We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self-dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two-dimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atomic Boolean algebras (CABA’s), and vertically by complexity of language. Complexity 0 contains only sets, CABA’s, and the inconsistent empty set. Complexity 1 admits noninteracting set-CABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts Stone-Pontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework. 1 Mathematics from matrices We organize much of mathematics into a single category Chu of Chu spaces, or games as Lafont and Streicher have called them [LS91]. A Chu space is just a matrix that we shall denote =|, but unlike the matrices of linear algebra, which serve as representations of linear transformations, Chu spaces serve as representations of the objects of mathematics, and their essence resides in how they transform. This organization permits a general two-dimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N00014-92-J-1974. 1 “Spectrum, ” the obvious candidate for this appliction, already has a standard meaning in Stone duality, namely the representation of the dual space of a lattice by its prime ideals. “A

Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not symmetrically. Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language. Event spaces resemble finite dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them i...

this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor,\Omega , linear implication, \Gammaffi, and the logical operator "exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We use capital Greek letters \Gamma; \Delta for sequences of formulae and A; B for single formulae. The Exchange rule simply allows the permutation of assumptions. The `! rules' have been given names by other authors. ! L\Gamma1 is called Weakening , ! L\Gamma2 Contraction, ! L\Gamma3 Dereliction and (! R ) Promotion

Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computer-aided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1

Abstract. This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration of cross-disciplinary connections. Among other results, we show that the notion of state in Scott’s information system corresponds precisely to that of formal concepts in FCA with respect to all finite Chu spaces, and the entailment relation corresponds to “association rules”. We introduce, moreover, the notion of approximable concept and show that approximable concepts represent algebraic lattices which are identical to Scott domains except the inclusion of a top element. This notion serves as a stepping stone in the recent work [Hitzler and Zhang, 2004] in which a new notion of morphism on formal contexts results in a category

this paper will be realizations. The category of Boolean operations and their property-preserving renamings is not self-dual since non-T 0 Chu spaces transpose to nonextensional ones. By the same reasoning the full subcategory consisting of T 0 operations, those with no properties a j b for distinct variables a; b, is self-dual. This is a very important fact: it means that to every full subcategory C of this self-dual category we may associate its dual as the image of C under the self-duality. This associates sets to complete atomic Boolean algebras, Boolean algebras to Stone spaces, distributive lattices to Stone-Priestley posets, semilattices to algebraic lattices, complete semilattices to themselves, and so on for many other familiar [Joh82] and not so familiar (self-duality of finite-dimensional vector spaces over GF (2)) instances of Stone duality We now illustrate the general idea with some examples.