Multiway Systems: Rewriting & Computation

• Multiway systems are abstract rewriting models that simultaneously apply all possible rules to produce branching evolution graphs encoding complete computation histories.
• Their structure is formalized through abstract rewriting systems and coalgebraic methods, enabling categorization into growth classes and modeling of higher homotopies.
• They underpin advances in theoretical physics, network protocols, concurrency, and quantum information by modeling causal structures and constructing quantum circuits.

A multiway system is an abstract rewriting formalism in which, starting from an initial state, every possible applicable rule is simultaneously applied at each step, producing a branching—often highly nondeterministic—evolution graph. Multiway systems serve as foundational structures in discrete mathematics, theoretical physics (notably in the Wolfram model), concurrency theory, quantum information, and network protocols. They encode the entire space of possible computation histories, causal relationships, and, under categorical enrichment, the hierarchy of homotopical structures and connections to geometric models.

1. Foundations: Abstract Structure and Rulial Space

The core structure of a multiway system is formalized as an abstract rewriting system (ARS), defined as a pair $(A,\to)$ where $A$ is a set of objects and $\to \subseteq A \times A$ is a binary rewrite relation. The multiway evolution graph $G_m$ is the directed acyclic graph whose vertices are $A$ and whose edges represent individual rewrite steps, $(A \to B) \Leftrightarrow (A,B) \in \to$. Finite paths in $G_m$ correspond to rewrite sequences $A \to A' \to \cdots \to B$ under $\to$ (Arsiwalla et al., 2021).

Rule applications are labeled by indices $\alpha \in \Lambda$, producing a family of relations and encoding the system's coalgebraic description: $$p \mapsto \{(\alpha, q) \in \Lambda \times A : p \to^\alpha q\}$$, making $(A,\to)$ an $F$-coalgebra for $F(X) = P(\Lambda \times X)$. The rulial space is the category of cospans $(L \leftarrow K \rightarrow R)$ of monomorphisms in an adhesive (or selectively adhesive) category $C$, representing the abstract space of all possible rewriting rules of fixed signature and carrying a monoidal structure from concurrency and parallelism theorems (Arsiwalla et al., 2021).

2. Homotopical Structure and $n$-Fold Categories

Multiway systems admit a hierarchy of higher homotopies via inclusions of additional bidirectional rewrite rules. Distinct paths $\pi_1, \pi_2$ from $A$ to $B$ in $G_m$ can be identified by introducing homotopy rule sets $H_1$, consisting of pairs $u_i \leftrightarrow v_i$ matching intermediate states along $\pi_1$ and $\pi_2$. Generalizing, $k$-th order homotopies are sets of bidirectional rules mapping $(k-1)$-cells to each other, yielding invertible $k$-cells between $(k-1)$-cells.

This structure is formalized categorically: the base category $M_0$ consists of objects (rewrite terms) and morphisms (1-paths). Introducing homotopy rules yields double categories ($n=2$ case), with higher $n$ yielding $n$-fold categories

$$M_{n-1} \Rightarrow\!\!\Rightarrow M_{n-2} \Rightarrow\!\!\Rightarrow \cdots \Rightarrow\!\!\Rightarrow M_1 \Rightarrow\!\!\Rightarrow M_0$$

with objects, 1-morphisms (paths), 2-morphisms (homotopies between paths), $\dots$, and $n$-morphisms (commutative $n$-hypercubes of rewrite steps). If all homotopy rules up to order $n$ are invertible, the structure is an $n$-fold groupoid; as $n \to \infty$, an $\infty$-groupoid emerges (Arsiwalla et al., 2021).

3. Growth Rates, Classes, and Computational Diversity

The state-space explosion in multiway systems is quantified by the growth function $\gamma_M(n)$, which counts the number of distinct states appearing exactly at step $n$. Asymptotic behavior is classified by tight upper and lower bounds, organizing multiway systems into "growth classes"—polynomial ($C_{pol}$), exponential ($C_{exp}$), intermediate ($C_{int}$), inverse polynomial/exp/int, and bounded/finite systems. String-based multiway systems can achieve arbitrarily slow growth (slower than any computable function's inverse) but never exceed exponential growth (Zeschke, 2021).

Multiway growth functions are computationally diverse: by combining atomic systems under algebraic sum and convolution, one constructs systems approximating $$x \to const, x^d, a^x, a^{\sqrt{x}}, \sqrt{x}, \ln x$$ and their combinations. This diversity enables the engineering of systems for prescribed growth rates within the closure of these elementary shapes (Zeschke, 2021).

4. Multiway Systems in Concurrency, Quantum Information, and Physics

Multiway systems generalize to frameworks in concurrency and quantum information. In categorical quantum information theory, multiway systems are implemented as double-pushout (DPO) rewriting in adhesive categories (Gorard et al., 2020). Branchial graphs—cross-sections of the multiway evolution graph—encode superpositional connectivity, corresponding in the quantum context to projective Hilbert space structure. Multiway rewriting with rulial composition induces a symmetric monoidal category, unifying the stacking of ZX-diagrams with parallel composition of rule sets (Gorard et al., 2020).

In the Wolfram model, multiway systems underlie causal networks and the emergence of spacetime geometry, with the $\infty$-groupoid correspondence enabling the mapping of computational evolution to manifold-like and gauge-theoretic structures. This framework supports constructivist approaches to quantum gravity and field theory, grounded purely in rule-based combinatorial computation (Arsiwalla et al., 2021).

5. Multiway Turing Machines, Integer Multiway Systems, and Universality

Minimal multiway Turing machines—defined by $(s, k, p)$: number of head states, tape colors, and rules—demonstrate the onset of computational irreducibility and universality at thresholds much lower than deterministic Turing machines (e.g., $s=1$, $k=2$, $p=3$). These machines generate complex, exponentially growing multiway graphs with causal invariance: all possible histories yield isomorphic causal event structures, paralleling Lorentz invariance (Wolfram, 2021).

Integer-based multiway systems, governed by affine rules $n \mapsto \{a n + b, c n + d\}$, exhibit emergent geometric and number-theoretic phenomena: tubes, grids, spirals, confluence, and fundamental undecidability. System behavior is classified by merges (confluence), Diophantine constraints, and arithmetic densities. These models serve as laboratories for exploring causality and emergent geometry in discrete physical models (Wolfram, 2021).

6. Multiway Systems in Network Protocols and Communication

Outside pure combinatorics, multiway systems arise as archetypal models in multi-way relay channels (MWRCs), cooperative communication protocols where $N$ users exchange messages through a relay. Achievable rate regions and coding architectures for the finite field MWRC are determined by constraints on the joint source distribution (almost balanced conditional mutual information, or ABCMI), enabling separated source-channel coding to achieve optimality in the presence of near-symmetry (Ong et al., 2012). Generalizations to multi-way buffer-aided relay selection (MW-Max-Link) in multi-antenna systems exploit multiway relay choices for diversity and rate improvements, employing selection metrics and buffer structures (Duarte et al., 2019).

7. Quantum Computational Models from Multiway Rewriting

Multiway rewriting systems—specifically string substitutions over cyclic character strings (Leibnizian strings)—yield explicit matrix representations of finite-dimensional quantum operators and circuits (Dündar et al., 23 Dec 2025). The S-matrix defined by the set of all paths between strings encodes quantum gate unitaries, including CNOT, Hadamard, and $\pi/8$ gates. These models abstract $N$-fermion systems and leverage discrete path integral constructions; Fermi-Dirac statistics for occupation numbers further suggest connections to quantum statistical mechanics.

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Multiway systems unify nondeterministic computation, concurrency modeling, quantum mechanical amplitude assignment, network communication, and categorical/topological formalisms under a common evolution-graph paradigm. Their mathematical richness has enabled advances in the categorical semantics of rewriting, physical modeling of fundamental processes, and the explicit construction of quantum circuits from purely combinatorial substrates. The emergence of higher homotopies, $\infty$-groupoid structures, and monoidal categories situates multiway systems as foundational objects at the intersection of computation, topology, algebra, and theoretical physics (Arsiwalla et al., 2021, Zeschke, 2021, Gorard et al., 2020, Dündar et al., 23 Dec 2025, Wolfram, 2021, Wolfram, 2021, Ong et al., 2012, Duarte et al., 2019).