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Domain Theory in Logical Form
 Annals of Pure and Applied Logic
, 1991
"... The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and system ..."
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Cited by 231 (10 self)
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The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:
Modeling Concurrency with Geometry
, 1991
"... The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer ..."
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Cited by 125 (13 self)
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The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true nfold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude...
PartialGaggles Applied to Logics with Restricted Structural Rules
 In Peter SchroederHeister and Kosta Dosen, editors, Substructural Logics
, 1991
"... Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in ..."
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Cited by 40 (1 self)
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Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in some position. (6) The family of operations OP is founded when there is a distinguished operator f 2 OP (the head) such that any other operator g 2 OP is a relative of f . Definition. A partialgaggle is a tonoid T = (X; ; OP), in which OP is a founded family. As examples, consider a p.o. residuated groupoid, with OP chosen to be any of the following families of operations (ffi is the head of the families of which it is a member): fffig, fffi; /g, fffi; !g, fffi; /;!g, f/g, f!g. Note that f!;/g does not formally fall under our definition since the trace of one is not directly the contrapositive of the trace of the other, even though the trace of each is a contrapositive of the trace of f...
A representation Theorem for Boolean Contact Algebras
, 2003
"... We prove a representation theorem for Boolean contact algebras which implies that the axioms for the Region Connection Calculus [20] (RCC) are complete for the class of subalgebras of the algebras of regular closed sets of weakly regular connected T 1 spaces. ..."
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Cited by 35 (13 self)
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We prove a representation theorem for Boolean contact algebras which implies that the axioms for the Region Connection Calculus [20] (RCC) are complete for the class of subalgebras of the algebras of regular closed sets of weakly regular connected T 1 spaces.
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
The Stone gamut: A coordinatization of mathematics
 In Logic in Computer Science
, 1995
"... We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone ..."
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Cited by 30 (13 self)
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We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional 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 setCABA 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 StonePontrjagin 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 twodimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N0001492J1974. 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
A Primer On Galois Connections
 York Academy of Science
, 1992
"... : We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to to ..."
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Cited by 29 (3 self)
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: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are wellknown to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 0601, 06A06 Secondary: 5401, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...
Modal Logics, Description Logics and Arithmetic Reasoning
 ARTIFICIAL INTELLIGENCE
, 1999
"... We introduce mathematical programming and atomic decomposition as the basic modal (TBox) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex a ..."
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Cited by 25 (3 self)
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We introduce mathematical programming and atomic decomposition as the basic modal (TBox) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex arithmetical conditions on sets of accessible worlds (role fillers). The atomic decomposition technique can deal with set constructors for modal parameters (role terms) and parameter (role) hierarchies specied in full propositional logic. Besides the standard modal operators, a number of other constructors can be added in a relatively straightforward way. Examples are graded modalities (qualified number restrictions) and also generalized quantiers like `most', `n%', `more' and `many'.
Event Spaces and Their Linear Logic
 In AMAST’91: Algebraic Methodology and Software Technology, Workshops in Computing
, 1991
"... 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 ..."
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Cited by 22 (9 self)
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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...