2,231 research outputs found
Monoids with tests and the algebra of possibly non-halting programs
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou
Bounded and unitary elements in pro-C^*-algebras
A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category of
topological *-algebras. From the perspective of non-commutative geometry,
pro-C^*-algebras can be seen as non-commutative k-spaces. An element of a
pro-C^*-algebra is bounded if there is a uniform bound for the norm of its
images under any continuous *-homomorphism into a C^*-algebra. The *-subalgebra
consisting of the bounded elements turns out to be a C^*-algebra. In this
paper, we investigate pro-C^*-algebras from a categorical point of view. We
study the functor (-)_b that assigns to a pro-C^*-algebra the C^*-algebra of
its bounded elements, which is the dual of the Stone-\v{C}ech-compactification.
We show that (-)_b is a coreflector, and it preserves exact sequences. A
generalization of the Gelfand-duality for commutative unital pro-C^*-algebras
is also presented.Comment: v2 (accepted
Algebraic Properties of Valued Constraint Satisfaction Problem
The paper presents an algebraic framework for optimization problems
expressible as Valued Constraint Satisfaction Problems. Our results generalize
the algebraic framework for the decision version (CSPs) provided by Bulatov et
al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties
and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP
languages to weighted algebras. We show that the difficulty of VCSP depends
only on the weighted variety generated by the associated weighted algebra.
Paralleling the results for CSPs we exhibit a reduction to cores and rigid
cores which allows us to focus on idempotent weighted varieties. Further, we
propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the
hardness direction and verify that it agrees with known results for VCSPs on
two-element sets [Cohen et al. 2006], finite-valued VCSPs [Thapper and Zivny
2013] and conservative VCSPs [Kolmogorov and Zivny 2013].Comment: arXiv admin note: text overlap with arXiv:1207.6692 by other author
Towards Autopoietic Computing
A key challenge in modern computing is to develop systems that address
complex, dynamic problems in a scalable and efficient way, because the
increasing complexity of software makes designing and maintaining efficient and
flexible systems increasingly difficult. Biological systems are thought to
possess robust, scalable processing paradigms that can automatically manage
complex, dynamic problem spaces, possessing several properties that may be
useful in computer systems. The biological properties of self-organisation,
self-replication, self-management, and scalability are addressed in an
interesting way by autopoiesis, a descriptive theory of the cell founded on the
concept of a system's circular organisation to define its boundary with its
environment. In this paper, therefore, we review the main concepts of
autopoiesis and then discuss how they could be related to fundamental concepts
and theories of computation. The paper is conceptual in nature and the emphasis
is on the review of other people's work in this area as part of a longer-term
strategy to develop a formal theory of autopoietic computing.Comment: 10 Pages, 3 figure
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
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