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The Stone gamut: A coordinatization of mathematics
 In Logic in Computer Science
, 1995
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Linear lambdaCalculus and Categorical Models Revisited
, 1992
"... 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 ..."
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Cited by 23 (0 self)
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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
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...
Linear logic for generalized quantum mechanics
 In Proc. Workshop on Physics and Computation (PhysComp'92
, 1993
"... 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 l ..."
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Cited by 16 (2 self)
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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 computeraided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1
Approximable concepts, Chu spaces, and information systems
"... 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 cros ..."
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Cited by 13 (8 self)
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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 crossdisciplinary 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 equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings.
Spaces: Complementarity and Uncertainty in Rational Mechanics
, 1994
"... 1 Introduction to Chu spaces 1.1 Basic notions A Boolean Chu space A = (X, =, A) consists of two sets X and A and a binary relation  = ⊆ X × A from X to A. We call the elements x, y,... of X states or opens, and the elements a, b,... of A points, propositions, or events. We ..."
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Cited by 10 (0 self)
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1 Introduction to Chu spaces 1.1 Basic notions A Boolean Chu space A = (X, =, A) consists of two sets X and A and a binary relation  = ⊆ X × A from X to A. We call the elements x, y,... of X states or opens, and the elements a, b,... of A points, propositions, or events. We
Summary
, 1993
"... In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties ..."
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In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties, such as closure under substitution, which others lack. We also study the process of reduction in all three logical formulations, including a detailed proof of cut elimination. Finally, we consider translations between Intuitionistic Logic (IL) and ILL. We then consider the linear term calculus, which arises from applying the CurryHoward correspondence to the natural deduction formulation. We show how the various proof theoretic formulations suggest reductions at the level of terms. The properties of strong normalization and confluence are proved for these reduction rules. We also consider mappings between the extended λcalculus and the linear term calculus. Next we consider a categorical model for ILL. We show how by considering the linear term calculus as an equational logic, we can derive a model: a Linear category. We consider two alternative models: firstly, one due to Seely and then one due to Lafont. Surprisingly, we find that Seely’s model is not sound, in that equal terms are not modelled with equal morphisms. We show how after adapting Seely’s model (by viewing it in a more abstract setting) it becomes a particular instance