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Categories for Computation in Context and Unified Logic: The "Intuitionist" Case
, 1997
"... In this paper we introduce the notion of contextual categories. These provide a categorical semantics for the modelling of computation in context, based on the idea of separating logical sequents into two zones, one representing the context over which the computation is occurring, the other the comp ..."
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In this paper we introduce the notion of contextual categories. These provide a categorical semantics for the modelling of computation in context, based on the idea of separating logical sequents into two zones, one representing the context over which the computation is occurring, the other the computation itself. The separation into zones is achieved via a bifunctor equipped with a tensorial strength. We show that a category with such a functor can be viewed as having an action on itself. With this interpretation, we obtain a fibration in which the base category consists of contexts, and the reindexing functors are used to change the context. We further observe that this structure also provides a framework for developing categorical semantics for Girard's Unified Logic, a key feature of which is to separate logical sequents into two zones, one in which formulas behave classically and one in which they behave linearly. This separation is analogous to the context/computation separation ...
Quantum Computing: A new Paradigm and it's Type Theory
 Lecture given at the Quantum Computing Seminar, Lehrstuhl Prof. Beth, Universität
, 1996
"... To use quantum mechanical behavior for computing has been proposed by Feynman. Shor gave an algorithm for the quantum computer which raised a big stream of research. This was because Shor's algorithm did reduce the yet assumed exponential complexity of the security relevant factorization problem, to ..."
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To use quantum mechanical behavior for computing has been proposed by Feynman. Shor gave an algorithm for the quantum computer which raised a big stream of research. This was because Shor's algorithm did reduce the yet assumed exponential complexity of the security relevant factorization problem, to a quadratic complexity if quantum computed. In the paper a short introduction to quantum mechanics can be found in the appendix. With this material the operation of the quantum computer, and the ideas of quantum logic will be explained. The focus will be the argument that a connection of quantum logic and linear logic is the right type theory for quantum computing. These ideas are inspired by Vaughan Pratt's view that the intuitionistic formulas argue about states (i.e physical quantum states) and linear formulas argue about state transformations (i.e computation steps). 1 Introduction A calculus for programs on quantum computers is strongly missed. Here we present the material t...
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
SECURITY ANALYSIS OF NETWORK PROTOCOLS: COMPOSITIONAL REASONING AND COMPLEXITYTHEORETIC FOUNDATIONS
, 2005
"... in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. John C. Mitchell(Principal Adviser) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of D ..."
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in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. John C. Mitchell(Principal Adviser) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.