Valental aspects of Peircean algebraic logic

AbstractThis paper describes a system of logic that has both an algebraic syntax and a graphical syntax and that may be regarded as a kind of semantic net. The paper analyses certain “valental” characteristics of terms and graphs of this logical system: characteristics defined in terms of the “valence”, or number of argument-places, in the terms and graphs. The most famous valental result is the so-called “Reduction Thesis” of Charles Sanders Peirce. The paper briefly explicates the author's proof, which is to be published in a forthcoming book, of this Reduction Thesis. Several additional valental results are proved, and the potential of PAL as a bridge between logic and topological graph theory is suggested

Components and acyclicity of graphs. An exercise in combining precision with concision

Central to algorithmic graph theory are the concepts of acyclicity and strongly connected components of a graph, and the related search algorithms. This article is about combining mathematical precision and concision in the presentation of these concepts. Concise formulations are given for, for example, the reflexive-transitive reduction of an acyclic graph, reachability properties of acyclic graphs and their relation to the fundamental concept of “definiteness”, and the decomposition of paths in a graph via the identification of its strongly connected components and a pathwise homomorphic acyclic subgraph. The relevant properties are established by precise algebraic calculation. The combination of concision and precision is achieved by the use of point-free relation algebra capturing the algebraic properties of paths in graphs, as opposed to the use of pointwise reasoning about paths between nodes in graphs

Network Rewriting I: The Foundation

A theory is developed which uses "networks" (directed acyclic graphs with some extra structure) as a formalism for expressions in multilinear algebra. It is shown that this formalism is valid for arbitrary PROPs (short for 'PROducts and Permutations category'), and conversely that the PROP axioms are implicit in the concept of evaluating a network. Ordinary terms and operads constitute the special case that the graph underlying the network is a rooted tree. Furthermore a rewriting theory for networks is developed. Included in this is a subexpression concept for which is given both algebraic and effective graph-theoretical characterisations, a construction of reduction maps from rewriting systems, and an analysis of the obstructions to confluence that can occur. Several Diamond Lemmas for this rewriting theory are given. In addition there is much supporting material on various related subjects. In particular there is a "toolbox" for the construction of custom orders on the free PROP, so that an order can be tailored to suit a specific rewriting system. Other subjects treated are the abstract index notation in a general PROP context and the use of feedbacks (sometimes called traces) in PROPs.Comment: 188 pages, numerous inlined figure

The complexity of approximating the complex-valued Potts model

We study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$, which corresponds to the case of the Ising model; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations.Comment: 58 pages. Changes on version 2: minor change