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N. Sabadini

Publications and source records attributed to N. Sabadini.

13 recordsLinked to original sources
arXiv2013

Blocked-braid Groups

We introduce and study a family of groups $\mathbf{BB}_n$, called the blocked-braid groups, which are quotients of Artin's braid groups $\mathbf{B}_n$, and have the corresponding symmetric groups $Σ_n$ as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of commutative Frobenius algebras and tangle algebras in braided strict monoidal categories. A fundamental equation true in $\mathbf{BB}_n$ is Dirac's Belt Trick; that torsion through $4π$ is equal to the identity. We show that $\mathbf{BB}_n$ is finite for $n=1,2$ and 3 but infinite for $n>3$.

math.CT↗
arXiv2013

Racks and blocked braids

In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or 4. We conjectured that the order was actually 4 but our methods in that paper, which involved introducing for any group G a braided monoidal category of tangled relations, were inadequate to demonstrate this fact. Subsequently Davide Maglia in unpublished work investigated exactly what part of the structure and properties of a group G are needed to permit the construction of a braided monoidal category with a tangle algebra and was able to distinguish blocked two-torsions from the identity. In this paper we present a simplification of his answer, which turns out to be related to the notion of rack. We show that if G is a rack then there is a braided monoidal category TRel_G generalizing that of the above paper. Further we introduce a variation of the notion of rack which we call irack which yields a tangle algebra in TRel_G. Iracks are in particular racks but have in addition to the operations abstracting group conjugation also a unary operation abstracting group inverse. Using iracks we obtain new invariants for tangles and blocked braids permitting us to present a proof of Maglia's result that a blocked double torsion is not the identity. This work was presented at the Conference in Memory of Aurelio Carboni, Milan, 24-26 June 2013.

math.CT↗
arXiv2011

Tangled Circuits

The theme of the paper is the use of commutative Frobenius algebras in braided strict monoidal categories in the study of varieties of circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras.

math.CT↗
arXiv2010

The compositional construction of Markov processes II

In an earlier paper we introduced a notion of Markov automaton, together with parallel operations which permit the compositional description of Markov processes. We illustrated by showing how to describe a system of n dining philosophers, and we observed that Perron-Frobenius theory yields a proof that the probability of reaching deadlock tends to one as the number of steps goes to infinity. In this paper we add sequential operations to the algebra (and the necessary structure to support them). The extra operations permit the description of hierarchical systems, and ones with evolving geometry.

math.CT↗
arXiv2009

The parallel composition of processes

We suggest that the canonical parallel operation of processes is composition in a well-supported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations arising from monoid objects in such a category, representing the fact that they are protocols based on an underlying broadcast communication.

math.CT↗
arXiv2009

A process algebra for the Span(Graph) model of concurrency

In this note we define a process algebra TCP (Truly Concurrent Processes) which corresponds closely with the automata model of concurrency based on Span(RGraph), the category of spans of reflexive graphs. In TCP, each process has a fixed set of interfaces. Actions are allowed to occur simultaneously on all the interfaces of a process. Asynchrony is modelled by the use of silent actions. Communication is anonymous: communication between two processes P and Q is described by an operation which connects some of the ports of P to some of the ports of Q; and a process can only communicate with other processes via its interfaces. The model is naturally equipped with a compositional semantics in terms of the operations in Span(RGraph) introduced in [5], and developed in [6, 7, 10].

math.CT↗
arXiv2009

The compositional construction of Markov processes

We describe an algebra for composing automata in which the actions have probabilities. We illustrate by showing how to calculate the probability of reaching deadlock in k steps in a model of the classical Dining Philosopher problem, and show, using the Perron-Frobenius Theorem, that this probability tends to 1 as k tends to infinity.

cs.LO↗
arXiv2008

On partita doppia

We give a precise mathematical account of partita doppia in terms of an algebraic structure on the bicategory of spans of reflexive graphs. (The paper was written in 1998.)

math.CT↗
arXiv2007

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.

math.CT↗
arXiv2000

On Automata with Boundary

We present a theory of automata with boundary for designing, modelling and analysing distributed systems. Notions of behaviour, design and simulation appropriate to the theory are defined. The problem of model checking for deadlock detection is discussed, and an algorithm for state space reduction in exhaustive search, based on the theory presented here, is described. Three examples of the application of the theory are given, one in the course of the development of the ideas and two as illustrative examples of the use of the theory.

cs.DC↗