Invertible & Reversible Automata

• Invertible and reversible automata are computational models where forward operations admit structured inverse forms, with distinctions such as global bijectivity versus local backward determinism.
• They are employed in various settings like cellular automata, finite automata, and Mealy automata to analyze computational behavior, communication bounds, and algebraic structures.
• Research highlights construction methods and complexity perspectives that balance local memory constraints with the need for a globally invertible or reversible operation.

Invertible and reversible automata form a family of automaton models in which forward evolution admits a formally constrained backward interpretation, but the exact constraint is model-dependent. In some settings, invertibility is a global bijectivity condition, while reversibility requires the inverse to remain inside the same automaton class; in others, reversibility is local backward determinism, and invertibility is the stronger requirement that each transition symbol act bijectively. This distinction appears across cellular automata on groups, finite automata, Mealy automata, causal graph dynamics, and higher-level rewriting models, and it is precisely this model dependence that organizes the modern theory (Yang, 29 Jun 2026, Radionova et al., 2024, Klimann et al., 2014, Arrighi et al., 2015).

1. Core notions and model-dependent meanings

A useful starting point is that “invertible” and “reversible” are not uniform across the literature. For cellular automata on groups, a cellular automaton $F:A^G\to A^G$ is bijective if it is a bijection of configuration spaces, and it is reversible if $F^{-1}$ is again a cellular automaton, hence has a finite memory description (Yang, 29 Jun 2026). For one-way reversible finite automata, reversibility means that each per-symbol transition map is injective, whereas one-way permutation automata require each per-symbol transition to be bijective on the state set (Radionova et al., 2024). For Mealy automata, invertible means each output map $\rho_q$ is a permutation of the alphabet, while reversible means each transition map $\delta_x$ is a permutation of the state set (Klimann et al., 2014). For causal graph dynamics, invertible dynamics means that the global evolution $F$ is a bijection, and reversible means that $F^{-1}$ is also a causal graph dynamics, namely shift-invariant, continuous, and bounded (Arrighi et al., 2015).

Model	Invertible	Reversible
Cellular automata on groups	Bijective global map $A^G\to A^G$	Inverse is again a cellular automaton
One-way finite automata	Per-symbol bijections on states	Per-symbol injective partial transitions
Mealy automata	Each $\rho_q\in\mathrm{Sym}(\Sigma)$	Each $\delta_x\in\mathrm{Sym}(Q)$
Causal graph dynamics	Global map $F$ is bijective	$F^{-1}$0 is also a CGD

A recurrent misconception is that bijectivity and reversibility should coincide automatically. That is true in several compact or finite-state settings, but it fails in others. Over finite alphabets, bijective cellular automata are reversible by the classical compactness argument, whereas over infinite alphabets bijectivity may fail to imply that the inverse has finite memory (Yang, 29 Jun 2026). In ANUCA, reversibility and invertibility are explicitly separated: reversibility is left-invertibility by an ANUCA with finite memory, while invertibility requires bijectivity together with an ANUCA inverse (Phung, 2022). This suggests that the decisive issue is not mere existence of an inverse as a set-theoretic map, but preservation of locality, bounded propagation, or backward determinism inside the ambient model.

2. Cellular automata on groups and the boundary set by local finiteness

For a group $F^{-1}$1 and alphabet $F^{-1}$2, a configuration is a map $F^{-1}$3, and the full shift $F^{-1}$4 carries the left shift action $F^{-1}$5. A global map $F^{-1}$6 is a cellular automaton if there exists a finite memory set $F^{-1}$7 and a local rule $F^{-1}$8 such that

$F^{-1}$9

This is the Curtis–Hedlund–Lyndon finite-memory framework, and reversibility means that the inverse $\rho_q$0 also admits such a finite-memory local rule (Yang, 29 Jun 2026).

The finite-alphabet case is classical: if $\rho_q$1 is finite, then $\rho_q$2 is compact in the product topology, cellular automata are continuous and shift-commuting, and every bijective cellular automaton has a continuous inverse which is again shift-commuting and therefore a cellular automaton. The 2026 characterization sharpens the infinite-alphabet case completely: a group $\rho_q$3 is locally finite if and only if, over every alphabet, every bijective cellular automaton $\rho_q$4 is reversible. Equivalently, if $\rho_q$5 is not locally finite, then for every infinite alphabet $\rho_q$6 there exists a bijective cellular automaton $\rho_q$7 whose inverse is not a cellular automaton (Yang, 29 Jun 2026).

The proof exhibits the obstruction explicitly. For non-locally finite $\rho_q$8, the counterexample already works over the countable alphabet

$\rho_q$9

with a rank track, a direction track, and a binary data track. The forward dynamics is triangular along finite directed chains of arbitrary length, and on each fibre it has the form $\delta_x$0, where $\delta_x$1 is pointwise nilpotent. Consequently the map is bijective, but the inverse has no uniform finite memory because recovering the value at the start of a chain requires reading the parity of outputs along the entire chain. The paper formulates the same phenomenon as an upper-triangular unipotent rule on finite directed chains, with the rank track ensuring finiteness at each site while allowing unbounded chain lengths across configurations (Yang, 29 Jun 2026).

This resolves Open Problem 2 from Ceccherini-Silberstein and Coornaert and removes the periodicity hypothesis from the negative direction. It also makes the group-theoretic boundary exact. Direct unions of finite groups and the Prüfer $\delta_x$2-group are locally finite and therefore satisfy “bijective implies reversible” for every alphabet, while $\delta_x$3, free groups, virtually $\delta_x$4 groups, and the first Grigorchuk group are not locally finite and therefore admit bijective non-reversible cellular automata over every infinite alphabet (Yang, 29 Jun 2026).

3. Non-uniformity, asynchrony, and phase-space inversion

Asynchronous non-uniform cellular automata (ANUCA) retain a common finite memory set $\delta_x$5, but replace a single local rule by a configuration of local rules $\delta_x$6, where $\delta_x$7. The global map is

$\delta_x$8

This framework is scheduler-free: non-uniformity is spatial rather than update-schedule based, and asynchrony is modeled through composition of ANUCA maps. The central stability notion is orbit-closure under translations of the rule configuration, $\delta_x$9, and the main theorem states that for countable $F$0, finite $F$1, and finite $F$2,

$F$3

Here reversibility means left-invertibility by an ANUCA with finite memory, and stable injectivity requires injectivity for every rule profile in $F$4 (Phung, 2022).

This framework separates notions that coincide for ordinary uniform finite-alphabet cellular automata. In particular, bijectivity need not imply reversibility: Example 14.2 gives a bijective but non-reversible ANUCA on $F$5 and $F$6. The paper also proves invertibility under structural hypotheses: if the rule configuration is asymptotic to a constant one, then injectivity implies invertibility on amenable groups, and injectivity of both the disturbed and constant systems implies invertibility on residually finite groups. For $F$7, stable injectivity together with bounded singularity also implies invertibility (Phung, 2022).

A distinct notion arises for asynchronous cellular automata with subset updates or single-site updates. Requiring unique predecessors is too strong: for nontrivial local rules it collapses to the identity automaton. The meaningful replacement is phase space invertibility, defined by reversal of the one-step edge relation in the phase-space graph. For purely asynchronous automata this means that there exists another asynchronous automaton $F$8 such that

$F$9

and for fully asynchronous automata the same equivalence is պահանջed for single-site updates. In the purely asynchronous case, invertibility is decidable in every dimension; in the fully asynchronous case it is decidable in one dimension. The paper also shows that every Turing machine can be simulated by a phase space invertible asynchronous cellular automaton (Wacker et al., 2012).

Taken together, these results relocate reversibility from simple bijectivity toward a more refined locality-preserving inverse semantics. In the non-uniform and asynchronous literature, what matters is preservation of the automaton format under inversion, often uniformly over translated rule profiles or phase-space edges.

4. Finite-state and language-theoretic models

For one-way finite automata, the basic distinction is between reversible automata and permutation automata. A 1RFA is a partial deterministic automaton such that for every input symbol $F^{-1}$0, the map $F^{-1}$1 is injective. A 1PerFA is stronger: each $F^{-1}$2 is a bijection on the state set. The expressive-power hierarchy established in 2024 is

$F^{-1}$3

Sweeping reversible automata are strictly stronger than one-way reversible automata, but their pass hierarchy collapses: three passes are always enough. Multiple initial states produce a proper hierarchy $F^{-1}$4, and $F^{-1}$5 is incomparable with sweeping reversible automata for every fixed $F^{-1}$6. In the unary case, $F^{-1}$7, $F^{-1}$8, and $F^{-1}$9 coincide in expressive power, while the inclusion of 1RFA into sRFA remains proper (Radionova et al., 2024).

Reversible Watson–Crick automata exploit a double-stranded input with a complementarity relation. The 2015 construction shows that one-way reversible Watson–Crick automata accept all regular languages, despite one-way reversibility constraints. The key mechanism is that the lower strand can encode an accepting transition sequence of a DFA while the run itself remains backward deterministic (Chatterjee et al., 2015). The state-complexity analysis refines this: for every NFA with $A^G\to A^G$0 states accepting a language $A^G\to A^G$1, there exists a reversible Watson–Crick automaton with non-injective complementarity relation accepting $A^G\to A^G$2 with $A^G\to A^G$3 states. For the family

$A^G\to A^G$4

the paper gives a construction with $A^G\to A^G$5 states and cites the lower bound that an NFA requires more than $A^G\to A^G$6 states (Chatterjee et al., 2020).

Two-party reversible computations over a shared input exhibit a different profile. In deterministic two-party Watson–Crick systems running in opposite directions, reversibility is equivalently global bijectivity on reachable configurations via deterministic backward steps. This model does not subsume all regular languages: the family $A^G\to A^G$7 provides regular languages not accepted by REV-PWK systems. At the same time, the model accepts non-semilinear and non-context-free languages, including the explicitly constructed language $A^G\to A^G$8 with $A^G\to A^G$9 communication and the language $\rho_q\in\mathrm{Sym}(\Sigma)$0. The communication-bounded reversible classes form a strict hierarchy

$\rho_q\in\mathrm{Sym}(\Sigma)$1

and emptiness, finiteness, inclusion, and equivalence are decidable under constant communication bounds but not semidecidable for $\rho_q\in\mathrm{Sym}(\Sigma)$2 and even $\rho_q\in\mathrm{Sym}(\Sigma)$3 communication (Kutrib et al., 2023).

Reversible one-way counter automata push backward determinism beyond finite-state devices. Here reversibility means that the global transition function on reachable configurations is injective and has a computable inverse via reverse transitions. The paper proves a superpolynomial-time separation between irreversible and reversible $\rho_q\in\mathrm{Sym}(\Sigma)$4-counter automata, an infinite and tight hierarchy for exponential time with respect to the number of counters, and widespread undecidability: for real-time reversible multi-counter automata with at least two counters, emptiness, finiteness, inclusion, equivalence, regularity, and context-freeness are not semidecidable (Kutrib et al., 2022).

5. Mealy automata, automaton groups, and bireversibility

For a Mealy automaton $\rho_q\in\mathrm{Sym}(\Sigma)$5, invertibility means that each output map $\rho_q\in\mathrm{Sym}(\Sigma)$6 is a permutation of $\rho_q\in\mathrm{Sym}(\Sigma)$7, and reversibility means that each transition map $\rho_q\in\mathrm{Sym}(\Sigma)$8 is a permutation of $\rho_q\in\mathrm{Sym}(\Sigma)$9. The dual automaton interchanges the roles of states and alphabet, and this duality is central in both algebraic and algorithmic arguments (Klimann et al., 2014). A stronger condition is bireversibility: in the 2025 formulation, a Mealy automaton $\delta_x\in\mathrm{Sym}(Q)$0 is bireversible if it is invertible and reversible and the map

$\delta_x\in\mathrm{Sym}(Q)$1

is a bijection (Francoeur, 13 Jul 2025).

The smallest reversible class already shows a rigid dichotomy. A reversible two-state Mealy automaton generates a semigroup that is either finite or free of rank $\delta_x\in\mathrm{Sym}(Q)$2. This yields decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata, and decidability of freeness for semigroups generated by two-state invertible-reversible Mealy automata. In the two-state invertible-reversible case, finiteness is equivalent to md-triviality, namely triviality of the iterated minimize-dualize reduction (Klimann, 2012).

At the next size level, connected 3-state invertible-reversible Mealy automata still exclude one of the classical torsion phenomena of automaton groups. A connected 3-state invertible-reversible Mealy automaton cannot generate an infinite Burnside group. Equivalently, whenever the generated group is infinite, it contains an element of infinite order. The proof is built around the labeled orbit tree of the dual automaton, the connection degree, a reduction edge, and $\delta_x\in\mathrm{Sym}(Q)$3-liftable paths; these combinatorial invariants force the existence of a branch with no label $\delta_x\in\mathrm{Sym}(Q)$4, and cyclically orbital words along such a branch have infinite order (Klimann et al., 2014).

A related torsion statement holds at semigroup level in a wider subclass. If an invertible reversible Mealy automaton has no bireversible connected component, then the automaton semigroup $\delta_x\in\mathrm{Sym}(Q)$5 is torsion-free. The proof again uses the labeled orbit tree, now together with self-liftable paths and a criterion equating finite order of $\delta_x\in\mathrm{Sym}(Q)$6 with bounded sizes of connected components along the powers $\delta_x\in\mathrm{Sym}(Q)$7 (Godin et al., 2014).

Bireversible automata connect automaton groups to graph automorphisms and commensurators. If $\delta_x\in\mathrm{Sym}(Q)$8 is finitely generated and $\delta_x\in\mathrm{Sym}(Q)$9 is a Cayley graph of $F$0, then

$F$1

can be expressed as a directed union of groups generated by bireversible automata. This yields strong consequences: every cyclic subgroup of a bireversible group is undistorted; if $F$2 is not locally finite, then $F$3 is residually infinite bireversible; several families, including infinite virtually nilpotent groups and $F$4 for $F$5, cannot be generated by bireversible automata; and the class of groups generated by bireversible automata is strictly contained in the class of groups generated by invertible and reversible automata. The Baumslag–Solitar groups $F$6, $F$7, provide the separating example: they are generated by invertible and reversible automata but not by bireversible automata because bireversible groups have undistorted cyclic subgroups (Francoeur, 13 Jul 2025).

6. Generalized reversible dynamics and structural semantics

Causal graph dynamics extend cellular automata from fixed lattices to connected, at-most-countable, bounded-degree graphs with finite vertex and edge labels. The global dynamics $F$8 acts on pointed graphs modulo isomorphism and is required to satisfy shift-invariance, continuity, and boundedness. In this setting, invertible dynamics means that $F$9 is a bijection, while reversibility means that the inverse is itself a causal graph dynamics with appropriate vertex tracking. The principal theorem states that invertible implies reversible: if a CGD is bijective on the compact configuration space determined by finite alphabets and bounded degree, then $F^{-1}$00 is again causal and shift-invariant (Arrighi et al., 2015).

The proof has two characteristic structural outputs. First, invertible CGD are almost vertex-preserving: beyond finitely many small finite graphs, the tracker $F^{-1}$01 is bijective. Second, every reversible CGD admits a block representation

$F^{-1}$02

where $F^{-1}$03 is a local mark gate and $F^{-1}$04 is its conjugate through a reversible extension $F^{-1}$05. This is a lattice-gas style finite-depth circuit decomposition by local reversible blocks. The paper also notes a limitation parallel to Kari’s phenomenon: there is no computable function bounding the inverse radius in terms of the forward radius (Arrighi et al., 2015).

A more syntactic route to reversibility appears in biorthogonal pattern-matching automata. A pattern-matching automaton

$F^{-1}$06

is orthogonal if it is non-ambiguous and left-linear. Its dual $F^{-1}$07 reverses rules and swaps the initial and final states. Biorthogonality means that both $F^{-1}$08 and $F^{-1}$09 are orthogonal, and then the step relation is a partial bijection with exact inverse

$F^{-1}$10

This yields step-by-step reversibility and a compositional compilation of high-level functional terms into reversible automata of size linear in the term size. The model is universal, but the logical boundary is sharp: the multiplicative–exponential fragment supports reversibility, whereas additives break biorthogonality and injectivity (Abramsky, 2011).

These generalized frameworks show that reversibility survives far beyond fixed-state-symbol models, provided the inverse preserves the same causal or syntactic structure. They also clarify that local invertibility need not come with simple quantitative control: the inverse may exist structurally while remaining hard to bound effectively.

7. Construction methods, optimization, and complexity-oriented perspectives

Reversibility is not only a restriction; it is also a design resource. One construction route uses conserved landscapes in one-dimensional periodic Boolean cellular automata. A marker CA flips the origin cell whenever its neighborhood matches one of a specified family of landscapes, and the Toffoli–Margolus incompatibility criterion guarantees local invertibility. The evolutionary study formulates the search for such reversible rules as an optimization problem over generating functions $F^{-1}$11, using GA and GP encodings, a compatibility objective

$F^{-1}$12

and the Hamming-weight objective $F^{-1}$13. The experiments show that conserved landscape reversibility forces $F^{-1}$14 to be very small relative to $F^{-1}$15, that balanced generating functions are maximally incompatible with reversibility, and that the discovered rules are involutive, so all cycles have length $F^{-1}$16 (Mariot et al., 2021).

A second constructive perspective uses reversible cellular automata as a scaffold for complex behavior. Starting from a reversible CA on $F^{-1}$17 states, the method appends $F^{-1}$18 new states at random while keeping the original reversible block intact, and then optionally applies permutations of states or row and column permutations in the evolution rule. Experimental results show that complexity can be obtained from reversible cellular automata by appending a proportion of about two times more states at random than the original number of states in the reversible automaton. Complexity appears to be commonly obtained from reversible cellular automata, and the reported yields include, for example, $F^{-1}$19 for $F^{-1}$20, $F^{-1}$21 for $F^{-1}$22, and $F^{-1}$23 for $F^{-1}$24 in the Welch-index-$F^{-1}$25 family (Seck-Tuoh-Mora et al., 2020).

These construction-oriented results indicate a broader methodological point. Reversibility often imposes rigid algebraic or local constraints, but those same constraints can supply stable periodic backgrounds, involutive subdynamics, or structurally certifiable search spaces. A plausible implication is that the modern study of invertible and reversible automata is best understood not as a single theory of “undoing computation,” but as a collection of locality-preserving inverse paradigms whose exact content depends on whether the ambient object is a shift space, a non-uniform rule profile, a finite-state recognizer, a transducer, a graph dynamics, or a rewriting system.