4,862 research outputs found
On flowchart theories Part I. The deterministic case
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.
Turing Automata and Graph Machines
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
Network algebra for synchronous dataflow
We develop an algebraic theory of synchronous dataflow networks. First, a
basic algebraic theory of networks, called BNA (Basic Network Algebra), is
introduced. This theory captures the basic algebraic properties of networks.
For synchronous dataflow networks, it is subsequently extended with additional
constants for the branching connections that occur between the cells of
synchronous dataflow networks and axioms for these additional constants. We
also give two models of the resulting theory, the one based on stream
transformers and the other based on processes as considered in process algebra.Comment: 24 page
Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
In flowchart languages, predicates play an interesting double role. In the
textual representation, they are often presented as conditions, i.e.,
expressions which are easily combined with other conditions (often via Boolean
combinators) to form new conditions, though they only play a supporting role in
aiding branching statements choose a branch to follow. On the other hand, in
the graphical representation they are typically presented as decisions,
intrinsically capable of directing control flow yet mostly oblivious to Boolean
combination. While categorical treatments of flowchart languages are abundant,
none of them provide a treatment of this dual nature of predicates. In the
present paper, we argue that extensive restriction categories are precisely
categories that capture such a condition/decision duality, by means of
morphisms which, coincidentally, are also called decisions. Further, we show
that having these categorical decisions amounts to having an internal logic:
Analogous to how subobjects of an object in a topos form a Heyting algebra, we
show that decisions on an object in an extensive restriction category form a De
Morgan quasilattice, the algebraic structure associated with the (three-valued)
weak Kleene logic . Full classical propositional logic can be
recovered by restricting to total decisions, yielding extensive categories in
the usual sense, and confirming (from a different direction) a result from
effectus theory that predicates on objects in extensive categories form Boolean
algebras. As an application, since (categorical) decisions are partial
isomorphisms, this approach provides naturally reversible models of classical
propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
Quantum Turing automata
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
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