79 research outputs found

    A finite axiomatization of flowchart schemes

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    Turing Automata and Graph Machines

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    Indexed monoidal algebras are introduced as an equivalent structure for self-dual compact closed categories, and a coherence theorem is proved for the category of such algebras. Turing automata and Turing graph machines are defined by generalizing the classical Turing machine concept, so that the collection of such machines becomes an indexed monoidal algebra. On the analogy of the von Neumann data-flow computer architecture, Turing graph machines are proposed as potentially reversible low-level universal computational devices, and a truly reversible molecular size hardware model is presented as an example

    Quantum Turing automata

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    A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The Joyal-Street-Verity Int construction is then used to extend this structure to a reversible bidirectional one.Comment: In Proceedings DCM 2012, arXiv:1403.757

    On flowchart theories Part I. The deterministic case

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    AbstractWe give a calculus for the classes of deterministic flowchart schemes with respect to the strong equivalence relation, similar to the calculus of the classes of polynomials with respect to the reduction of similar terms. The algebraic structure involved is a strong iteration theory, i.e., an iteration theory (defined by Bloom, Elgot, and Wright, SIAM J. Comput. 9 (1980), 525–540) satisfying a “functorial dagger implication.

    A note on the axiomatization of iteration theories

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    Acta Cybernetica : Volume 9. Number 4.

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    On the completeness of the traced monoidal category axioms in (Rel,+)

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    It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced symmetric monoidal axioms by two simple axioms, which capture the additive nature of trace in this category. The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms. In practical terms this means that if two biaccessible flowchart schemes are not isomorphic, then there exists an interpretation of the schemes by partial injections which distinguishes them

    Covering morphisms and unique minimal D-schemes

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