Two polygraphic presentations of Petri nets

This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is totally equivalent to Petri nets, but lacks any graphical intuition. The second one considers places as 1-dimensional cells and transitions as 2-dimensional ones: this translation recovers a graphical meaning but raises many difficulties since it uses explicit permutations. Finally, the third translation sees places as degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is a setting equivalent to Petri nets, equipped with a graphical interpretation.Comment: 28 pages, 24 figure

Polygraphs for termination of left-linear term rewriting systems

We present a methodology for proving termination of left-linear term rewriting systems (TRSs) by using Albert Burroni's polygraphs, a kind of rewriting systems on algebraic circuits. We translate the considered TRS into a polygraph of minimal size whose termination is proven with a polygraphic interpretation, then we get back the property on the TRS. We recall Yves Lafont's general translation of TRSs into polygraphs and known links between their termination properties. We give several conditions on the original TRS, including being a first-order functional program, that ensure that we can reduce the size of the polygraphic translation. We also prove sufficient conditions on the polygraphic interpretations of a minimal translation to imply termination of the original TRS. Examples are given to compare this method with usual polynomial interpretations.Comment: 15 page

Computing Critical Pairs in 2-Dimensional Rewriting Systems

International audienceRewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories, which can be seen as a generalization of term rewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2-categories

Higher-dimensional categories with finite derivation type

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs

Cofibrant complexes are free

We define a notion of cofibration among n-categories and show that the cofibrant objects are exactly the free ones, that is those generated by polygraphs.Comment: 16 page

The three dimensions of proofs

In this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting.Comment: 38 pages, 50 figure

Towards 3-Dimensional Rewriting Theory

String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs

Investigation into the acupuncture and meridian system

The Meridian system conceived by the ancient Chinese has been described and referenced for more than a thousand years. The Meridians meaning paths are the main trunks that run longitudinally within the body. The Meridian system consists of about 400 acupuncture nodes and 20 Meridian channels connecting most of these points. It deals with the routing and distribution of certain signals that may affect physiological functions. It integrates meridians, tissues and organs into a complex system. Initially, modeling of the acupuncture system is investigated. The physical effect of injecting an acupuncture needle at a node is suggested by an equivalent model of a current (voltage) source based on a simple Faraday disk generator concept. The motion of the needle due to hand manipulation in the presence of Earth\u27s magnetic fields acts as a Faraday\u27s dynamo and causes accumulation of positive (negative) charges at the tip of the needle. Due to clockwise (counter clockwise) rotation, further increase of accumulated charges at the tip results in their release in the form of an equivalent current (voltage) source. This effect has been enhanced by connecting a variable frequency source on a needle inserted into one of the nodes of the meridian system. Voltage variations at the adjacent nodes along the same meridian are measured and the relative connectivity has been observed to verify the concept of a network. It is observed that the induced voltages are proportional to the corresponding path lengths, and further more, the existence paths are found to be frequency dependent. An equivalent transmission line model is suggested. The presence of minute electrical currents also suggests that there is magnetic field along the meridian and therefore the inclusion of series inductance is appropriate. This has already been confirmed by SQUID measurements carried out and reported by [Lo 2003]. The presence of the inductive (resistive) path suggests that capacitive effects due to accompanying electric fields have to be included as shunt capacitance in the equivalent model. It shows that distributed resistance and inductance plus the shunt capacitance perfectly simulate the equivalent transmission line that is essential for signal propagation along the meridians of the acupuncture system. Measurements carried out indicate the presence of lossy resistive paths along the meridian consisting of three nodes. This has been carried out in an acupuncture clinic and two human subjects are subjected to testing on three different occasions. Sinusoidal signals in the frequency range between 20 to 80 Hz are used with different amplitudes, and strengths of propagated signals are measured to verify the existence of the electrical transmission path along that meridian. Additional hypothesis is made suggesting that the cluster water wire can be used to model the pathways of the acupuncture system. One of the reasons for this approach is that cluster water wires are ideal to model tiny nano-size capillaries. They may be present but their presences have not been verified yet physically, even through the SQUID measurements confirm the flow of minute currents along the acupuncture meridians. Petri net formulation has been developed as an attractive alternative to model bionetwork consisting of acupuncture nodes and meridians. However, validating this assumption requires an extensive measurement to be carried out, which is beyond the currently available capabilities and resources. Future work includes much more accurate modeling of pathways and nodes on each meridian, their coupling with each other. Further frequency dependent system identification in terms of equivalent parameters and their coupling behavior in the complex network, i.e., Petri net formation is required to solve the unexplained acupuncture meridian system. The presence of 20 meridians involving more than 400 nodes suggests that the acupuncture system is ideal to model a biological network

Termination orders for 3-dimensional rewriting

This paper studies 3-polygraphs as a framework for rewriting on two-dimensional words. A translation of term rewriting systems into 3-polygraphs with explicit resource management is given, and the respective computational properties of each system are studied. Finally, a convergent 3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In order to prove these results, it is explained how to craft a class of termination orders for 3-polygraphs.Comment: 30 pages, 35 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