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A Categorical Quantum Logic
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax ca ..."
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Cited by 22 (5 self)
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We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.
Types for Quantum Computation
, 2007
"... This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract qua ..."
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Cited by 11 (5 self)
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This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in lowlevel formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly
Expanding the realm of systematic proof theory
"... Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language ..."
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Cited by 4 (2 self)
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Abstract. This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionisticsubstructural axioms and singleconclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language of classical linear logic without exponentials. We then give a systematic procedure to transform axioms up to the level P ′ 3 of the hierarchy into inference rules in multipleconclusion (hyper)sequent calculi, which enjoy cutelimination under a certain condition. This allows a systematic treatment of logics which could not be dealt with in the previous approach. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P ′ 3. The case study of Abelian and ̷Lukasiewicz logic is outlined. 1
HigherOrder Encodings with Constructors
, 2008
"... As programming languages become more complex, there is a growing call in the research community for machinechecked proofs about programming languages. A key obstacle to this goal is in formalizing name binding, where a new name is created in a limited scope. Name binding is used in almost every pro ..."
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Cited by 1 (0 self)
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As programming languages become more complex, there is a growing call in the research community for machinechecked proofs about programming languages. A key obstacle to this goal is in formalizing name binding, where a new name is created in a limited scope. Name binding is used in almost every programming language to refer to the formal arguments to a function. For example, the function f (x) = x ∗ 2, which doubles its argument, binds the name x for its formal argument. Though this concept is intuitively straightforward, it is complex to define precisely because of the intended properties of name binding. For example, the above function is considered “syntactically equivalent ” to f (y) = y ∗ 2. It is the goal of this dissertation to posit a new technique for encoding name binding, called HigherOrder Encodings with Constructors or HOEC. HOEC encodes name binding with a construct called the νabstraction, which binds new constructors in a limited scope. These constructors can then be used to encode names. νabstractions already have the required properties of name bindings, so name binding need only be ii formalized once, in the definition of the νabstraction. The user thus then gets name