Reflexive Composition of Elementary State Machines, with an Application to the Reversal of Cellular Automata Rule 90

TL;DR

The paper investigates a lattice of two-state, two-symbol finite-state machines updated by reflexive composition, unveiling a rich repertoire of dynamics that include CA-like evolution and time-reversed trajectories. By systematically enumerating 256 elementary machines, it shows that state-reporting machines align with CA rules such as $Rule\ 90$ under suitable boundary conditions, while certain mirror machines realize the inverse dynamics and, via a state–message transposition, reveal a deep symmetry between processing and carrying information. Key findings include M45's CA-like mapping and its boundary treatment, M44's reversible, distance-spanning dynamics, and the discovery that M54/M60 realize $\text{Rule }90$ in reverse, with the time direction emerging from how states and messages are treated as operators and operands. These results suggest a fundamental time-symmetry in the system's formulation and open avenues for exploring preimage dynamics and time reversal in discrete, information-processing models. The work highlights how a simple, symmetric formalism can yield complex, fractal-like trajectories and deeper insights into the relationship between state and message in computation.

Abstract

We explore the dynamics of a one-dimensional lattice of state machines on two states and two symbols sequentially updated via a process of "reflexive composition." The space of 256 machines exhibits a variety of behavior, including substitution, reversible "billiard ball" dynamics, and fractal nesting. We show that one machine generates the Sierpinski Triangle and, for a subset of boundary conditions, is isomorphic to cellular automata Rule 90 in Wolfram's naming scheme. More surprisingly, two other machines follow trajectories that map to Rule 90 in reverse. Whereas previous techniques have been developed to uncover preimages of Rule 90, this is the first study to produce such inverse dynamics naturally from the formalism itself. We argue that the system's symmetric treatment of state and message underlies its expressive power.

Figures (17)

• Figure 1: Three steps of FSM with $Q = \{\textit{S0}, \textit{S1}\}$ and $\mathscr{A} = \{0, 1\}$. State-transition function $\delta$ and output function $\varphi$ are defined in \ref{['M45']}(a). Dark nodes and bold edges indicate the current state and transition, respectively, in each frame, with input and output streams shown below.
• Figure 2: FSM with identifier M61. States $\textit{S0}$ and $\textit{S1}$ and message values $\textit{0}$ and $\textit{1}$ are each mapped to the bits 0 and 1, respectively, resulting in the bit string ($\texttt{00111101})_2 = 61$.
• Figure 3: Four steps (vertical) of state composition (horizontal) applied to a two cell lattice of M45 for input sequence $(\textit{0},\textit{1},\textit{0},\textit{1})$. Directionality is alternated at each step. Dark nodes and bold edges indicate the current state and transition, respectively. Dashed lines show the composition of inputs to outputs at each step; row outputs are discarded.
• Figure 4: (a) Top: Two-state, two-symbol state machine identified as M45. Bottom: table of state machine transitions and derivation of naming ($(\texttt{00101101})_2 = 45$). Bottom: table of state machine transitions. (b) Four steps of the state composition trajectory of a five-cell M45 lattice with null-input boundary conditions and initial states $(\textit{0}, \textit{1}, \textit{0}, \textit{0}, \textit{1})$.
• Figure 5: Trajectories of the full collection of 256 2-state, 2-symbol machines from initial centered pixel on 19-cell lattice with input-0 boundary conditions. For visibility only even rows are shown. Black = S0, White = S1.