Logic programming is a declarative programming paradigm that finds extensive use in the field of Artificial Intelligence (AI). As a result, it has become a valuable tool used in university courses for teaching students AI techniques. Besides Prolog language, the more recent Answer Set Programming (ASP) language turns out to be a powerful tool for developing advanced applications due to the expressiveness of the language and the availability of efficient solving systems. Unfortunately, the output of ASP solvers can be difficult to interpret, since it is a set of atoms, often long and verbose. This is most true in the case of students learning the language or in the case of experts developing applications for complex real-world problems. For these reasons, the ability to produce, when possible, a graphical representation of the solver output becomes useful to ensure easier interpretation of the results. In this paper we present ASPECT, a sub-language of ASP in which the user can directly define, in an intuitive and declarative way, a graphical representation of the answer set. The ASPECT atoms can be converted into the popular LaTeX markup language to produce vector graphics. The documents produced by ASPECT are easy to embed in documents such as scientific articles, course handouts and presentations. Also, the development of user-friendly interfaces is critical for wider use of similar technologies in the industrial sector as well. Moreover, ASPECT is also extended to deal with temporal information, and provide graphical animations of answer sets that enclose the temporal dimension, such as in planning problems. Finally, we advocate the use of ASPECT to create complex and animated presentations starting from a declarative specification.
{"title":"ASPECT: Answer Set rePresentation as vEctor graphiCs in laTex","authors":"Alessandro Bertagnon, Marco Gavanelli","doi":"10.1093/logcom/exae042","DOIUrl":"https://doi.org/10.1093/logcom/exae042","url":null,"abstract":"Logic programming is a declarative programming paradigm that finds extensive use in the field of Artificial Intelligence (AI). As a result, it has become a valuable tool used in university courses for teaching students AI techniques. Besides Prolog language, the more recent Answer Set Programming (ASP) language turns out to be a powerful tool for developing advanced applications due to the expressiveness of the language and the availability of efficient solving systems. Unfortunately, the output of ASP solvers can be difficult to interpret, since it is a set of atoms, often long and verbose. This is most true in the case of students learning the language or in the case of experts developing applications for complex real-world problems. For these reasons, the ability to produce, when possible, a graphical representation of the solver output becomes useful to ensure easier interpretation of the results. In this paper we present ASPECT, a sub-language of ASP in which the user can directly define, in an intuitive and declarative way, a graphical representation of the answer set. The ASPECT atoms can be converted into the popular LaTeX markup language to produce vector graphics. The documents produced by ASPECT are easy to embed in documents such as scientific articles, course handouts and presentations. Also, the development of user-friendly interfaces is critical for wider use of similar technologies in the industrial sector as well. Moreover, ASPECT is also extended to deal with temporal information, and provide graphical animations of answer sets that enclose the temporal dimension, such as in planning problems. Finally, we advocate the use of ASPECT to create complex and animated presentations starting from a declarative specification.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"112 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203808","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 a sequent calculus, $textbf{M}_{textbf{S}}$, the minimal structural logic, which includes all structural rules while excluding operational ones. Despite its limited calculus, $textbf{M}_{textbf{S}}$ unexpectedly shares a property with intuitionistic logic and modal logics between $textsf{S1}$ and $textsf{S5}$: it lacks sound and complete finitely-valued (deterministic) semantics. Mirroring Gödel’s and Dugundji’s findings, we demonstrate that $textbf{M}_{textbf{S}}$ does possess a natural finitely-valued non-deterministic semantics. In fact, we show that $textbf{M}_{textbf{S}}$ is sound and complete with respect to any semantics belonging to a natural class of maximally permissive non-deterministic matrices. We close by examining the case of subsystems of $textbf{M}_{textbf{S}}$, including the “structural kernels” of the strict-tolerant and tolerant-strict logics $textbf{ST}$ and $textbf{TS}$, and strengthen this result to also preclude finitely-valued deterministic semantics with respect to variable designated value frameworks.
{"title":"A Gödel-Dugundji-style theorem for the minimal structural logic","authors":"Pawel Pawlowski, Thomas M Ferguson, Ethan Gertler","doi":"10.1093/logcom/exae045","DOIUrl":"https://doi.org/10.1093/logcom/exae045","url":null,"abstract":"This paper introduces a sequent calculus, $textbf{M}_{textbf{S}}$, the minimal structural logic, which includes all structural rules while excluding operational ones. Despite its limited calculus, $textbf{M}_{textbf{S}}$ unexpectedly shares a property with intuitionistic logic and modal logics between $textsf{S1}$ and $textsf{S5}$: it lacks sound and complete finitely-valued (deterministic) semantics. Mirroring Gödel’s and Dugundji’s findings, we demonstrate that $textbf{M}_{textbf{S}}$ does possess a natural finitely-valued non-deterministic semantics. In fact, we show that $textbf{M}_{textbf{S}}$ is sound and complete with respect to any semantics belonging to a natural class of maximally permissive non-deterministic matrices. We close by examining the case of subsystems of $textbf{M}_{textbf{S}}$, including the “structural kernels” of the strict-tolerant and tolerant-strict logics $textbf{ST}$ and $textbf{TS}$, and strengthen this result to also preclude finitely-valued deterministic semantics with respect to variable designated value frameworks.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"70 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203861","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}
Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar [4] and Halldén [30], these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $textsf{B}_{text{e}}^{square }$ and $textsf{PWK}^{Box }_{text{e}} $, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases $textsf{B}_{text{e}}$ and $textsf{PWK}_{text{e}}$, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $textsf{PWK}_{text{e}}$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.
{"title":"Modal weak Kleene logics: axiomatizations and relational semantics","authors":"S Bonzio, N Zamperlin","doi":"10.1093/logcom/exae046","DOIUrl":"https://doi.org/10.1093/logcom/exae046","url":null,"abstract":"Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar [4] and Halldén [30], these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $textsf{B}_{text{e}}^{square }$ and $textsf{PWK}^{Box }_{text{e}} $, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases $textsf{B}_{text{e}}$ and $textsf{PWK}_{text{e}}$, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $textsf{PWK}_{text{e}}$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203810","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 note we extend a remarkable result of Brauer (2024, Journal of Logic and Computation) concerning propositional Classical Core Logic. We show that it holds also at first order. This affords a soundness and completeness result for Classical Core Logic. The $mathbb{C}^{+}$-provable sequents are exactly those that are uniform substitution instances of perfectly valid sequents, i.e. sequents that are valid and that need every one of their sentences in order to be so. Brauer (2020, Review of Symbolic Logic, 13, 436–457) showed that the notion of perfect validity itself is unaxiomatizable. In the Appendix we use his method to show that our notion of relevant validity in Tennant (2024, Philosophia Mathematica) is likewise unaxiomatizable. It would appear that the taking of substitution instances is an essential ingredient in the construction of a semantical relation of consequence that will be axiomatizable—and indeed, by the rules of proof for Classical Core Logic.
{"title":"Perfect proofs at first order","authors":"Neil Tennant","doi":"10.1093/logcom/exae033","DOIUrl":"https://doi.org/10.1093/logcom/exae033","url":null,"abstract":"In this note we extend a remarkable result of Brauer (2024, Journal of Logic and Computation) concerning propositional Classical Core Logic. We show that it holds also at first order. This affords a soundness and completeness result for Classical Core Logic. The $mathbb{C}^{+}$-provable sequents are exactly those that are uniform substitution instances of perfectly valid sequents, i.e. sequents that are valid and that need every one of their sentences in order to be so. Brauer (2020, Review of Symbolic Logic, 13, 436–457) showed that the notion of perfect validity itself is unaxiomatizable. In the Appendix we use his method to show that our notion of relevant validity in Tennant (2024, Philosophia Mathematica) is likewise unaxiomatizable. It would appear that the taking of substitution instances is an essential ingredient in the construction of a semantical relation of consequence that will be axiomatizable—and indeed, by the rules of proof for Classical Core Logic.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We design and study various topological semantics for the diamond-free intuitionistic modal logic $textsf{iS4}$, an intuitionistic analogue of $textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.
{"title":"Intuitionistic S4 as a logic of topological spaces","authors":"Jim de Groot, Ian Shillito","doi":"10.1093/logcom/exae030","DOIUrl":"https://doi.org/10.1093/logcom/exae030","url":null,"abstract":"We design and study various topological semantics for the diamond-free intuitionistic modal logic $textsf{iS4}$, an intuitionistic analogue of $textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141933324","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}
Mario Alviano, Ly Ly Trieu, Tran Cao Son, Marcello Balduccini
Explainable artificial intelligence (XAI) aims at addressing complex problems by coupling solutions with reasons that justify the provided answer. In the context of Answer Set Programming (ASP) the user may be interested in linking the presence or absence of an atom in an answer set to the logic rules involved in the inference of the atom. Such explanations can be given in terms of directed acyclic graphs (DAGs). This article reports on the advancements in the development of the XAI system xASP by revising the main foundational notions and by introducing new ASP encodings to compute minimal assumption sets, explanation sequences, and explanation DAGs. DAGs are shown to the user in an interactive form via the xASP navigator application, also introduced in this work.
{"title":"The XAI system for answer set programming xASP2","authors":"Mario Alviano, Ly Ly Trieu, Tran Cao Son, Marcello Balduccini","doi":"10.1093/logcom/exae036","DOIUrl":"https://doi.org/10.1093/logcom/exae036","url":null,"abstract":"Explainable artificial intelligence (XAI) aims at addressing complex problems by coupling solutions with reasons that justify the provided answer. In the context of Answer Set Programming (ASP) the user may be interested in linking the presence or absence of an atom in an answer set to the logic rules involved in the inference of the atom. Such explanations can be given in terms of directed acyclic graphs (DAGs). This article reports on the advancements in the development of the XAI system xASP by revising the main foundational notions and by introducing new ASP encodings to compute minimal assumption sets, explanation sequences, and explanation DAGs. DAGs are shown to the user in an interactive form via the xASP navigator application, also introduced in this work.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869246","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}
Carmine Dodaro, Giuseppe Galatà, Martin Gebser, Marco Maratea, Cinzia Marte, Marco Mochi, Marco Scanu
The Operating Room Scheduling (ORS) problem deals with the optimization of daily operating room surgery schedules. It is a challenging problem subject to many constraints, like to determine the starting time of different surgeries and allocating the required resources, including the availability of beds in different units. In the past years, Answer Set Programming (ASP) has been successfully employed for addressing and solving the ORS problem. Despite its importance, due to the inherent difficulty of retrieving real data, all the analyses on ORS ASP encodings have been performed on synthetic data so far. In this paper, first we present a new, improved ASP encoding for the ORS problem. Then, we deal with the real case of ASL1 Liguria, an Italian health authority operating through three hospitals, and present adaptations of the ASP encodings to deal with the real-world data. Further, we analyse the resulting encodings on hospital scheduling data by ASL1 Liguria. Results on some scenarios show that the ASP solutions produce satisfying schedules also when applied to such challenging, real data.1
手术室调度（ORS）问题涉及手术室每日手术时间表的优化。这是一个具有挑战性的问题，受到许多约束条件的限制，如确定不同手术的开始时间和分配所需资源，包括不同科室的床位供应情况。在过去几年中，答案集编程（ASP）已被成功用于处理和解决 ORS 问题。尽管其重要性不言而喻，但由于检索真实数据的固有困难，迄今为止，所有关于 ORS ASP 编码的分析都是在合成数据上进行的。在本文中，我们首先针对 ORS 问题提出了一种新的、改进的 ASP 编码。然后，我们处理了 ASL1 Liguria 的真实案例（ASL1 Liguria 是意大利的一个卫生机构，通过三家医院运营），并介绍了 ASP 编码的调整，以处理真实世界的数据。此外，我们还对 ASL1 Liguria 的医院调度数据进行了分析。一些场景的结果表明，ASP 解决方案在应用于此类具有挑战性的真实数据时，也能产生令人满意的日程安排1。
{"title":"Operating Room Scheduling via Answer Set Programming: improved encoding and test on real data","authors":"Carmine Dodaro, Giuseppe Galatà, Martin Gebser, Marco Maratea, Cinzia Marte, Marco Mochi, Marco Scanu","doi":"10.1093/logcom/exae041","DOIUrl":"https://doi.org/10.1093/logcom/exae041","url":null,"abstract":"The Operating Room Scheduling (ORS) problem deals with the optimization of daily operating room surgery schedules. It is a challenging problem subject to many constraints, like to determine the starting time of different surgeries and allocating the required resources, including the availability of beds in different units. In the past years, Answer Set Programming (ASP) has been successfully employed for addressing and solving the ORS problem. Despite its importance, due to the inherent difficulty of retrieving real data, all the analyses on ORS ASP encodings have been performed on synthetic data so far. In this paper, first we present a new, improved ASP encoding for the ORS problem. Then, we deal with the real case of ASL1 Liguria, an Italian health authority operating through three hospitals, and present adaptations of the ASP encodings to deal with the real-world data. Further, we analyse the resulting encodings on hospital scheduling data by ASL1 Liguria. Results on some scenarios show that the ASP solutions produce satisfying schedules also when applied to such challenging, real data.1","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"155 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869242","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}
Valentina Gliozzi, Gian Luca Pozzato, Gabriele Tessore, Alberto Valese
In this paper we present our final solution to the problem of designing an efficient theorem prover for Conditional Logics with the selection function semantics. Conditional Logics recently have received a renewed attention and have found several applications in knowledge representation and artificial intelligence. In order to provide an efficient theorem prover for Conditional Logics, we introduce labelled sequent calculi for the logics characterized by well-established axioms systems including the axiom of strong centering CS, the axiom of conditional identity ID, the axiom of conditional modus ponens MP, as well as the conditional third excluded middle CEM, rejected by Lewis but endorsed by Stalnaker, as well as for the whole cube of extensions. The proposed calculi revise and improve the calculi SeqS introduced in Olivetti et al. (2007, ACM Trans. Comput. Logics, 8). We also present an implementation of these calculi in SWI Prolog, including a graphical interface in Python as well as standard heuristics and refinements that allow us to obtain an efficient theorem prover for the logics under consideration. Moreover, we present some statistics about the performances of the theorem prover, which are promising and significantly better than those of its predecessor CondLean, an implementation of the calculi SeqS.
{"title":"Sequent calculi and an efficient theorem prover for conditional logics with selection function semantics","authors":"Valentina Gliozzi, Gian Luca Pozzato, Gabriele Tessore, Alberto Valese","doi":"10.1093/logcom/exae037","DOIUrl":"https://doi.org/10.1093/logcom/exae037","url":null,"abstract":"In this paper we present our final solution to the problem of designing an efficient theorem prover for Conditional Logics with the selection function semantics. Conditional Logics recently have received a renewed attention and have found several applications in knowledge representation and artificial intelligence. In order to provide an efficient theorem prover for Conditional Logics, we introduce labelled sequent calculi for the logics characterized by well-established axioms systems including the axiom of strong centering CS, the axiom of conditional identity ID, the axiom of conditional modus ponens MP, as well as the conditional third excluded middle CEM, rejected by Lewis but endorsed by Stalnaker, as well as for the whole cube of extensions. The proposed calculi revise and improve the calculi SeqS introduced in Olivetti et al. (2007, ACM Trans. Comput. Logics, 8). We also present an implementation of these calculi in SWI Prolog, including a graphical interface in Python as well as standard heuristics and refinements that allow us to obtain an efficient theorem prover for the logics under consideration. Moreover, we present some statistics about the performances of the theorem prover, which are promising and significantly better than those of its predecessor CondLean, an implementation of the calculi SeqS.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"109 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869245","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 consistency of a theory means that each of its formal derivations $D_{0}, D_{1}, D_{2}, ldots $ is free of contradictions. For Peano Arithmetic PA, after the standard coding of derivations by numerals, PA-consistency is directly represented by the consistency scheme $textsf{Con}^{S}_{textsf{PA}}$, which is a series of arithmetical statements ‘$n$ is not a code of a derivation of $ (0=1)$’ for numerals $n=0,1,2,ldots $. We note that the consistency formula $textsf{Con}_{textsf{PA}}$, $forall x$ ‘$x$ is not a code of a derivation of $(0=1)$,’ is strictly stronger in PA than PA-consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula $textsf{Con}_{textsf{PA}}$ but rather with the consistency scheme $textsf{Con}^{S}_{textsf{PA}}$, which adequately represents PA-consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA-consistency in the form $textsf{Con}^{S}_{textsf{PA}}$ in PA. These findings show that PA proves its consistency whereas, by Gödel’s second incompleteness theorem, PA cannot prove its uniform consistency.
一个理论的一致性意味着它的每一个形式推导 $D_{0}, D_{1}, D_{2}, ldots $ 都没有矛盾。对于皮亚诺算术 PA，在用数字对导数进行标准编码之后，PA-一致性直接由一致性方案 $textsf{Con}^{S}_{textsf{PA}}$来表示，它是一系列针对数字 $n=0,1,2,ldots$的算术声明'$n$不是 $ (0=1)$ 的导数的编码'。我们注意到一致性公式$textsf{Con}_{textsf{PA}}$, $forall x$ '$x$不是$(0=1)$的派生代码'，在PA中严格强于PA-一致性，并且对应于另一些性质，我们称之为统一一致性。在研究 PA 中一致性的可证明性时，我们不应使用一致性公式 $textsf{Con}_{textsf{PA}}$ ，而应使用一致性方案 $textsf{Con}^{S}_{textsf{PA}}$ ，它充分代表了 PA 一致性。本文引入了希尔伯特启发的理论中无穷序列公式的证明概念，并证明了 PA 中 $textsf{Con}^{S}_{textsf{PA}}$ 形式的 PA 一致性。这些发现表明 PA 证明了其一致性，而根据哥德尔第二不完备性定理，PA 无法证明其统一一致性。
{"title":"Serial properties, selector proofs and the provability of consistency","authors":"Sergei Artemov","doi":"10.1093/logcom/exae034","DOIUrl":"https://doi.org/10.1093/logcom/exae034","url":null,"abstract":"The consistency of a theory means that each of its formal derivations $D_{0}, D_{1}, D_{2}, ldots $ is free of contradictions. For Peano Arithmetic PA, after the standard coding of derivations by numerals, PA-consistency is directly represented by the consistency scheme $textsf{Con}^{S}_{textsf{PA}}$, which is a series of arithmetical statements ‘$n$ is not a code of a derivation of $ (0=1)$’ for numerals $n=0,1,2,ldots $. We note that the consistency formula $textsf{Con}_{textsf{PA}}$, $forall x$ ‘$x$ is not a code of a derivation of $(0=1)$,’ is strictly stronger in PA than PA-consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula $textsf{Con}_{textsf{PA}}$ but rather with the consistency scheme $textsf{Con}^{S}_{textsf{PA}}$, which adequately represents PA-consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA-consistency in the form $textsf{Con}^{S}_{textsf{PA}}$ in PA. These findings show that PA proves its consistency whereas, by Gödel’s second incompleteness theorem, PA cannot prove its uniform consistency.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"55 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777113","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}
JosÉ Luis Castiglioni, Víctor FernÁndez, Héctor Federico Mallea, HernÁn Javier San MartÍn
Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subintuitionistic logics, i.e. logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose equivalent algebraic semantics is given by these classes of algebras. Several expansions of these calculi are also studied together with some interesting properties of them.
{"title":"On subreducts of subresiduated lattices and some related logics","authors":"JosÉ Luis Castiglioni, Víctor FernÁndez, Héctor Federico Mallea, HernÁn Javier San MartÍn","doi":"10.1093/logcom/exad042","DOIUrl":"https://doi.org/10.1093/logcom/exad042","url":null,"abstract":"Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subintuitionistic logics, i.e. logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose equivalent algebraic semantics is given by these classes of algebras. Several expansions of these calculi are also studied together with some interesting properties of them.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"167 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777112","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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