on relation algebras and power allegories, using relational approximation operators on multirelations developed in a companion article. We relate Nerode and Wijesekera’s box operator with a relational approximation operator for multirelations and two related operators that approximate multirelations by different kinds of deterministic multirelations. We provide an algebraic soundness proof of Goldblatt’s axioms for concurrent dynamic logic and one for a multirelational Hoare logic based on Nerode and Wijesekera’s box as applications.
{"title":"Modal algebra of multirelations","authors":"Hitoshi Furusawa, Walter Guttmann, Georg Struth","doi":"10.1093/logcom/exae023","DOIUrl":"https://doi.org/10.1093/logcom/exae023","url":null,"abstract":"We formalize the modal operators from the concurrent dynamic logics of Peleg, Nerode and Wijesekera in a multirelational algebraic language based on relation algebras and power allegories, using relational approximation operators on multirelations developed in a companion article. We relate Nerode and Wijesekera’s box operator with a relational approximation operator for multirelations and two related operators that approximate multirelations by different kinds of deterministic multirelations. We provide an algebraic soundness proof of Goldblatt’s axioms for concurrent dynamic logic and one for a multirelational Hoare logic based on Nerode and Wijesekera’s box as applications.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194641","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}
This paper introduces two logic frameworks for the study of SIR (Susceptible-Infected-Recovered) and SIRS compartmental epidemic models, one based on Linear Temporal Logic and the other on Computation Tree Logic. We provide a short literature overview on compartmental models and other related works using logics, and then define our logics with their respective axiomatizations, and demonstrate their soundness and completeness proofs.
{"title":"Temporal logics for compartmental models","authors":"Vitor Machado, Mario Benevides","doi":"10.1093/logcom/exae021","DOIUrl":"https://doi.org/10.1093/logcom/exae021","url":null,"abstract":"This paper introduces two logic frameworks for the study of SIR (Susceptible-Infected-Recovered) and SIRS compartmental epidemic models, one based on Linear Temporal Logic and the other on Computation Tree Logic. We provide a short literature overview on compartmental models and other related works using logics, and then define our logics with their respective axiomatizations, and demonstrate their soundness and completeness proofs.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"16 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931627","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}
Attack principles have been introduced in semi-abstract argumentation frameworks and, in the present work, we interpret them in sequent-based argumentation frameworks. Thus, we investigate the role of minimality and consistency of the support set of an argument. Through the notion of preservation of strength, we introduce a formal criterion to sort out the attack principles; isolate the more “acceptable” ones, i.e. those easier to justify; and recover a new argumentative semantics for the non-classical logic that arises from dropping the rules $(neg , r)$, $(land , r)$ and $(supset , l)$ from Gentzen’s classical sequent calculus for classical logic $textsf{LK}$.
{"title":"Attack principles in sequent-based argumentation theory","authors":"Esther Anna Corsi","doi":"10.1093/logcom/exad080","DOIUrl":"https://doi.org/10.1093/logcom/exad080","url":null,"abstract":"Attack principles have been introduced in semi-abstract argumentation frameworks and, in the present work, we interpret them in sequent-based argumentation frameworks. Thus, we investigate the role of minimality and consistency of the support set of an argument. Through the notion of preservation of strength, we introduce a formal criterion to sort out the attack principles; isolate the more “acceptable” ones, i.e. those easier to justify; and recover a new argumentative semantics for the non-classical logic that arises from dropping the rules $(neg , r)$, $(land , r)$ and $(supset , l)$ from Gentzen’s classical sequent calculus for classical logic $textsf{LK}$.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575773","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 paper, a new approach to the issue of extra-logical information within analytic (i.e. obeying the sub-formula property) sequent systems is introduced. We prove that incorporating extra-logical axioms into a purely logical system can preserve analyticity, provided these axioms belong to a suitable class of formulas that can be decomposed into a set of equivalent initial sequents and are permutable over the cut rule. Our approach is applicable not only to first-order classical and intuitionistic logics, but also to substructural logics. Furthermore, we establish a limit for the augmented systems under analysis: exceeding the boundaries of their respective classes of extra-logical axioms leads to either a loss of analyticity or a loss of structural properties.
{"title":"Analyticity with extra-logical information","authors":"Mario Piazza, Matteo Tesi","doi":"10.1093/logcom/exae013","DOIUrl":"https://doi.org/10.1093/logcom/exae013","url":null,"abstract":"In this paper, a new approach to the issue of extra-logical information within analytic (i.e. obeying the sub-formula property) sequent systems is introduced. We prove that incorporating extra-logical axioms into a purely logical system can preserve analyticity, provided these axioms belong to a suitable class of formulas that can be decomposed into a set of equivalent initial sequents and are permutable over the cut rule. Our approach is applicable not only to first-order classical and intuitionistic logics, but also to substructural logics. Furthermore, we establish a limit for the augmented systems under analysis: exceeding the boundaries of their respective classes of extra-logical axioms leads to either a loss of analyticity or a loss of structural properties.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"72 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575909","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}
The automated planning community has developed a defacto standard planning language called PDDL. Using the PDDL tools, the reliability of PDDL descriptions can only be posterior examined. However, the Event-B method supports a rich refinement technique that is mathematically proven. Indeed, the Event-B method relies on a first-order predicates language and sets (including mathematical functions and relations) to model the data and on a simple action language to model the treatments. A model described in Event-B includes static modeling elements (sets, constants, axioms and theorems) and dynamic modeling elements (variables, invariant properties and events). Moreover, the Event-B method allows the step-by-step correct construction of Event-B models. To specify and solve the planning problems, a development process based on the combination of Event-B and PDDL is proposed. Our development process favors the obtaining of reliable PDDL description from an ultimate Event-B model using our Event-B2PDDL Eclipse plugin. Our process is successfully experimented on the sliding puzzle game.
{"title":"A correct-by-construction approach for development of reliable planning problems","authors":"Sabrine Ammar, Taoufik Sakka Rouis, Mohamed Tahar Bhiri","doi":"10.1093/logcom/exae016","DOIUrl":"https://doi.org/10.1093/logcom/exae016","url":null,"abstract":"The automated planning community has developed a defacto standard planning language called PDDL. Using the PDDL tools, the reliability of PDDL descriptions can only be posterior examined. However, the Event-B method supports a rich refinement technique that is mathematically proven. Indeed, the Event-B method relies on a first-order predicates language and sets (including mathematical functions and relations) to model the data and on a simple action language to model the treatments. A model described in Event-B includes static modeling elements (sets, constants, axioms and theorems) and dynamic modeling elements (variables, invariant properties and events). Moreover, the Event-B method allows the step-by-step correct construction of Event-B models. To specify and solve the planning problems, a development process based on the combination of Event-B and PDDL is proposed. Our development process favors the obtaining of reliable PDDL description from an ultimate Event-B model using our Event-B2PDDL Eclipse plugin. Our process is successfully experimented on the sliding puzzle game.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"47 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575850","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 paper, I will develop a $lambda $-term calculus, $lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.
{"title":"Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations","authors":"Sara Ayhan","doi":"10.1093/logcom/exae014","DOIUrl":"https://doi.org/10.1093/logcom/exae014","url":null,"abstract":"In this paper, I will develop a $lambda $-term calculus, $lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"108 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575772","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 paper, I argue against the thesis that the meaning of ‘computability’ is logic-dependent. I do this from a category-theoretic perspective. Applying a method due to Mortensen and Lavers [26], I show that we can dualize the internal logic of the effective topos, in order to obtain a model of paraconsistent computability theory. Since the dualization leaves the structural properties of universal constructions in the topos unchanged, in particular the properties of the natural numbers object, I conclude that, at least in this case, changing the logic does not change our characterization of computability.
{"title":"Changing the logic without changing the subject: the case of computability","authors":"Francisco N MartÍnez-Aviña","doi":"10.1093/logcom/exae015","DOIUrl":"https://doi.org/10.1093/logcom/exae015","url":null,"abstract":"In this paper, I argue against the thesis that the meaning of ‘computability’ is logic-dependent. I do this from a category-theoretic perspective. Applying a method due to Mortensen and Lavers [26], I show that we can dualize the internal logic of the effective topos, in order to obtain a model of paraconsistent computability theory. Since the dualization leaves the structural properties of universal constructions in the topos unchanged, in particular the properties of the natural numbers object, I conclude that, at least in this case, changing the logic does not change our characterization of computability.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"439 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575749","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}
Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko
We present an axiomatization of the fuzzy bi-Gödel modal logic ${textbf{K}textsf{biG}}^{textsf{f}}$ formulated in the language containing $triangle $ (Baaz Delta operator) and treating $-!-!< $ (co-implication) as the defined connective. We also consider two paraconsistent relatives of ${textbf{K}textsf{biG}}^{textsf{f}}$ — $textbf{K}textsf{G}^{2pm textsf{f}}$ and $textsf{G}^{2pm textsf{f}}_{blacksquare ,blacklozenge }$. These logics are defined on fuzzy frames with two valuations $e_{1}$ and $e_{2}$ standing for the support of truth and falsity, respectively, and equipped with two fuzzy relations$R^{+}$ and $R^{-}$ used to determine supports of truth and falsity of modal formulas. We construct embeddings of $textbf{K}textsf{G}^{2pm textsf{f}}$ and $textsf{G}^{2pm textsf{f}}_{blacksquare ,blacklozenge }$ into ${textbf{K}textsf{biG}}^{textsf{f}}$ and use them to obtain the characterization of $textbf{K}textsf{G}^{2}$- and $textsf{G}^{2}_{blacksquare ,blacklozenge }$-definable frames. Moreover, we study the transfer of ${textbf{K}textsf{biG}}^{textsf{f}}$ formulas into $textbf{K}textsf{G}^{2pm textsf{f}}$, i.e., formulas that are ${textbf{K}textsf{biG}}^{textsf{f}}$-valid on mono-relational frames $mathfrak{F}$ and $mathfrak{F}^{prime}$ iff they are $textbf{K}textsf{G}^{2pm textsf{f}}$-valid on their bi-relational counterparts. Finally, we establish $textsf{PSpace}$-completeness of all considered logics.
{"title":"Fuzzy bi-Gödel modal logic and its paraconsistent relatives","authors":"Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko","doi":"10.1093/logcom/exae011","DOIUrl":"https://doi.org/10.1093/logcom/exae011","url":null,"abstract":"We present an axiomatization of the fuzzy bi-Gödel modal logic ${textbf{K}textsf{biG}}^{textsf{f}}$ formulated in the language containing $triangle $ (Baaz Delta operator) and treating $-!-!&lt; $ (co-implication) as the defined connective. We also consider two paraconsistent relatives of ${textbf{K}textsf{biG}}^{textsf{f}}$ — $textbf{K}textsf{G}^{2pm textsf{f}}$ and $textsf{G}^{2pm textsf{f}}_{blacksquare ,blacklozenge }$. These logics are defined on fuzzy frames with two valuations $e_{1}$ and $e_{2}$ standing for the support of truth and falsity, respectively, and equipped with two fuzzy relations$R^{+}$ and $R^{-}$ used to determine supports of truth and falsity of modal formulas. We construct embeddings of $textbf{K}textsf{G}^{2pm textsf{f}}$ and $textsf{G}^{2pm textsf{f}}_{blacksquare ,blacklozenge }$ into ${textbf{K}textsf{biG}}^{textsf{f}}$ and use them to obtain the characterization of $textbf{K}textsf{G}^{2}$- and $textsf{G}^{2}_{blacksquare ,blacklozenge }$-definable frames. Moreover, we study the transfer of ${textbf{K}textsf{biG}}^{textsf{f}}$ formulas into $textbf{K}textsf{G}^{2pm textsf{f}}$, i.e., formulas that are ${textbf{K}textsf{biG}}^{textsf{f}}$-valid on mono-relational frames $mathfrak{F}$ and $mathfrak{F}^{prime}$ iff they are $textbf{K}textsf{G}^{2pm textsf{f}}$-valid on their bi-relational counterparts. Finally, we establish $textsf{PSpace}$-completeness of all considered logics.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575771","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 1998 R. Downey formulated a problem: to describe a property $P$ of classical order types, which guarantees that if $mathcal{L}$ is a low linear order and $P$ holds for the order type of $mathcal{L}$ then $mathcal{L}$ is isomorphic to a computable linear order. We find a new such property $P$. Also, we give an upper bound on a complexity of an isomorphism between computable and low copies and show that this bound is sharp.
{"title":"Low scattered linear orders","authors":"Andrey Frolov, Maxim Zubkov","doi":"10.1093/logcom/exae008","DOIUrl":"https://doi.org/10.1093/logcom/exae008","url":null,"abstract":"In 1998 R. Downey formulated a problem: to describe a property $P$ of classical order types, which guarantees that if $mathcal{L}$ is a low linear order and $P$ holds for the order type of $mathcal{L}$ then $mathcal{L}$ is isomorphic to a computable linear order. We find a new such property $P$. Also, we give an upper bound on a complexity of an isomorphism between computable and low copies and show that this bound is sharp.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201208","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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