Markovian Moore Machine

• A Markovian Moore machine is a formalism that fuses deterministic Moore automata with stochastic Markov dynamics and thermodynamic principles to compute resource usage and error rates.
• It models state transitions as time-inhomogeneous Markov chains with dynamic states encoded by energy levels, integrating explicit thermodynamic constraints.
• The approach yields rigorous analytic expressions for statistical properties, ergodic limits, output measures, and coding map characteristics in both deterministic and probabilistic settings.

A Markovian Moore machine is a formalism uniting Moore automata theory with Markovian dynamics and coding of Markov measures. The model appears in two principal guises: (1) as a deterministic Moore machine presented as a time-inhomogeneous Markov chain with explicit thermodynamic realizability, and (2) as an automaton acting as a coding map (block code) on bi-infinite Markovian sequences, inducing push-forward measures whose properties can be precisely characterized. Both perspectives yield rigorous analytic expressions for resource usage, error rates, statistical properties, and ergodic limits.

1. Formal Definition and Structure

A Moore machine is defined as a 6-tuple $(Q, \Sigma, \Delta, \delta, \lambda, q_0)$, where $Q$ is a finite set of internal (label) states, $\Sigma$ is a finite input alphabet, $\Delta$ is a finite output alphabet, $\delta:Q\times \Sigma\to Q$ is a deterministic transition function, $\lambda:Q\to\Delta$ is the output function, and $q_0\in Q$ is the initial state. The output at each step depends only on the current state, not the current input.

In the Markovian incarnation, each Moore machine state is realized as a node of a Markov chain, extending the configuration space with “dynamic” states to encode transitions. Explicitly, for every state and input, there is a unique dynamic state $D^{i\rightarrow \delta(i,a)}_a$. This leads to $M$ label states and $Mn$ dynamic states for $M=|Q|$ and $n=|\Sigma|$, forming a time-inhomogeneous, input-driven Markov chain (Chu et al., 2018).

2. Markovian Coding Maps and Measures

Given an input Markov measure $\mu$ on $A^{\mathbb N}$ (with $A$ a finite alphabet), a Moore coding map acts as a block code: $$\tau_{M'}(x_0x_1x_2\cdots)\mapsto \omega(q_0)\omega(q_1)\omega(q_2)\cdots$$ with $q_0$ the initial state and $q_{n+1}=\delta(q_n, x_n)$. For each infinite input, the output stream is determined entirely by the deterministic Moore dynamics (Grigorchuk et al., 2021).

If the coding map is applied to inputs distributed according to a stationary, irreducible Markov chain, then the push-forward (output) measure $\nu=\tau_{M'*}\mu$ can either be absolutely continuous, mutually singular, or (in exceptional cases) preserved relative to the input measure, depending on the structure and activity of the automaton.

3. Thermodynamically Consistent Moore Machines

The physical realization described in (Chu et al., 2018) models the Moore machine as a network of energy levels. Each label state $L_i$ is assigned $E=0$, while dynamic states can have $E=-\Delta E$ (active for current input) or $E=+\Delta E$ (inactive). The instantaneous transition rates $k_{ij}(t)$ respect local detailed balance via $k_{ij}(t) = u\exp[-\beta(E_j(t) - E_i(t))]$, with $u$ an attempt frequency and $\beta$ the inverse temperature.

Transitions are restricted by high energy barriers except along permissible machine pathways. Each change in tape symbol drives a reconfiguration of the energy landscape, so the system is always driven by precise, input-dependent manipulations, resulting in a time-inhomogeneous Markov process.

4. Resource Costs, Error Probabilities, and Cycle Statistics

Each update cycle—reading an input symbol, flipping state, and re-setting for the next input—consists of two atomic thermodynamic operations: an $N_Q$-it flip and an $n$-it set, with $N_Q$ the maximum dynamic-state in-degree per label. The average work and entropy production per cycle are given by

$$\langle \Delta W \rangle = 2\ln(k_+/k_-) + O\left((k_-/k_+)\ln(k_+/k_-)\right)\geq 0,$$

where $k_+ = e^{\Delta E}$ and $k_- = e^{-\Delta E}$. The error probability per cycle is approximately $p_{\text{err}}\simeq ((1+N_Q)k_-)/k_+$ in the high-barrier regime (Chu et al., 2018).

Characteristic times to equilibration for each atomic operation are derived from the master equation eigenvalues: $$\tau_{\text{flip}}^{-1} = k_+ + N_Q k_-;\quad \tau_{\text{set}}^{-1}\simeq k_+,\ \text{for } k_+\gg k_-N.$$ The time per elementary cycle, $\tau_{\text{cycle}}$, scales as $2N_M/k_+$ with $N_M$ denoting the number of relaxation multiples for high fidelity.

An explicit example with $Q=\{A,B,C\}$ and $\Sigma=\{0,1\}$ gives, for $k_+=10^3$, $k_-=1$, $N_M=10$, work per cycle $\approx14\ k_BT$, time per cycle $\approx0.02\ u^{-1}$, and error rate $\approx 0.4\%$.

5. Statistical Properties of Output Processes

The Moore coding map applied to Markov inputs induces well-defined statistical properties on the outputs:

• Ergodicity: If the original chain and the automaton are irreducible and strongly connected, the lifted process on $Q\times A$ is ergodic and mixing ($\psi$–mixing).
• Frequencies (LLN): For any output symbol $b\in B$, in the output sequence,

$$\lim_{n\to\infty}\frac{1}{n}\#\{\,0\leq k<n:(\tau(x))_k=b\} = \sum_{\substack{q\in Q,\,i\in A \ \lambda(q,i) = b}} t_{(q,i)},$$

where $t$ is the stationary vector of the lifted chain.

• Central Limit Theorem: For any observable on the output sequence depending on finitely many coded symbols, the output satisfies a CLT as $n\to\infty$ provided the pair-chain is $\psi$–mixing.

Entropy of the output measure always satisfies $h(\nu)\leq h(\mu)$, with strict drop equal to the conditional entropy $H(i_1|b_0\cdots b_{k-1})$ when coding memory is $k$ (Grigorchuk et al., 2021).

6. Absolute Continuity, Singularity, and Invariance Criteria

The image measure $\nu$ is absolutely continuous with respect to the input measure $\mu$ if the coding map has polynomial activity growth, formalized via the Radon–Nikodym derivative: $$\frac{d\nu}{d\mu}(x) = \sum_{w\in V_{\max}} \frac{\mu(\tau^{-1}([w]))}{\mu([w])}\,\mathbf{1}_{wA^{\mathbb N}}(x),$$ where $V_{\max}$ is a finite set of maximal “bad prefixes.” This criterion ensures that measure zero sets in the input remain of measure zero under push-forward (Grigorchuk et al., 2021).

For invertible, strongly connected automata on $A$, either all cylinder measures are preserved (thus $\nu = \mu$ identically), or $\nu$ and $\mu$ are mutually singular, which is revealed by a different limiting frequency for some finite block word under the action of the coding map.

7. Protocols and Implementation Considerations

The thermodynamically consistent implementation of a Markovian Moore machine progresses in four protocol steps for each input symbol:

1. Flip: Extract from current dynamic to correct label state.
2. Rearrange Barriers: Change activation of barriers to reflect new in/out topology.
3. Setter: Prepare the active dynamic state set for the next symbol.
4. Restore Barriers: Reset configuration to begin the next cycle.

This protocol enforces the time-inhomogeneous driving necessary for correct computation, error suppression, and energy minimization.

Summary Table

Aspect	Markovian Moore Machine (thermodynamic) (Chu et al., 2018)	Moore Coding of Markov Measure (Grigorchuk et al., 2021)
State Structure	Label + Dynamic states implemented as energy wells	Discrete state set, block code action
Dynamics	Time-inhomogeneous Markov chain, driven by input	Push-forward/coding map on Markov process
Resource/Energy Cost	Explicitly quantified per cycle	Not applicable
Statistical Properties	Error rate, relaxation time, average work	Ergodicity, entropy, limiting frequencies
Continuity/Mixing	Deterministic pathway, Markov stochasticity	Output inherits mixing/ergodicity

The Markovian Moore machine unites automata theory and stochastic process analysis, enabling explicit computation of thermodynamic, statistical, and measure-theoretic properties for both physical and information-theoretic realizations (Chu et al., 2018, Grigorchuk et al., 2021).