IA Scholar Query: A sequent calculus for relation algebras.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 20 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Fibered Universal Algebra for First-Order Logics
https://scholar.archive.org/work/d23fhht6azdtrm2yvv547ufroq
We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing "algebraic" semantics for nonclassical first-order logics. This framework includes a natural notion of substitution, which allows first-order logics to be considered as structural closure operators just as propositional logics are in abstract algebraic logic. We then establish an extension of the homomorphism theorem from universal algebra for generalized prop-categories and characterize two natural closure operators on the prop-categorical semantics. The first closes a class of structures (which are interpreted as morphisms of prop-categories) under the satisfaction of their common first-order theory and the second closes a class of prop-categories under their associated first-order consequence. It turns out these closure operators have characterizations that closely mirror Birkhoff's characterization of the closure of a class of algebras under the satisfaction of their common equational theory and Blok and J\'onsson's characterization of closure under equational consequence, respectively. These "algebraic" characterizations of the first-order closure operators are unique to the prop-categorical semantics and do not have analogs, for example, in the Tarskian semantics for classical first-order logic. The prop-categories we consider are much more general than traditional intuitionistic prop-categories or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, to our knowledge, our results are still new even when restricted to these special classes of prop-categories.Colin Bloomfield, Yoshihiro Maruyamawork_d23fhht6azdtrm2yvv547ufroqSun, 20 Nov 2022 00:00:00 GMTCompleteness of Nominal PROPs
https://scholar.archive.org/work/xuj6fp4ukrcypj5j4lixv2xwou
We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.Samuel Balco, Alexander Kurzwork_xuj6fp4ukrcypj5j4lixv2xwouSat, 19 Nov 2022 00:00:00 GMTSome Remarks on Counting Propositional Logic
https://scholar.archive.org/work/pcpz757n2rbl7o55tfnq7w52tu
Counting propositional logic was recently introduced in relation to randomized computation and shown able to logically characterize the full counting hierarchy. In this paper we aim to clarify the intuitive meaning and expressive power of its univariate fragment. On the one hand, we provide an effective procedure to measure the probability of counting formulas. On the other, we make the connection between this logic and stochastic experiments explicit, proving that the counting language can simulate any (and only) event associated with dyadic distributions.Melissa Antonelliwork_pcpz757n2rbl7o55tfnq7w52tuWed, 16 Nov 2022 00:00:00 GMTComparing Calculi for First-Order Infinite-Valued Łukasiewicz Logic and First-Order Rational Pavelka Logic
https://scholar.archive.org/work/a6vktkexirfvdhktkuadsj3gia
We consider first-order infinite-valued Łukasiewicz logic and its expansion, first-order rational Pavelka logic RPL∀. From the viewpoint of provability, we compare several Gentzen-type hypersequent calculi for these logics with each other and with Hájek's Hilbert-type calculi for the same logics. To facilitate comparing previously known calculi for the logics, we define two new analytic calculi for RPL∀ and include them in our comparison. The key part of the comparison is a density elimination proof that introduces no cuts for one of the hypersequent calculi considered.Alexander S. Gerasimovwork_a6vktkexirfvdhktkuadsj3giaWed, 16 Nov 2022 00:00:00 GMTAncient Natural Deduction
https://scholar.archive.org/work/mwbdqz5qnre4tav37mkut6kewa
In this note we explore certain key aspects of ancient logic focusing on Aristotle's Topics and Prior Analytics, Galen's Institutio Logica and Proclus' commentary of the first book of Euclid. We argue that the ancients were in possession of the main rules of natural deduction and foreshadowed certain constructivist principles. The conclusion that emerges is that, contrary to what is sometimes believed, Aristotle' s theory of the categorical syllogism represented but a small portion of the achievements of ancient logic, shown here to be adequate for formalising a substantial portion of mathematics as well as scientific and philosophical reasoning.Clarence Lewis Protinwork_mwbdqz5qnre4tav37mkut6kewaTue, 15 Nov 2022 00:00:00 GMTUnder Lock and Key: A Proof System for a Multimodal Logic
https://scholar.archive.org/work/amfb2eeekfemfeamsxbsl3msb4
We present a proof system for a multimodal logic, based on our previous work on a multimodal Martin-Loef type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e. a small 2-category. The logic is extended to a lambda calculus, establishing a Curry-Howard correspondence.G. A. Kavvos, Daniel Gratzerwork_amfb2eeekfemfeamsxbsl3msb4Fri, 11 Nov 2022 00:00:00 GMTZeta Functions and the (Linear) Logic of Markov Processes
https://scholar.archive.org/work/2hovn5h4fje6rknxvupmdnicom
The author introduced models of linear logic known as "Interaction Graphs" which generalise Girard's various geometry of interaction constructions. In this work, we establish how these models essentially rely on a deep connection between zeta functions and the execution of programs, expressed as a cocycle. This is first shown in the simple case of graphs, before begin lifted to dynamical systems. Focussing on probabilistic models, we then explain how the notion of graphings used in Interaction Graphs captures a natural class of sub-Markov processes. We then extend the realisability constructions and the notion of zeta function to provide a realisability model of second-order linear logic over the set of all (discrete-time) sub-Markov processes.Thomas Seillerwork_2hovn5h4fje6rknxvupmdnicomTue, 08 Nov 2022 00:00:00 GMTHHLPy: Practical Verification of Hybrid Systems using Hoare Logic
https://scholar.archive.org/work/ajs5s3v72bbyjdmqjqhxpeapzu
We present a tool for verification of hybrid systems expressed in the sequential fragment of HCSP (Hybrid Communicating Sequential Processes). The tool permits annotating HCSP programs with pre- and postconditions, invariants, and proof rules for reasoning about ordinary differential equations. Verification conditions are generated from the annotations following the rules of hybrid Hoare logic. We designed labeling and highlighting mechanisms to distinguish and visualize different verification conditions. The tool is implemented in Python and has a web-based user interface. We evaluated the effectiveness of the tool on translations of Simulink/Stateflow models and on KeYmaera X benchmarks.Huanhuan Sheng, Alexander Bentkamp, Bohua Zhanwork_ajs5s3v72bbyjdmqjqhxpeapzuMon, 31 Oct 2022 00:00:00 GMTA fundamental non-classical logic
https://scholar.archive.org/work/nbt4qkxni5ee3pnktefqhjhgfe
We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.Wesley H. Hollidaywork_nbt4qkxni5ee3pnktefqhjhgfeSun, 30 Oct 2022 00:00:00 GMTSequent calculi of finite dimension
https://scholar.archive.org/work/7c7kv76mz5fsvdsq6o7c7gj43m
In recent work, the authors introduced the notion of n-dimensional Boolean algebra and the corresponding propositional logic nCL. In this paper, we introduce a sequent calculus for nCL and we show its soundness and completeness.Antonio Bucciarelli and Antonio Ledda and Francesco Paoli and Antonino Salibrawork_7c7kv76mz5fsvdsq6o7c7gj43mWed, 26 Oct 2022 00:00:00 GMTA drag-and-drop proof tactic
https://scholar.archive.org/work/jkiytfd6w5af7owqje3yjnx7xu
We explore the features of a user interface where formal proofs can be built through gestural actions. In particular, we show how proof construction steps can be associated to drag-and-drop actions. We argue that this can provide quick and intuitive proof construction steps. This work builds on theoretical tools coming from deep inference. It also resumes and integrates some ideas of the former proof-by-pointing project.Pablo Donatowork_jkiytfd6w5af7owqje3yjnx7xuFri, 21 Oct 2022 00:00:00 GMTQualitative reasoning in a two-layered framework
https://scholar.archive.org/work/asppcxppdnaxjaut3bekb2dpd4
The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.Marta Bilkova and Sabine Frittella and Daniil Kozhemiachenko and Ondrej Majerwork_asppcxppdnaxjaut3bekb2dpd4Mon, 17 Oct 2022 00:00:00 GMTProving, Refuting, Improving—Looking for a Theorem
https://scholar.archive.org/work/5iwruzz2ungltozhri3hkmomti
Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B,and propose a procedure that should result in the new statement if A′, then B′,which is similar to the starting one, but provable. We think that this procedure, based on the extrapolation method, can be considered a basic methodological tool applicable to prove–refute–improve any conjecture. This new notion, extrapolation, presents a dual counterpart of the well-known interpolation introduced in traditional logic sixty-five years ago.Branislav Boričićwork_5iwruzz2ungltozhri3hkmomtiSat, 15 Oct 2022 00:00:00 GMTWhen programs have to watch paint dry
https://scholar.archive.org/work/3ev55fzryneqbaz54hbtum44gi
We explore type systems and programming abstractions for the safe use of resources. In particular, we investigate how to use types to modularly specify and check when programs are allowed to use their resources, e.g., when programming a robot arm on a production line, it is crucial that painted parts are given enough time to dry before assembly. We capture such temporal resources using a time-graded variant of Fitch-style modal type systems, develop a corresponding modally typed, effectful core calculus, and equip it with a graded-monadic denotational semantics illustrated by a concrete presheaf model. Our calculus also includes temporally-aware graded algebraic effects and effect handlers. The former are given a novel temporal treatment, where operations' specifications include their execution times, and their continuations know that an operation's worth of additional time has passed before they start executing, making it possible to safely access further temporal resources in them, and where effect handlers have to respect this temporal discipline.Danel Ahmanwork_3ev55fzryneqbaz54hbtum44giFri, 14 Oct 2022 00:00:00 GMTProofs and Refutations for Intuitionistic and Second-Order Logic (Extended Version)
https://scholar.archive.org/work/akg6uq3kvreutge4nl74kvugdy
The lambda-PRK-calculus is a typed lambda-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend lambda-PRK to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order lambda-PRK, and we study canonicity results.Pablo Barenbaum, Teodoro Freundwork_akg6uq3kvreutge4nl74kvugdyThu, 13 Oct 2022 00:00:00 GMTSemantical Analysis of the Logic of Bunched Implications
https://scholar.archive.org/work/5y3b4gmr6ncaxmfedwwhlceztm
We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a meta-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth's clause for disjunction instead of Kripke's). It is this complexity that motivates us to use BI as a case-study for this approach to semantics.Alexander V. Gheorghiu, David J. Pymwork_5y3b4gmr6ncaxmfedwwhlceztmTue, 11 Oct 2022 00:00:00 GMTClassifying topoi in synthetic guarded domain theory
https://scholar.archive.org/work/i4lk57mvzjgc5oftl6sviitnxe
Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed in the semantics of programming languages. In order to unify the accounts of guarded recursion and coinduction, several authors have enriched SGDT with multiple "clocks" parameterizing different time-streams, leading to more complex and difficult to understand topos models. Until now these topoi have been understood very concretely qua categories of presheaves, and the logico-geometrical question of what theories these topoi classify has remained open. We show that several important topos models of SGDT classify very simple geometric theories, and that the passage to various forms of multi-clock guarded recursion can be rephrased more compositionally in terms of the lower bagtopos construction of Vickers and variations thereon due to Johnstone. We contribute to the consolidation of SGDT by isolating the universal property of multi-clock guarded recursion as a modular construction that applies to any topos model of single-clock guarded recursion.Daniele Palombi, Jonathan Sterlingwork_i4lk57mvzjgc5oftl6sviitnxeMon, 10 Oct 2022 00:00:00 GMTRelational Models for the Lambek Calculus with Intersection and Constants
https://scholar.archive.org/work/3xtfwjho3ndplcw6ytylkah6mi
We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.Stepan L. Kuznetsovwork_3xtfwjho3ndplcw6ytylkah6miSun, 02 Oct 2022 00:00:00 GMTNested Session Types
https://scholar.archive.org/work/cdzjx4x355eyjn7slpugmdj6di
Session types statically describe communication protocols between concurrent message-passing processes. Unfortunately, parametric polymorphism even in its restricted prenex form is not fully understood in the context of session types. In this article, we present the metatheory of session types extended with prenex polymorphism and, as a result, nested recursive datatypes. Remarkably, we prove that type equality is decidable by exhibiting a reduction to trace equivalence of deterministic first-order grammars. Recognizing the high theoretical complexity of the latter, we also propose a novel type equality algorithm and prove its soundness. We observe that the algorithm is surprisingly efficient and, despite its incompleteness, sufficient for all our examples. We have implemented our ideas by extending the Rast programming language with nested session types. We conclude with several examples illustrating the expressivity of our enhanced type system.Ankush Das, Henry Deyoung, Andreia Mordido, Frank Pfenningwork_cdzjx4x355eyjn7slpugmdj6diFri, 30 Sep 2022 00:00:00 GMTFocusing on Liquid Refinement Typing
https://scholar.archive.org/work/dqxe6jc5jvblhjzv7ddobqfs2i
We present a foundation systematizing, in a way that works for any evaluation order, the variety of mechanisms for SMT constraint generation found in index refinement and liquid type systems. Using call-by-push-value, we design a polarized subtyping relation allowing us to prove that our logically focused typing algorithm is sound, complete, and decidable, even in cases seemingly likely to produce constraints with existential variables. We prove type soundness with respect to an elementary domain-theoretic denotational semantics. Soundness implies, relatively simply, our system's totality and logical consistency.Dimitrios J. Economou and Neel Krishnaswami and Jana Dunfieldwork_dqxe6jc5jvblhjzv7ddobqfs2iMon, 26 Sep 2022 00:00:00 GMT