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Stone-Type Dualities for Separation Logics
Logical Methods in Computer Science Volume 15, Issue 1, 2019, pp. 27:1–27:51 Submitted Oct. 10, 2017 https://lmcs.episciences.org/ Published Mar. 14, 2019 STONE-TYPE DUALITIES FOR SEPARATION LOGICS SIMON DOCHERTY a AND DAVID PYM a;b a University College London, London, UK e-mail address: [email protected] b The Alan Turing Institute, London, UK e-mail address: [email protected] Abstract. Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because | in addition to elegant abstraction | they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. -
Quantum Heaps, Cops and Heapy Categories
Mathematical Communications 12(2007), 1-9 1 Quantum heaps, cops and heapy categories Zoran Skodaˇ ∗ Abstract. A heap is a structure with a ternary operation which is intuitively a group with a forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in a monoidal category which is in the case of vector spaces to a quantum heap about what a coalge- bra is to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly, the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category. Key words: quantum algebra, rings, algebras AMS subject classifications: 20N10, 16A24 Received January 25, 2007 Accepted February 1, 2007 1. Classical background: heaps A reference for this section is [1]. 1.1 Before forgetting: group as heap. Let G be a group. Then the ternary operation t : G × G × G → G given by t(a, b, c)=ab−1c, (1) satisfies the following relations: t(b, b, c)=c = t(c, b, b) (2) t(a, b, t(c, d, e)) = t(t(a, b, c),d,e) Aheap(H, t) is a pair of nonempty set H and a ternary operation t : H × H × H satisfying relation (2). A morphism f :(H, t) → (H,t) of heaps is a set map f : H → H satisfying t ◦ (f × f × f)=f ◦ t. -
An Introduction to Category Theory and Categorical Logic
An Introduction to Category Theory and Categorical Logic Wolfgang Jeltsch Category theory An Introduction to Category Theory basics Products, coproducts, and and Categorical Logic exponentials Categorical logic Functors and Wolfgang Jeltsch natural transformations Monoidal TTU¨ K¨uberneetika Instituut categories and monoidal functors Monads and Teooriaseminar comonads April 19 and 26, 2012 References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and comonads Monads and comonads References References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and Monads and comonads comonads References References An Introduction to From set theory to universal algebra Category Theory and Categorical Logic Wolfgang Jeltsch I classical set theory (for example, Zermelo{Fraenkel): I sets Category theory basics I functions from sets to sets Products, I composition -
Ordered Groupoids and the Holomorph of an Inverse Semigroup
ORDERED GROUPOIDS AND THE HOLOMORPH OF AN INVERSE SEMIGROUP N.D. GILBERT AND E.A. MCDOUGALL ABSTRACT. We present a construction for the holomorph of an inverse semi- group, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that pre- serve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids. INTRODUCTION The holomorph of a group G is the semidirect product of Hol(G) = Aut(G) n G of G and its automorphism group (with the natural action of Aut(G) on G). The embedding of Aut(G) into the symmetric group ΣG on G extends to an embedding of Hol(G) into ΣG where g 2 G is identified with its (right) Cayley representation ρg : a 7! ag. Then Hol(G) is the normalizer of G in ΣG ([15, Theorem 9.17]). A second interesting characterization of Hol(G) is due to Baer [1] (see also [3] and [15, Exercise 520]). The heap operation on G is the ternary operation defined by ha; b; ci = ab−1c. Baer shows that a subset of G is closed under the heap operation if and only if it is a coset of some subgroup, and that the subset of ΣG that preserves h· · ·i is precisely Hol(G): that is, if σ 2 ΣG, then for all a; b; c; 2 G we have ha; b; ciσ = haσ; bσ; cσi if and only if σ 2 Hol(G). -
Categorical Semantics of Constructive Set Theory
Categorical semantics of constructive set theory Beim Fachbereich Mathematik der Technischen Universit¨atDarmstadt eingereichte Habilitationsschrift von Benno van den Berg, PhD aus Emmen, die Niederlande 2 Contents 1 Introduction to the thesis 7 1.1 Logic and metamathematics . 7 1.2 Historical intermezzo . 8 1.3 Constructivity . 9 1.4 Constructive set theory . 11 1.5 Algebraic set theory . 15 1.6 Contents . 17 1.7 Warning concerning terminology . 18 1.8 Acknowledgements . 19 2 A unified approach to algebraic set theory 21 2.1 Introduction . 21 2.2 Constructive set theories . 24 2.3 Categories with small maps . 25 2.3.1 Axioms . 25 2.3.2 Consequences . 29 2.3.3 Strengthenings . 31 2.3.4 Relation to other settings . 32 2.4 Models of set theory . 33 2.5 Examples . 35 2.6 Predicative sheaf theory . 36 2.7 Predicative realizability . 37 3 Exact completion 41 3.1 Introduction . 41 3 4 CONTENTS 3.2 Categories with small maps . 45 3.2.1 Classes of small maps . 46 3.2.2 Classes of display maps . 51 3.3 Axioms for classes of small maps . 55 3.3.1 Representability . 55 3.3.2 Separation . 55 3.3.3 Power types . 55 3.3.4 Function types . 57 3.3.5 Inductive types . 58 3.3.6 Infinity . 60 3.3.7 Fullness . 61 3.4 Exactness and its applications . 63 3.5 Exact completion . 66 3.6 Stability properties of axioms for small maps . 73 3.6.1 Representability . 74 3.6.2 Separation . 74 3.6.3 Power types . -
Logic and Categories As Tools for Building Theories
Logic and Categories As Tools For Building Theories Samson Abramsky Oxford University Computing Laboratory 1 Introduction My aim in this short article is to provide an impression of some of the ideas emerging at the interface of logic and computer science, in a form which I hope will be accessible to philosophers. Why is this even a good idea? Because there has been a huge interaction of logic and computer science over the past half-century which has not only played an important r^olein shaping Computer Science, but has also greatly broadened the scope and enriched the content of logic itself.1 This huge effect of Computer Science on Logic over the past five decades has several aspects: new ways of using logic, new attitudes to logic, new questions and methods. These lead to new perspectives on the question: What logic is | and should be! Our main concern is with method and attitude rather than matter; nevertheless, we shall base the general points we wish to make on a case study: Category theory. Many other examples could have been used to illustrate our theme, but this will serve to illustrate some of the points we wish to make. 2 Category Theory Category theory is a vast subject. It has enormous potential for any serious version of `formal philosophy' | and yet this has hardly been realized. We shall begin with introduction to some basic elements of category theory, focussing on the fascinating conceptual issues which arise even at the most elementary level of the subject, and then discuss some its consequences and philosophical ramifications. -
Notes on Categorical Logic
Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour. -
A Bunched Logic for Conditional Independence
A Bunched Logic for Conditional Independence Jialu Bao Simon Docherty University of Wisconsin–Madison University College London Justin Hsu Alexandra Silva University of Wisconsin–Madison University College London Abstract—Independence and conditional independence are fun- For instance, criteria ensuring that algorithms do not discrim- damental concepts for reasoning about groups of random vari- inate based on sensitive characteristics (e.g., gender or race) ables in probabilistic programs. Verification methods for inde- can be formulated using conditional independence [8]. pendence are still nascent, and existing methods cannot handle conditional independence. We extend the logic of bunched impli- Aiming to prove independence in probabilistic programs, cations (BI) with a non-commutative conjunction and provide a Barthe et al. [9] recently introduced Probabilistic Separation model based on Markov kernels; conditional independence can be Logic (PSL) and applied it to formalize security for several directly captured as a logical formula in this model. Noting that well-known constructions from cryptography. The key ingre- Markov kernels are Kleisli arrows for the distribution monad, dient of PSL is a new model of the logic of bunched implica- we then introduce a second model based on the powerset monad and show how it can capture join dependency, a non-probabilistic tions (BI), in which separation is interpreted as probabilistic analogue of conditional independence from database theory. independence. While PSL enables formal reasoning about Finally, we develop a program logic for verifying conditional independence, it does not support conditional independence. independence in probabilistic programs. The core issue is that the model of BI underlying PSL provides no means to describe the distribution of one set of variables I. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
A Constructive Approach to the Resource Semantics of Substructural Logics
A Constructive Approach to the Resource Semantics of Substructural Logics Jason Reed and Frank Pfenning Carnegie Mellon University Pittsburgh, Pennsylvania, USA Abstract. We propose a constructive approach to the resource seman- tics of substructural logics via proof-preserving translations into a frag- ment of focused first-order intuitionistic logic with a preorder. Using these translations, we can obtain uniform proofs of cut admissibility, identity expansion, and the completeness of focusing for a variety of log- ics. We illustrate our approach on linear, ordered, and bunched logics. Key words: Resource Semantics, Focusing, Substructural Logics, Con- structive Logic 1 Introduction Substructural logics derive their expressive power from subverting our usual intuitions about hypotheses. While assumptions in everyday reasoning are things can reasonably be used many times, used zero times, and used in any order, hypotheses in linear logic [Gir87] must be used exactly once, in affine logic at most once, in relevance logic at least once, in ordered logics [Lam58,Pol01] in a specified order, and in bunched logic [OP99] they must obey a discipline of occurrence and disjointness more complicated still. The common device these logics employ to enforce restrictions on use of hypotheses is a structured context. The collection of active hypotheses is in linear logic a multiset, in ordered logic a list, and in bunched logic a tree. Similar to, for instance, display logic [Bel82] we aim to show how to unify diverse substructural logics by isolating and reasoning about the algebraic properties of their context’s structure. Unlike these other approaches, we so do without introducing a new logic that itself has a sophisticated notion of structured context, but instead by making use of the existing concept of focused proofs [And92] in a more familiar nonsubstructural logic. -
Toric Heaps, Cyclic Reducibility, and Conjugacy in Coxeter Groups 3
TORIC HEAPS, CYCLIC REDUCIBILITY, AND CONJUGACY IN COXETER GROUPS SHIH-WEI CHAO AND MATTHEW MACAULEY ABSTRACT. As a visualization of Cartier and Foata’s “partially commutative monoid” theory, G.X. Viennot introduced heaps of pieces in 1986. These are essentially labeled posets satisfying a few additional properties. They naturally arise as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version, motivated by the idea of taking a heap and wrapping it into a cylinder. We call this object a toric heap, as we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. To define the concept of a toric extension, we develop a morphism in the category of toric heaps. We study toric heaps in Coxeter theory, in view of the fact that a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. This allows us to formalize and study a framework of cyclic reducibility in Coxeter theory, and apply it to model conjugacy. We introduce the notion of torically reduced, which is stronger than being cyclically reduced for group elements. This gives rise to a new class of elements called torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems. -
Categorical Logic from a Categorical Point of View
Categorical logic from a categorical point of view Michael Shulman DRAFT for AARMS Summer School 2016 Version of July 28, 2016 2 ii Contents Prefacei 0 Introduction3 0.1 Appetizer: inverses in group objects................3 0.2 On syntax and free objects.....................8 0.3 On type theory and category theory................ 12 0.4 Expectations of the reader...................... 16 1 Unary type theories 19 1.1 Posets................................. 19 1.2 Categories............................... 23 1.2.1 Primitive cuts......................... 24 1.2.2 Cut admissibility....................... 29 1.3 Meet-semilattices........................... 39 1.3.1 Sequent calculus for meet-semilattices........... 40 1.3.2 Natural deduction for meet-semilattices.......... 44 1.4 Categories with products...................... 47 1.5 Categories with coproducts..................... 56 1.6 Universal properties and modularity................ 61 1.7 Presentations and theories...................... 63 1.7.1 Group presentations..................... 64 1.7.2 Category presentations.................... 66 1.7.3 ×-presentations........................ 68 1.7.4 Theories............................ 75 2 Simple type theories 85 2.1 Towards multicategories....................... 85 2.2 Introduction to multicategories................... 89 2.3 Multiposets and monoidal posets.................. 93 2.3.1 Multiposets.......................... 93 2.3.2 Sequent calculus for monoidal posets............ 95 2.3.3 Natural deduction for monoidal posets........... 99 2.4