47 research outputs found
String Diagrammatic Trace Theory
We extend the theory of formal languages in monoidal categories to the
multi-sorted, symmetric case, and show how this theory permits a graphical
treatment of topics in concurrency. In particular, we show that Mazurkiewicz
trace languages are precisely symmetric monoidal languages over monoidal
distributed alphabets. We introduce symmetric monoidal automata, which define
the class of regular symmetric monoidal languages. Furthermore, we prove that
Zielonka's asynchronous automata coincide with symmetric monoidal automata over
monoidal distributed alphabets. Finally, we apply the string diagrams for
symmetric premonoidal categories to derive serializations of traces.Comment: Paper accepted for MFCS 202
String Diagrammatic Electrical Circuit Theory
We develop a comprehensive string diagrammatic treatment of electrical
circuits. Building on previous, limited case studies, we introduce controlled
sources and meters as elements, and the impedance calculus, a powerful toolbox
for diagrammatic reasoning on circuit diagrams. We demonstrate the power of our
approach by giving comprehensive proofs of several textbook results, including
the superposition theorem and Th\'evenin's theorem.Comment: 13 pages + appendices. Accepted for ACT202
Resumptions, Weak Bisimilarity and Big-Step Semantics for While with Interactive I/O: An Exercise in Mixed Induction-Coinduction
We look at the operational semantics of languages with interactive I/O
through the glasses of constructive type theory. Following on from our earlier
work on coinductive trace-based semantics for While, we define several big-step
semantics for While with interactive I/O, based on resumptions and
termination-sensitive weak bisimilarity. These require nesting inductive
definitions in coinductive definitions, which is interesting both
mathematically and from the point-of-view of implementation in a proof
assistant.
After first defining a basic semantics of statements in terms of resumptions
with explicit internal actions (delays), we introduce a semantics in terms of
delay-free resumptions that essentially removes finite sequences of delays on
the fly from those resumptions that are responsive. Finally, we also look at a
semantics in terms of delay-free resumptions supplemented with a silent
divergence option. This semantics hinges on decisions between convergence and
divergence and is only equivalent to the basic one classically.
We have fully formalized our development in Coq.Comment: In Proceedings SOS 2010, arXiv:1008.190
Monoidal Width
We introduce monoidal width as a measure of complexity for morphisms in
monoidal categories. Inspired by well-known structural width measures for
graphs, like tree width and rank width, monoidal width is based on a notion of
syntactic decomposition: a monoidal decomposition of a morphism is an
expression in the language of monoidal categories, where operations are
monoidal products and compositions, that specifies this morphism. Monoidal
width penalises the composition operation along ``big'' objects, while it
encourages the use of monoidal products. We show that, by choosing the correct
categorical algebra for decomposing graphs, we can capture tree width and rank
width. For matrices, monoidal width is related to the rank. These examples
suggest monoidal width as a good measure for structural complexity of processes
modelled as morphisms in monoidal categories.Comment: Extended version of arXiv:2202.07582 and arXiv:2205.0891
Equational Characterization of Covariant-Contravariant Simulation and Conformance Simulation Semantics
Covariant-contravariant simulation and conformance simulation generalize
plain simulation and try to capture the fact that it is not always the case
that "the larger the number of behaviors, the better". We have previously
studied their logical characterizations and in this paper we present the
axiomatizations of the preorders defined by the new simulation relations and
their induced equivalences. The interest of our results lies in the fact that
the axiomatizations help us to know the new simulations better, understanding
in particular the role of the contravariant characteristics and their interplay
with the covariant ones; moreover, the axiomatizations provide us with a
powerful tool to (algebraically) prove results of the corresponding semantics.
But we also consider our results interesting from a metatheoretical point of
view: the fact that the covariant-contravariant simulation equivalence is
indeed ground axiomatizable when there is no action that exhibits both a
covariant and a contravariant behaviour, but becomes non-axiomatizable whenever
we have together actions of that kind and either covariant or contravariant
actions, offers us a new subtle example of the narrow border separating
axiomatizable and non-axiomatizable semantics. We expect that by studying these
examples we will be able to develop a general theory separating axiomatizable
and non-axiomatizable semantics.Comment: In Proceedings SOS 2010, arXiv:1008.190
String Diagram Rewrite Theory III: Confluence with and without Frobenius
In this paper we address the problem of proving confluence for string diagram
rewriting, which was previously shown to be characterised combinatorically as
double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For
standard DPO rewriting without interfaces, confluence for terminating rewrite
systems is, in general, undecidable. Nevertheless, we show here that confluence
for DPOI, and hence string diagram rewriting, is decidable. We apply this
result to give effective procedures for deciding local confluence of symmetric
monoidal theories with and without Frobenius structure by critical pair
analysis. For the latter, we introduce the new notion of path joinability for
critical pairs, which enables finitely many joins of a critical pair to be
lifted to an arbitrary context in spite of the strong non-local constraints
placed on rewriting in a generic symmetric monoidal theory
Span(Graph): a Canonical Feedback Algebra of Open Transition Systems
We show that Span(Graph)*, an algebra for open transition systems introduced
by Katis, Sabadini and Walters, satisfies a universal property. By itself, this
is a justification of the canonicity of this model of concurrency. However, the
universal property is itself of interest, being a formal demonstration of the
relationship between feedback and state. Indeed, feedback categories, also
originally proposed by Katis, Sabadini and Walters, are a weakening of traced
monoidal categories, with various applications in computer science. A state
bootstrapping technique, which has appeared in several different contexts,
yields free such categories. We show that Span(Graph)* arises in this way,
being the free feedback category over Span(Set). Given that the latter can be
seen as an algebra of predicates, the algebra of open transition systems thus
arises - roughly speaking - as the result of bootstrapping state to that
algebra. Finally, we generalize feedback categories endowing state spaces with
extra structure: this extends the framework from mere transition systems to
automata with initial and final states.Comment: 48 pages, 33 figures, journal versio
Interacting Frobenius Algebras are Hopf
Theories featuring the interaction between a Frobenius algebra and a Hopf
algebra have recently appeared in several areas in computer science: concurrent
programming, control theory, and quantum computing, among others. Bonchi,
Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive
law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the
opposite approach, and show that interacting Frobenius algebras form Hopf
algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the
underlying object---the so-called phase group---and investigate the effects of
finite dimensionality of the underlying model. We recover the system of Bonchi
et al as a subtheory in the prime power dimensional case, but the more general
theory does not arise from a distributive law.Comment: 32 pages; submitte
A non-interleaving process calculus for multi-party synchronisation
We introduce the wire calculus. Its dynamic features are inspired by Milner's
CCS: a unary prefix operation, binary choice and a standard recursion
construct. Instead of an interleaving parallel composition operator there are
operators for synchronisation along a common boundary and non-communicating
parallel composition. The (operational) semantics is a labelled transition
system obtained with SOS rules. Bisimilarity is a congruence with respect to
the operators of the language. Quotienting terms by bisimilarity results in a
compact closed category
General Reversibility
The first and the second author introduced reversible ccs (rccs) in order to model concurrent computations where certain actions are allowed to be reversed. Here we show that the core of the construction can be analysed at an abstract level, yielding a theorem of pure category theory which underlies the previous results. This opens the way to several new examples; in particular we demonstrate an application to Petri nets.