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An Algebraic View of Structural Induction
 Computer Science Logic 1994, number 933 in Lect. Notes Comp. Sci
, 1994
"... We propose a uniform, categorytheoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B . Regarding ..."
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We propose a uniform, categorytheoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B . Regarding a predicate logic as a fibration p : P!B over B , we consider a logical predicate lifting of T to the total category P. Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the `truth' predicate functor ? : B !Ppreserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the `stability' property of the induction principle. 1...
Quantum and classical structures in nondeterministic computation
 Proceedings of Quanum Interaction 2009, Lecture
"... Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspon ..."
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Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of nonstandard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an onticepistemic gap, as it provides no interface to these nonstandard quantum structures. 1
Additional Key Words and Phrases: Stream circuits
"... This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category into the category of preorders and monotone relations. We g ..."
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This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category into the category of preorders and monotone relations. We give several examples of how our theory generalises usual Hoare logics (partial correctness of while programs, partial correctness of pointer programs), and provide some case studies on how it can be used to develop new Hoare logics (runtime analysis of while programs and stream circuits).