IA Scholar Query: Free Algebras for Gödel-Löb Provability Logic.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 03 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Crisp bi-Gödel modal logic and its paraconsistent expansion
https://scholar.archive.org/work/3huwl47hvfbknfvvg46nuu3c6i
In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-Gödel modal logic . We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on [0,1]. We also consider a paraconsistent expansion of with a De Morgan negation which we dub . We devise a Hilbert-style calculus for this logic and, as a consequence of a conservative translation from to , prove its completeness w.r.t. crisp Kripke models with two valuations over [0,1] connected via . For these two logics, we establish that their decidability and validity are 𝖯𝖲𝖯𝖠𝖢𝖤-complete. We also study the semantical properties of and . In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in 𝐊 (the classical modal logic) and the crisp Gödel modal logic ^c. We show that, among others, all Sahlqvist formulas and all formulas ϕ→χ where ϕ and χ are monotone, define the same classes of frames in 𝐊 and ^c.Marta Bilkova and Sabine Frittella and Daniil Kozhemiachenkowork_3huwl47hvfbknfvvg46nuu3c6iThu, 03 Nov 2022 00:00:00 GMTFinding the limit of incompleteness II
https://scholar.archive.org/work/eewivur5sjgs3occmv7emdi6yq
This work is motivated from finding the limit of the applicability of the first incompleteness theorem (G1). A natural question is: can we find a minimal theory for which G1 holds? The answer of this question depends on our definition of minimality. We first show that the Turing degree structure of recursively enumerable theories for which G1 holds is as complex as the structure of recursively enumerable Turing degrees. Then we examine the interpretation degree structure of recursively enumerable theories weaker than the theory 𝐑 with respect to interpretation for which G1 holds, and answer all questions about this structure in our published paper. We have two general characterizations which tell us under what conditions there are no minimal recursively enumerable theories with some property with respect to interpretation. As an application, we propose the theory version of recursively inseparable theories, tRI theories, and show that there are no minimal tRI theories with respect to interpretation: for any tRI theory, we can effectively find a strictly weaker tRI theory with respect to interpretation.Yong Chengwork_eewivur5sjgs3occmv7emdi6yqMon, 31 Oct 2022 00:00:00 GMTTopological duality for distributive lattices, and applications
https://scholar.archive.org/work/xnnm6lpovzeg7euoh4xupsxexm
This material will be published by Cambridge University Press as "Topological Duality for Distributive Lattices: Theory and Applications" by Mai Gehrke and Sam van Gool. This pre-publication is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. \copyright Mai Gehrke and Sam van Gool This book is a course in Stone-Priestley duality theory, with applications to logic and theoretical computer science. Our target audience are graduate students and researchers in mathematics and computer science. Our aim is to get in a fairly full palette of duality tools as directly and quickly as possible, then to illustrate and further elaborate these tools within the setting of three emblematic applications: semantics of propositional logics, domain theory in logical form, and the theory of profinite monoids for the study of regular languages and automata. This pre-publication contains the first part of the book, a graduate level 'crash course' in duality theory as it is practiced now, and a chapter on applications to domain theory.Mai Gehrke, Sam van Goolwork_xnnm6lpovzeg7euoh4xupsxexmFri, 26 Aug 2022 00:00:00 GMTA Totally Predictable Outcome: An Investigation of Traversals of Infinite Structures
https://scholar.archive.org/work/r3x3v6xxnzcxdf4udfraduoimu
Functors with an instance of the Traversable type class can be thought of as data structures which permit a traversal of their elements. This has been made precise by the correspondence between traversable functors and finitary containers (also known as polynomial functors) -- established in the context of total, necessarily terminating, functions. However, the Haskell language is non-strict and permits functions that do not terminate. It has long been observed that traversals can at times in fact operate over infinite lists, for example in distributing the Reader applicative. The result of such a traversal remains an infinite structure, however it nonetheless is productive -- i.e. successive amounts of finite computation yield either termination or successive results. To investigate this phenomenon, we draw on tools from guarded recursion, making use of equational reasoning directly in Haskell.Gershom Bazermanwork_r3x3v6xxnzcxdf4udfraduoimuWed, 20 Jul 2022 00:00:00 GMTTranslational Embeddings via Stable Canonical Rules
https://scholar.archive.org/work/jgbs75l7lbbotczwwupajpyfbu
This paper presents a new uniform method for studying modal companions of superintuitionistic deductive systems and related notions, based on the machinery of stable canonical rules. Using our method, we obtain an alternative proof of the Blok-Esakia theorem both for logics and for rule systems, and prove an analogue of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalises smoothly to richer signatures. We illustrate this by applying our techniques to prove analogues of the Blok-Esakia theorem (for both logics and rule systems) and of the Dummett-Lemmon conjecture (for rule systems) in the setting of tense companions of bi-superintuitionistic deductive systems. We also use our techniques to prove that the lattice of rule systems (logics) extending the modal intuitionistic logic 𝙺𝙼 and the lattice of rule systems (logics) extending the provability logic 𝙶𝙻 are isomorphic.Nick Bezhanishvili, Antonio Maria Cleaniwork_jgbs75l7lbbotczwwupajpyfbuFri, 17 Jun 2022 00:00:00 GMTUntangled: A Complete Dynamic Topological Logic
https://scholar.archive.org/work/bmlt2gs2lzftfdovkbu6p5yete
Dynamic topological logic (𝐃𝐓𝐋) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by Fernández-Duque that the natural set of axioms for 𝐃𝐓𝐋 is incomplete, but he provided a complete axiomatisation in an extended language. In this paper, we consider dynamic topological logic over scattered spaces, which are topological spaces where every nonempty subspace has an isolated point. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of 𝐃𝐓𝐋 based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete.David Fernández-Duque, Yoàv Montacutework_bmlt2gs2lzftfdovkbu6p5yeteMon, 18 Apr 2022 00:00:00 GMTGeometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472)
https://scholar.archive.org/work/gnnydk6kx5adlmegxfiwolipge
At least from a practical and contemporary angle, the time-honoured question about the extent of intuitionistic mathematics rather is to which extent any given proof is effective, which proofs of which theorems can be rendered effective, and whether and how numerical information such as bounds and algorithms can be extracted from proofs. All this is ideally done by manipulating proofs mechanically or by adequate metatheorems, which includes proof translations, automated theorem proving, program extraction from proofs, proof analysis and proof mining. The question should thus be put as: What is the computational content of proofs? Guided by this central question, the present Dagstuhl seminar puts a special focus on coherent and geometric theories and their generalisations. These are not only widespread in mathematics and non-classical logics such as temporal and modal logics, but also a priori amenable for constructivisation, e.g., by Barr's Theorem, and last but not least particularly suited as a basis for automated theorem proving. Specific topics include categorical semantics for geometric theories, complexity issues of and algorithms for geometrisation of theories including speed-up questions, the use of geometric theories in constructive mathematics including finding algorithms, proof-theoretic presentation of sheaf models and higher toposes, and coherent logic for automatically readable proofs.Thierry Coquand, Hajime Ishihara, Sara Negri, Peter M. Schusterwork_gnnydk6kx5adlmegxfiwolipgeMon, 11 Apr 2022 00:00:00 GMTComputability in constructive type theory
https://scholar.archive.org/work/6nindq52jjdxfe3jnudoeds4yi
We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice's theorem, the Myhill isomorphism theorem, and the existence of Post's simple and hypersimple predicates relying on no other axioms such as Markov's principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type → is L-computable. This thesis is a product of more than seven years of research. Since then, my journey through academia has brought me to many great places and introduced me to many great people. I feel that the final product of the journey -this thesis -is a result of a big collaboration of all the people who journeyed with me. Ein riesiger Dank geht an meine Familie. Papa, ich hab die letzten Jahre häufig gemerkt wie viel ich von dir gelernt habe, und wie viel ich noch lernen kann. Die Leichtigkeit, die du zeigst, während du jeden Morgen so früh aufstehst, und die Energie, die du trotzdem für dein Umfeld hast, sind mir ein Vorbild. Mama, deine Liebe, dein Interesse, deine Unterstützung, dein Verständnis und dein Vertrauen waren unabdingbar für mich. Joshua, danke dass du mich in FIFA bisweilen hast gewinnen lassen. Opa, es macht mich immer noch traurig, dass du meinen Werdegang nicht begleiten konntest. Danke für das Fördern meiner Neugier und dein Vertrauen in meine Fähigkeiten ab dem ersten Tag. Ich wäre ohne euch alle nie an den Punkt gekommen, überhaupt diese Dissertation zu schreiben. Mi familia peruana, gracias por todo.Yannick Forster, Universität Des Saarlandeswork_6nindq52jjdxfe3jnudoeds4yiTue, 05 Apr 2022 00:00:00 GMTRealising Intensional S4 and GL Modalities
https://scholar.archive.org/work/rbvs2orvjfh3nhslqt7yncixla
There have been investigations into type-theoretic foundations for metaprogramming, notably Davies and Pfenning's (2001) treatment in S4 modal logic, where code evaluating to values of type A is given the modal type Code A (□A in the original paper). Recently Kavvos (2017) extended PCF with Code A and intensional recursion, understood as the deductive form of the GL (Gödel-Löb) axiom in provability logic, but the resulting type system is logically inconsistent. Inspired by staged computation, we observe that a term of type Code A is, in general, code to be evaluated in a next stage, whereas S4 modal type theory is a special case where code can be evaluated in the current stage, and the two types of code should be discriminated. Consequently, we use two separate modalities ⊠ and □ to model S4 and GL respectively in a unified categorical framework while retaining logical consistency. Following Kavvos' (2017) novel approach to the semantics of intensionality, we interpret the two modalities in the P-category of assemblies and trackable maps. For the GL modality □ in particular, we use guarded type theory to articulate what it means by a "next" stage and to model intensional recursion by guarded recursion together with Kleene's second recursion theorem. Besides validating the S4 and GL axioms, our model better captures the essence of intensionality by refuting congruence (so that two extensionally equal terms may not be intensionally equal) and internal quoting (both A → □A and A → ⊠A). Our results are developed in (guarded) homotopy type theory and formalised in Agda.Liang-Ting Chen, Hsiang-Shang Ko, Florin Manea, Alex Simpsonwork_rbvs2orvjfh3nhslqt7yncixlaThu, 27 Jan 2022 00:00:00 GMT