J. Log. Comput.Pub Date : 2021-05-06DOI: 10.1093/LOGCOM/EXAB020
Sonia Marin, Marianela Morales, Lutz Straßburger
{"title":"A fully labelled proof system for intuitionistic modal logics","authors":"Sonia Marin, Marianela Morales, Lutz Straßburger","doi":"10.1093/LOGCOM/EXAB020","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAB020","url":null,"abstract":"\u0000 Labelled proof theory has been famously successful for modal logics by mimicking their relational semantics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionistic modal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequent system for intuitionistic modal logics such that there is not only one but two relation symbols appearing in sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logics and one for the pre-order relation associated with the Kripke semantics for intuitionistic logic. This puts our system in close correspondence with the standard birelational Kripke semantics for intuitionistic modal logics. As a consequence, it can be extended with arbitrary intuitionistic Scott–Lemmon axioms. We show soundness and completeness, together with an internal cut elimination proof, encompassing a wider array of intuitionistic modal logics than any existing labelled system.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114772903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2021-04-23DOI: 10.1093/logcom/exac049
Marlou M. Gijzen, H. Ishihara, Tatsuji Kawai
{"title":"Reflexive combinatory algebras","authors":"Marlou M. Gijzen, H. Ishihara, Tatsuji Kawai","doi":"10.1093/logcom/exac049","DOIUrl":"https://doi.org/10.1093/logcom/exac049","url":null,"abstract":"\u0000 We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer–Scott axiom of combinatory models, which indeed allows us to characterize an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatization with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterized as strongly reflexive combinatory algebras that are stable. Moreover, there is a canonical construction of a lambda algebra from a strongly reflexive combinatory algebra. The resulting axiomatization of lambda algebras by the seven axioms for strong reflexivity together with those for stability is shown to correspond to the axiomatization of lambda algebras due to Selinger (2002, J. Funct. Program., 12, 549–566).","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131039188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2021-04-16DOI: 10.1093/logcom/exab015
R. Kuznets, Björn Lellmann
{"title":"Interpolation for intermediate logics via injective nested sequents","authors":"R. Kuznets, Björn Lellmann","doi":"10.1093/logcom/exab015","DOIUrl":"https://doi.org/10.1093/logcom/exab015","url":null,"abstract":"\u0000 We introduce a novel, semantically inspired method of constructing nested sequent calculi for propositional intermediate logics. Applying recently developed methods for proving Craig interpolation to these nested sequent calculi, we obtain constructive proofs of the interpolation property for most non-trivial interpolable intermediate logics, as well as Lyndon interpolation for Gödel logic. Finally, we provide a prototype implementation combining proof search and countermodel construction.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131782146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2021-04-12DOI: 10.1093/LOGCOM/EXAB017
D. Galmiche, D. Méry
{"title":"Labelled cyclic proofs for separation logic","authors":"D. Galmiche, D. Méry","doi":"10.1093/LOGCOM/EXAB017","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAB017","url":null,"abstract":"\u0000 Separation logic (SL) is a logical formalism for reasoning about programs that use pointers to mutate data structures. It is successful for program verification as an assertion language to state properties about memory heaps using Hoare triples. Most of the proof systems and verification tools for ${textrm{SL}}$ focus on the decidable but rather restricted symbolic heaps fragment. Moreover, recent proof systems that go beyond symbolic heaps are purely syntactic or labelled systems dedicated to some fragments of ${textrm{SL}}$ and they mainly allow either the full set of connectives, or the definition of arbitrary inductive predicates, but not both. In this work, we present a labelled proof system, called ${textrm{G}_{textrm{SL}}}$, that allows both the definition of cyclic proofs with arbitrary inductive predicates and the full set of SL connectives. We prove its soundness and show that we can derive in ${textrm{G}_{textrm{SL}}}$ the built-in rules for data structures of another non-cyclic labelled proof system and also that ${textrm{G}_{textrm{SL}}}$ is strictly more powerful than that system.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"42 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130660432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2021-03-29DOI: 10.1093/LOGCOM/EXAB014
Camillo Fiorentini, Mauro Ferrari
{"title":"A forward internal calculus for model generation in S4","authors":"Camillo Fiorentini, Mauro Ferrari","doi":"10.1093/LOGCOM/EXAB014","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAB014","url":null,"abstract":"\u0000 We propose an internal calculus to check the satisfiability of a set of formulas in ${boldsymbol {S4}}$. Our calculus directly supports model extraction and is designed so to implement a forward proof-search strategy that can be understood as a top-down construction of a model. We prove that the extracted models have minimal height.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125818583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2021-03-26DOI: 10.1093/LOGCOM/EXAB013
Bartosz Więckowski
{"title":"Intuitionistic multi-agent subatomic natural deduction for belief and knowledge","authors":"Bartosz Więckowski","doi":"10.1093/LOGCOM/EXAB013","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAB013","url":null,"abstract":"\u0000 This paper proposes natural deduction systems for the representation of inferences in which several agents participate in deriving conclusions about what they believe or know, where belief and knowledge are understood in an intuitionistic sense. Multi-agent derivations in these systems may involve relatively complex belief (resp. knowledge) constructions which may include forms of nested, reciprocal, shared, distributed or universal belief/knowledge as well as attitudes de dicto/re/se. The systems consist of two main components: multi-agent belief bases which assign to each agent a subatomic system that represents the agent’s beliefs concerning atomic sentences and a set of multi-agent labelled rules for logically compound formulae. Derivations in these systems normalize. Moreover, normal derivations possess the subexpression property (a refinement of the subformula property) which makes them fully analytic. Relying on the normalization result, a proof-theoretic approach to the semantics of the intensional operators for intuitionistic belief/knowledge is presented which explains their meaning entirely by appeal to the structure of derivations. Importantly, this proof-theoretic semantics is autarkic with respect to its foundations as the systems (unlike, e.g. external/labelled proof systems which internalize possible worlds truth conditions) are not defined on the basis of a possible worlds semantics. Detailed applications to a logical puzzle (McCarthy’s three wise men puzzle) and to a semantical difficulty (Geach’s problem of intentional identity), respectively, illustrate the systems. The paper also provides comparisons with other approaches to intuitionistic belief/knowledge and multi-agent natural deduction.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126513346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2020-12-30DOI: 10.1093/logcom/exaa083
D. Willard
{"title":"About the characterization of a fine line that separates generalizations and boundary-case exceptions for the Second Incompleteness Theorem under semantic tableau deduction","authors":"D. Willard","doi":"10.1093/logcom/exaa083","DOIUrl":"https://doi.org/10.1093/logcom/exaa083","url":null,"abstract":"\u0000 Our previous research showed that the semantic tableau deductive methodology of Fitting and Smullyan permits boundary-case exceptions to the second incompleteness theorem, if multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableau methodologies prove a schema of theorems verifying all instances of the law of the excluded middle. But if one promotes this schema of theorems into formalized logical axioms, then the meaning of the pronoun of ‘I’, used by our self-referencing engine, changes quite sharply. Our partial evasions of the second incompleteness theorem shall then come to a complete halt.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115371716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2020-12-29DOI: 10.1093/logcom/exaa079
Yasir Mahmood, A. Meier, Johannes Schmidt
{"title":"Parameterized complexity of abduction in Schaefer's framework","authors":"Yasir Mahmood, A. Meier, Johannes Schmidt","doi":"10.1093/logcom/exaa079","DOIUrl":"https://doi.org/10.1093/logcom/exaa079","url":null,"abstract":"\u0000 Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version, asking for sets of variables as explanations, we study, besides the problem of wether there exists a set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to a given size bound). In this paper, we present a thorough two-dimensional classification of these problems: the first dimension is regarding the parameterized complexity under a wealth of different parameterizations, and the second dimension spans through all possible Boolean fragments of these problems in Schaefer’s constraint satisfaction framework with co-clones (T. J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, R.J. Lipton, W.A. Burkhard, W.J. Savitch, E.P. Friedman, A.V. Aho eds, pp. 216–226. ACM, 1978). Thereby, we almost complete the parameterized complexity classification program initiated by Fellows et al. (The parameterized complexity of abduction. In Proceedings of the Twenty-Sixth AAAI Conference on Articial Intelligence, July 22–26, 2012, Toronto, Ontario, Canada, J. Homann, B. Selman eds. AAAI Press, 2012), partially building on the results by Nordh and Zanuttini (What makes propositional abduction tractable. Artificial Intelligence, 172, 1245–1284, 2008). In this process, we outline a fine-grained analysis of the inherent parameterized intractability of these problems and pinpoint their FPT parts. As the standard algebraic approach is not applicable to our problems, we develop an alternative method that makes the algebraic tools partially available again.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128261329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2020-12-23DOI: 10.1093/logcom/exaa070
N. Ackerman, Cameron E. Freer, Rehana Patel
{"title":"On computable aspects of algebraic and definable closure","authors":"N. Ackerman, Cameron E. Freer, Rehana Patel","doi":"10.1093/logcom/exaa070","DOIUrl":"https://doi.org/10.1093/logcom/exaa070","url":null,"abstract":"\u0000 We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $varSigma ^0_{n+2}$ sets. We further show that these bounds are tight.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134358855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Log. Comput.Pub Date : 2020-12-19DOI: 10.1093/logcom/exaa075
H. Kushida
{"title":"Constructive truth and falsity in Peano arithmetic","authors":"H. Kushida","doi":"10.1093/logcom/exaa075","DOIUrl":"https://doi.org/10.1093/logcom/exaa075","url":null,"abstract":"\u0000 Artemov (2019, The provability of consistency) offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer–Heyting–Kolmogorov semantics and its formalization, the logic of proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano arithmetic. For this purpose, we generalize the constructive falsity to $n$-constructive falsity in Peano arithmetic where $n$ is any positive natural number. Based on this generalization, we also analyse the logical status of well-known Gödelean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences and Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor $n$-constructively false for any $n$.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129438269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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