Bireversible Automata

• Bireversible automata are finite-state Mealy automata with invertible dynamics in both state transitions and letter actions, ensuring full duality and rigidity.
• They serve as a framework for constructing complex groups, yielding models such as lamplighter groups and Klein four groups through explicit affine and square-complex constructions.
• Their connections to square complexes, Wang tilings, and boundary dynamics provide deep insights into geometric group theory and computational properties.

Searching arXiv for recent and foundational papers on bireversible automata. arXiv search: bireversible automata lamplighter square complexes boundary action changing alphabet Bireversible automata are finite-state Mealy automata in which the dynamics are invertible in both the state and letter directions: each state acts by a permutation of the alphabet, each letter induces a permutation of the state set, and the combined map

$$\delta:Q\times A\to A\times Q,\qquad (q,a)\mapsto (\lambda(q,a),\rho(q,a))$$

is bijective (Francoeur, 13 Jul 2025). Within automaton-group theory this is a particularly rigid class. Bireversible automata simultaneously support dual automata, dual groups, normal-form descriptions of associated fundamental groups, and geometric interpretations via complete square complexes and tree automorphisms (Bondarenko et al., 2017).

1. Definitions and equivalent formulations

A Mealy automaton is a finite-state transducer $\mathcal M=(A,Q,\lambda,\rho)$, where $A$ is the alphabet, $Q$ the state set, $\lambda:Q\times A\to A$ the output map, and $\rho:Q\times A\to Q$ the transition map (Francoeur, 13 Jul 2025). It is invertible if for every $q\in Q$, the map $\lambda_q:A\to A$ is a bijection, and reversible if for every $a\in A$, the map $\rho_a:Q\to Q$ is a bijection. It is bireversible if it is invertible, reversible, and $\mathcal M=(A,Q,\lambda,\rho)$0 is bijective; equivalently, it is invertible and both it and its inverse automaton are reversible (Francoeur, 2022).

The dual automaton is obtained by interchanging states and letters. In concrete calculations, bireversibility is often verified through duality and inversion. For the $\mathcal M=(A,Q,\lambda,\rho)$1-state, $\mathcal M=(A,Q,\lambda,\rho)$2-letter automaton generating $\mathcal M=(A,Q,\lambda,\rho)$3, it is enough to check that the automaton $\mathcal M=(A,Q,\lambda,\rho)$4, its dual $\mathcal M=(A,Q,\lambda,\rho)$5, and the dual of its inverse $\mathcal M=(A,Q,\lambda,\rho)$6 are invertible; from this one obtains invertibility, reversibility, coreversibility, and hence bireversibility (Bondarenko et al., 2015). A stronger global formulation appears in the square-complex setting: one may require that all eight automata obtained by repeated dualization and inversion are well-defined and deterministic/complete (Bondarenko et al., 2017).

This terminology excludes a common simplification. Invertibility plus reversibility does not, by itself, capture bireversibility; the inverse automaton must also be reversible, or equivalently $\mathcal M=(A,Q,\lambda,\rho)$7 must be bijective (Francoeur, 13 Jul 2025). That extra symmetry is what makes bireversible automata unusually tractable and unusually constrained.

2. Square complexes, dual groups, and algebraic packages

From a finite automaton $\mathcal M=(A,Q,\lambda,\rho)$8 with state set $\mathcal M=(A,Q,\lambda,\rho)$9 and alphabet $A$0, one builds a one-vertex square complex $A$1 with one loop for each state, one loop for each letter, and one square for each transition $A$2. Its fundamental group has presentation

$A$3

(Bondarenko et al., 2017). In the bireversible case this construction is equivalent to the theory of complete directed $A$4 square complexes with one vertex: a complete directed $A$5 square complex with one vertex corresponds exactly to a bireversible automaton, and conversely (Bondarenko et al., 2017).

The square-complex viewpoint yields several equivalent characterizations. For a bireversible automaton, the associated bipartite graph is complete bipartite, the Wang tile set is $A$6-way deterministic, the square complex is a complete square complex, it is non-positively curved, and its universal cover is the direct product of two trees (Bondarenko et al., 2017). This places bireversible automata at a junction of self-similar group theory, nonpositive curvature, and symbolic dynamics.

A second algebraic package arises from the fundamental group

$A$7

For bireversible automata, the subgroups generated by $A$8 and by $A$9 are free, every element of $Q$0 has a unique normal-form decomposition, and the automaton group and dual group are obtained by quotienting by the largest normal subgroups lying in those free factors (Francoeur, 13 Jul 2025). In the parallel square-complex language, if $Q$1 is the maximal normal subgroup of $Q$2 contained in the subgroup generated by $Q$3, then $Q$4, and similarly for the dual automaton group on the alphabet side (Bondarenko et al., 2017).

These correspondences are not only descriptive. They produce structural consequences for residual finiteness. If $Q$5 is bireversible and has either two states or binary alphabet, then infiniteness of the automaton group $Q$6 implies that $Q$7 is non-residually finite (Bondarenko et al., 2017). In this sense, small bireversible automata can encode geometrically rigid yet residually pathological groups.

3. Small explicit examples

A decisive example is the automaton with states $Q$8 over alphabet $Q$9, given by the wreath recursions

$\lambda:Q\times A\to A$0

Its dual is equivalent to the original automaton under the correspondence $\lambda:Q\times A\to A$1, $\lambda:Q\times A\to A$2, $\lambda:Q\times A\to A$3, so the automaton is self-dual up to relabeling. The inverse-state recursions are also explicit, and the resulting automaton is invertible, reversible, coreversible, and therefore bireversible (Bondarenko et al., 2015).

The group-theoretic core of that example is the element

$\lambda:Q\times A\to A$4

which satisfies

$\lambda:Q\times A\to A$5

hence has order $\lambda:Q\times A\to A$6 (Bondarenko et al., 2015). The normal subgroup

$\lambda:Q\times A\to A$7

is normal, abelian, and every nontrivial element of $\lambda:Q\times A\to A$8 has order $\lambda:Q\times A\to A$9. Together with the fact that $\rho:Q\times A\to Q$0 has infinite order, this yields

$\rho:Q\times A\to Q$1

(Bondarenko et al., 2015). The example is notable because a very small automaton—$\rho:Q\times A\to Q$2 states and $\rho:Q\times A\to Q$3 letters—already realizes a lamplighter group while remaining bireversible and self-dual.

A second explicit model is a $\rho:Q\times A\to Q$4-state, $\rho:Q\times A\to Q$5-letter bireversible automaton with alphabet $\rho:Q\times A\to Q$6, states $\rho:Q\times A\to Q$7, and wreath recursion

$\rho:Q\times A\to Q$8

where $\rho:Q\times A\to Q$9 is the transposition $q\in Q$0 (Ahmed et al., 2018). Its automaton group is

$q\in Q$1

Writing

$q\in Q$2

one obtains a Klein four subgroup $q\in Q$3, and the conjugation action of $q\in Q$4 shifts the lamp configurations (Ahmed et al., 2018).

That same example admits an affine description on the ring of formal power series $q\in Q$5. For

$q\in Q$6

the generators $q\in Q$7 are realized by affine transformations of this form, linking bireversible automata to affine tree automorphisms and formal power series dynamics (Ahmed et al., 2018).

4. Lamplighter constructions and their scope

A broad family of lamplighter constructions is obtained from a nontrivial finite abelian group $q\in Q$8, written additively, by identifying both the state set and the alphabet with $q\in Q$9. The automaton $\lambda_q:A\to A$0 is defined by

$\lambda_q:A\to A$1

Its generators satisfy the wreath recursion

$\lambda_q:A\to A$2

and the resulting automaton group is

$\lambda_q:A\to A$3

(Nowak et al., 2023). This automaton is reversible for every finite abelian $\lambda_q:A\to A$4, self-dual, and bireversible if and only if $\lambda_q:A\to A$5 is odd. In the corresponding square-complex language, bireversibility reduces to solvability of

$\lambda_q:A\to A$6

for all $\lambda_q:A\to A$7, equivalently to the existence of square roots for all elements (Nowak et al., 2023).

A different program uses affine transformations of power-series rings over finite commutative rings. For

$\lambda_q:A\to A$8

the associated automaton $\lambda_q:A\to A$9 has alphabet $a\in A$0, state set

$a\in A$1

transition map

$a\in A$2

and output map

$a\in A$3

If $a\in A$4 is a unit, then $a\in A$5; moreover, $a\in A$6 is reversible iff $a\in A$7, its inverse is reversible iff $a\in A$8, and it is bireversible iff both $a\in A$9 and $\rho_a:Q\to Q$0 are units (Skipper et al., 2018). In this affine framework, bireversible realizability is governed by the additive structure of the ring: one obtains bireversible models precisely when the $\rho_a:Q\to Q$1-Sylow subgroup has no cyclic $\rho_a:Q\to Q$2-summand occurring with multiplicity $\rho_a:Q\to Q$3 (Skipper et al., 2018).

That ring-theoretic restriction is not universal. A later construction proves that for every non-trivial finite abelian group $\rho_a:Q\to Q$4, there exists a bireversible automaton generating

$\rho_a:Q\to Q$5

(Francoeur, 2022). The construction uses the twisted Cayley machine $\rho_a:Q\to Q$6 with states $\rho_a:Q\to Q$7, alphabet $\rho_a:Q\to Q$8, and

$\rho_a:Q\to Q$9

For finite groups this automaton is bireversible, and in the abelian case its automaton group is exactly the lamplighter group (Francoeur, 2022). This indicates that odd-order criteria or $\mathcal M=(A,Q,\lambda,\rho)$00-Sylow criteria are specific to particular construction schemes rather than absolute obstructions to bireversible realization.

5. Boundary dynamics, Wang tilings, and commensurators

For an invertible automaton $\mathcal M=(A,Q,\lambda,\rho)$01, the generated group acts on the boundary $\mathcal M=(A,Q,\lambda,\rho)$02, and one may study the stabilizer map

$\mathcal M=(A,Q,\lambda,\rho)$03

A point is singular if this map is not continuous there. The singular set has measure zero for every invertible automaton (D'Angeli et al., 2016). In the bireversible case the situation sharpens: singularity is equivalent to the existence of a nontrivial boundary stabilizer, so $\mathcal M=(A,Q,\lambda,\rho)$04 if and only if stabilizers are trivial at every boundary point (D'Angeli et al., 2016). Moreover, if a reversible invertible automaton generates a group without singular points, then the automaton must actually be bireversible (D'Angeli et al., 2016).

The same paper relates automata to Wang tilings. To a Mealy automaton $\mathcal M=(A,Q,\lambda,\rho)$05 one associates a tileset $\mathcal M=(A,Q,\lambda,\rho)$06, and $\mathcal M=(A,Q,\lambda,\rho)$07 is $\mathcal M=(A,Q,\lambda,\rho)$08-way deterministic if and only if $\mathcal M=(A,Q,\lambda,\rho)$09 is bireversible (D'Angeli et al., 2016). Periodic tilings correspond to commuting pairs $\mathcal M=(A,Q,\lambda,\rho)$10 with

$\mathcal M=(A,Q,\lambda,\rho)$11

This gives a direct bridge from bireversible dynamics to deterministic tilings, periodicity, and undecidability phenomena (D'Angeli et al., 2016).

A more recent viewpoint identifies bireversible automata with commensurators of Cayley graphs. In the free-group case, the bireversible automorphisms of the oriented tree $\mathcal M=(A,Q,\lambda,\rho)$12 are exactly

$\mathcal M=(A,Q,\lambda,\rho)$13

and for an arbitrary finitely generated marked group $\mathcal M=(A,Q,\lambda,\rho)$14 with $\mathcal M=(A,Q,\lambda,\rho)$15, one has

$\mathcal M=(A,Q,\lambda,\rho)$16

where $\mathcal M=(A,Q,\lambda,\rho)$17 is the set of bireversible automorphisms of $\mathcal M=(A,Q,\lambda,\rho)$18 (Francoeur, 13 Jul 2025). The same paper shows that $\mathcal M=(A,Q,\lambda,\rho)$19 is a directed union of groups generated by bireversible automata. From this it derives strong rigidity consequences: every cyclic subgroup of a bireversible group is undistorted, and the class $\mathcal M=(A,Q,\lambda,\rho)$20 of groups generated by bireversible automata is strictly smaller than the class $\mathcal M=(A,Q,\lambda,\rho)$21 generated by invertible and reversible automata, with Baumslag–Solitar groups $\mathcal M=(A,Q,\lambda,\rho)$22 for $\mathcal M=(A,Q,\lambda,\rho)$23 providing the separation (Francoeur, 13 Jul 2025).

These rigidity statements also give exclusions. Several families are shown not to be bireversible, including infinite virtually nilpotent groups, certain irreducible lattices in semisimple Lie groups, uniform lattices in $\mathcal M=(A,Q,\lambda,\rho)$24, $\mathcal M=(A,Q,\lambda,\rho)$25 for $\mathcal M=(A,Q,\lambda,\rho)$26, topologically rigid hyperbolic groups, certain hyperbolic groups with boundary a sphere or Sierpiński carpet, and fundamental groups of closed irreducible oriented $\mathcal M=(A,Q,\lambda,\rho)$27-manifolds with nontrivial geometric decomposition (Francoeur, 13 Jul 2025).

6. Changing alphabets, small-state limits, and open directions

The notion of bireversibility extends beyond fixed alphabets to automata over a changing alphabet

$\mathcal M=(A,Q,\lambda,\rho)$28

where each level $\mathcal M=(A,Q,\lambda,\rho)$29 has its own finite alphabet $\mathcal M=(A,Q,\lambda,\rho)$30. Such an automaton is given by level-dependent maps

$\mathcal M=(A,Q,\lambda,\rho)$31

and is called bi-reversible if it is invertible and both the automaton and its inverse are reversible levelwise (Woryna, 2017). This framework preserves the essential symmetry of the Mealy setting while allowing the ambient rooted tree to vary with the level.

In the two-state case, the decisive parameter is boundedness of the alphabet sequence. A non-abelian free group $\mathcal M=(A,Q,\lambda,\rho)$32 can be generated by a $\mathcal M=(A,Q,\lambda,\rho)$33-state bi-reversible automaton over $\mathcal M=(A,Q,\lambda,\rho)$34 if and only if $\mathcal M=(A,Q,\lambda,\rho)$35 is unbounded (Woryna, 2017). When the changing alphabet is constantly binary, $\mathcal M=(A,Q,\lambda,\rho)$36 for all $\mathcal M=(A,Q,\lambda,\rho)$37, the entire class $\mathcal M=(A,Q,\lambda,\rho)$38 consists exactly of

$\mathcal M=(A,Q,\lambda,\rho)$39

(Woryna, 2017). Thus even mild changes in the alphabet model can dramatically alter the range of possible groups.

Current research emphasizes how restrictive bireversibility really is. On one side lie explicit constructive families: small automata generating lamplighter groups, affine realizations over power-series rings, and self-dual models with strong symmetry. On the other side lie rigidity theorems, non-realizability results, and geometric obstructions (Francoeur, 2022, Francoeur, 13 Jul 2025). Two open problems highlighted in this literature are especially central: the Grigorchuk–Savchuk problem asking for infinite bireversible automaton groups with trivial stabilizers everywhere on the boundary (D'Angeli et al., 2016), and the question

$\mathcal M=(A,Q,\lambda,\rho)$40

posed explicitly in the commensurator framework (Francoeur, 13 Jul 2025). Together these problems mark the boundary between explicit construction and global structural limitation in the theory of bireversible automata.