76 research outputs found
Synthesis and Analysis of Petri Nets from Causal Specifications
Petri nets are one of the most prominent system-level formalisms for the specification of causality in concurrent, distributed, or multi-agent systems. This formalism is abstract enough to be analyzed using theoretical tools, and at the same time, concrete enough to eliminate ambiguities that would arise at implementation level. One interesting feature of Petri nets is that they can be studied from the point of view of true concurrency, where causal scenarios are specified using partial orders, instead of approaches based on interleaving.
On the other hand, message sequence chart (MSC) languages, are a standard formalism for the specification of causality from a purely behavioral perspective. In other words, this formalism specifies a set of causal scenarios between actions of a system, without providing any implementation-level details about the system.
In this work, we establish several new connections between MSC languages and Petri nets, and show that several computational problems involving these formalisms are decidable. Our results fill some gaps in the literature that had been open for several years. To obtain our results we develop new techniques in the realm of slice automata theory, a framework introduced one decade ago in the study of the partial order behavior of bounded Petri nets. These techniques can also be applied to establish connections between Petri nets and other well studied behavioral formalisms, such as the notion of Mazurkiewicz trace languages.publishedVersio
Tropical Fourier-Motzkin elimination, with an application to real-time verification
We introduce a generalization of tropical polyhedra able to express both
strict and non-strict inequalities. Such inequalities are handled by means of a
semiring of germs (encoding infinitesimal perturbations). We develop a tropical
analogue of Fourier-Motzkin elimination from which we derive geometrical
properties of these polyhedra. In particular, we show that they coincide with
the tropically convex union of (non-necessarily closed) cells that are convex
both classically and tropically. We also prove that the redundant inequalities
produced when performing successive elimination steps can be dynamically
deleted by reduction to mean payoff game problems. As a complement, we provide
a coarser (polynomial time) deletion procedure which is enough to arrive at a
simply exponential bound for the total execution time. These algorithms are
illustrated by an application to real-time systems (reachability analysis of
timed automata).Comment: 29 pages, 8 figure
Process versus Unfolding Semantics for Place/Transition Petri Nets
In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game," one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc, a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification
On rank functions for heaps
Motivated by work of Stembridge, we study rank functions for Viennot's heaps
of pieces. We produce a simple and sufficient criterion for a heap to be a
ranked poset and apply the results to the heaps arising from fully commutative
words in Coxeter groups.Comment: 18 pages AMSTeX, 3 figure
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi
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