[Paper Review] Network Rewriting I: The Foundation

Lars Hellström|arXiv (Cornell University)|Apr 11, 2012

Logic, programming, and type systems7 references3 citations

TL;DR

This paper introduces a network rewriting framework based on directed acyclic graphs (networks) to formalize expressions in multilinear algebra, generalizing traditional tree-based terms to handle non-commutative and tensorial structures. It establishes that PROP axioms are inherently encoded in network evaluation, develops a rewriting theory with algebraic and graph-theoretic characterizations of subexpressions, and proves Diamond Lemmas while identifying novel obstructions to confluence such as 'wrap ambiguity'.

ABSTRACT

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.

Motivation & Objective

• To develop a formalism for multilinear algebra expressions that generalizes tree-based terms to handle non-commutative and tensorial structures.
• To show that the axioms of a PROP are implicitly encoded in the evaluation of networks.
• To construct a rewriting theory for networks with algebraic and graph-theoretic characterizations of subexpressions.
• To analyze obstructions to confluence, including a novel class of ambiguities termed 'wrap ambiguity'.
• To provide tools for custom ordering of free PROPs to support tailored rewriting systems.

Proposed method

• Uses directed acyclic graphs with additional structure (networks) as formal expressions in multilinear algebra.
• Defines subexpressions via both algebraic conditions and graph-theoretic properties such as reachability and connected components.
• Constructs reduction maps from rewriting systems using transference types to track structural dependencies.
• Introduces a toolbox for defining custom orders on free PROPs to suit specific rewriting systems.
• Applies abstract index notation and feedback (trace) operations within the general PROP framework.
• Proves multiple Diamond Lemmas to ensure confluence under certain conditions, particularly for well-behaved rewriting systems.

Experimental results

Research questions

• RQ1 How can multilinear algebra expressions be formalized beyond rooted tree structures to accommodate tensor products and non-commutative operations?
• RQ2 What is the precise relationship between the axioms of a PROP and the evaluation of network-based expressions?
• RQ3 How can a rewriting system for networks be constructed such that subexpressions are both algebraically and graph-theoretically well-defined?
• RQ4 What novel types of non-confluence obstructions arise in network rewriting, and how do they differ from classical overlap and inclusion ambiguities?
• RQ5 Can custom orders on free PROPs be systematically constructed to support termination and confluence in rewriting systems?

Key findings

• The formalism of networks captures arbitrary PROP structures, and the PROP axioms are implicitly embedded in the network evaluation process.
• The theory generalizes traditional term rewriting by allowing non-tree graphs, enabling representation of structures such as quantum circuits and Hopf algebras.
• A new class of ambiguity—'wrap ambiguity'—is identified, arising when rules wrap around each other without overlapping, leading to irreducible but non-equivalent reductions.
• The subexpression concept is characterized both algebraically (via homomorphisms) and graph-theoretically (via reachability and component structure).
• Reduction maps are constructed from rewriting systems using transference types, enabling precise tracking of structural dependencies during rewriting.
• Several Diamond Lemmas are proven, providing sufficient conditions for confluence in the network rewriting framework.