---
title: Returning Finite Automata
url: https://www.emergentmind.com/topics/returning-finite-automata
type: topic
---

# Returning Finite Automata

Returning finite automata are finite-state devices organized around a return operation that interrupts ordinary one-pass scanning. In the contemporary literature, the expression denotes several non-equivalent models rather than a single canonical machine: one-dimensional automata with translucent input letters restart from the left end of the remaining word after certain steps; picture automata on rectangular or hexagonal inputs repeatedly return to a fixed side or frontier; graph-walking automata are called returning when they may accept only after coming back to the initial node; and returning parallel communicating systems reset a queried component to its initial state. What unifies these models is a reset-like control discipline, but the formal role of “returning” depends on the ambient geometry and transition semantics [2606.26683] [2509.12077] [2011.14856] [1405.5600].

## 1. Terminological scope

The literature uses “returning finite automata” in several technically distinct senses. The following usages are all explicit.

| Context | Model | Meaning of “returning” |
|---|---|---|
| Translucent-letter word automata | \(DFAwtl\), \(NFAwtl\) | After consuming a visible symbol, the head restarts from the left end of the remaining input [2606.26683] |
| Rectangular picture automata | RFA | Each row is scanned left to right; after the row ends, the head jumps to the beginning of the next row [2509.12077] |
| Graph-walking automata | returning GWA | Acceptance is allowed only in the initial node, formally \(F\subseteq Q\times \Sigma_0\) [2011.14856] |
| Parallel communicating systems | returning DRCPCFA | When a component is queried, its state is transmitted and the sender is reset to its initial state [1405.5600] |
| Hexagonal picture automata | GHRFA | Each frontier is scanned in a fixed direction, then the scan restarts on the next frontier in the same direction [2508.07779] |

This terminological dispersion is not accidental. Each model uses a finite control together with a structured input-access discipline, and “returning” names the specific reset mechanism built into that discipline.

## 2. Returning automata with translucent input letters

In the one-dimensional theory of finite automata with translucent input letters, a machine has a state-dependent translucency mapping \(\tau:Q\to 2^\Sigma\). In state \(q\), symbols in \(\tau(q)\) are translucent and are skipped; the machine processes the first symbol to the right that is not translucent in \(q\). A non-returning nondeterministic automaton with translucent letters is written
\[
M=\langle Q,\Sigma,q_0,\triangleleft,\tau,\delta\rangle,
\]
with partial transition function
\[
\delta:Q\times (\Sigma\cup\{\triangleleft\})\to 2^Q\cup\{accept\}.
\]
Configurations have the form \((u q v)\) or \(accept\), and the initial configuration on input \(w\) is \((q_0 w)\). The input is a shrinking word: visible symbols are deleted, translucent symbols are skipped, and the machine does not rewrite symbols [2606.26683].

The returning mode is obtained by changing the successor relation. If
\[
v=xay,\qquad x\in\tau(q)^*,\ a\notin \tau(q),\ p\in\delta(q,a),
\]
then the returning transition is
\[
(u q x a y)\ \vdash\ (p\,u x y).
\]
Thus the machine skips the translucent prefix \(x\), consumes the first visible symbol \(a\), changes state to \(p\), and restarts from the left end of the shortened word \(uxy\). If the whole visible suffix is translucent, so that \(v\in\tau(q)^*\), then the machine sees the endmarker \(\triangleleft\); it may accept or change state and begin another sweep from the left. The deterministic variants \(DFAwtl\) and \(nrDFAwtl\) satisfy \(|\delta(q,a)|\le 1\) for all \(q\) and \(a\). In the general returning deterministic model used here, a \(DFAwtl\) may continue a computation by another sweep when the endmarker is seen.

Returning finite automata with translucent letters were introduced by Nagy and Otto, while the non-returning variant was introduced by Mráz and Otto. The 2026 analysis of open problems does not re-found the theory, but it sharpens one boundary for the returning families: \(L(DFAwtl)\) and \(L(NFAwtl)\) are not closed under inverse non-erasing homomorphism. The witness language
\[
L_{cd}=\{\,uv\mid u\in\{a,c\}^*,\ v\in\{b,c,d\}^*,\ |uv|_c=|v|_d\,\}
\]
belongs to \(L(DFAwtl)\); under the homomorphism \(h(a)=ac\), \(h(b)=bd\), \(h(\#)=\#\), its inverse image is
\[
h^{-1}(L_{cd})=\{\,a^n\# b^m\mid n=m\,\},
\]
which lies outside both returning families. The same paper records that reversal for \(L(NFAwtl)\) remains open, while emptiness for the returning variants is known to be decidable from earlier work [2606.26683].

## 3. Repetitive and non-returning variants

A later refinement separates two design choices that had previously been bundled together in the non-returning models: whether the head returns to the left end after deletion, and whether seeing the endmarker may trigger further computation. The new repetitive models \(RDFAwtl\) and \(RNFAwtl\) are returning automata with translucent letters that, on seeing \(\lhd\), may change state and continue on the same remaining tape instead of being forced to halt immediately. Formally, an \(RNFAwtl\) is
\[
A=(Q,\Sigma,\lhd,\tau,I,\delta),
\]
where \(\delta(q,\lhd)\) may be a nonempty subset of \(Q\), \(Accept\), or \(Reject\). The new endmarker clause is
\[
qw\cdot \lhd \vdash_A q'w\cdot \lhd
\]
when \(w\in(\tau(q))^*\) and \(q'\in\delta(q,\lhd)\) [2409.06975].

This isolates a genuine deterministic intermediate class. The paper proves
\[
\mathcal L(DFAwtl)\subsetneq \mathcal L(RDFAwtl)\subsetneq \mathcal L(nr\mbox{-}DFAwtl).
\]
The strictness of the first inclusion is witnessed by
\[
L_{\vee,c}= \{\,w\in\{a,b,c\}^*\mid |w|_c=1 \text{ and } |w|_a=|w|_b\,\}\ \cup\ \{\,w\in\{a,b\}^*\mid 2|w|_a=|w|_b\,\},
\]
which is accepted by an \(RDFAwtl\) but not by any \(DFAwtl\). The second strictness uses
\[
L_2=\{a^n b^n\mid n\ge 0\},
\]
which is accepted by an \(nr\mbox{-}DFAwtl\) but not by any \(NFAwtl\), and hence not by any \(RDFAwtl\). On the nondeterministic side the effect disappears:
\[
\mathcal L(RNFAwtl)=\mathcal L(NFAwtl).
\]

The same decomposition clarifies that non-returning motion alone is not the source of the stronger non-returning families. If one forbids repetitive endmarker behavior as well, the resulting \(nr\mbox{-}nr\mbox{-}NFAwtl\) and \(nr\mbox{-}nr\mbox{-}DFAwtl\) recognize exactly the regular languages. A plausible implication is that, in the translucent-letter setting, the large expressivity gap is created by the interaction of non-returning head movement with repetitive restart at the endmarker, not by either feature in isolation.

## 4. Picture-scanning returning automata

In picture-language theory, a returning finite automaton is a scanning device over bordered two-dimensional inputs. For rectangular pictures, a picture \(P\in\Sigma^{m,n}\) is surrounded by a border symbol \(\#\) to form \(\hat P\), and an RFA is
\[
A=(Q,\Sigma,\delta,q_0,F,\#),
\]
with \(\delta\subseteq Q\times (\Sigma\cup\{\#\})\times Q\) in the nondeterministic case or \(\delta:Q\times (\Sigma\cup\{\#\})\to Q\) in the deterministic case. The boundary picture is serialized by a fixed scanning strategy, and acceptance reduces to ordinary finite-state acceptance of the serialized string. The defining operational point is that an RFA scans each row left to right and, after finishing a row, makes a “big jump” back to the beginning of the next row. In this literature, “returning” names that repeated jump-to-line-start behavior [2509.12077].

Several sharp classification results are known for these picture RFAs. RFA and boustrophedon automata recognize the same picture language family:
\[
\mathrm{RFA}=\mathrm{BFA}.
\]
Determinism and nondeterminism also coincide for both models. On one-dimensional pictures \(P\in \Sigma^{m,1}\), RFAs recognize exactly the regular string languages. The same paper places RFA inside a DAG-automata framework: under the input-agnostic encoding \(₹\), deterministic DAG automata recognize exactly the same picture languages as RFAs, expressed by
\[
\textbf{DDA } =_{₹}\textbf{ RFA}.
\]
The RFA scan order therefore corresponds to a string-DAG encoding of the picture, rather than to a more genuinely two-dimensional dependency structure. This explains why row-aligned languages such as horizontal stripes are easy, while the diagonal language \(L_{\diagdown}\) is not recognizable by an RFA in the standard scanning direction [2509.12077].

A hexagonal analogue extends the same scanning idea to hexagonal pictures. A General Hexagonal Returning Finite Automaton scans one frontier of a bordered hexagonal picture in a fixed direction, then moves to the next frontier and resumes scanning in that same direction. The canonical direction is
\[
d_{HRFA}= \begin{pmatrix} s\downarrow&\CircleArrowright \end{pmatrix}.
\]
The central comparison theorem is that hexagonal returning and hexagonal boustrophedon automata define the same language family:
\[
L(HBFA)=L(HRFA).
\]
The proof simulates alternating boustrophedon motion inside the finite control of a returning automaton. The paper also establishes extensive equivalences among the twelve directional modes by rotations and reflections of the hexagonal plane [2508.07779].

## 5. Two-head two-dimensional returning automata

A more powerful two-dimensional variant is the \(2\)-head returning finite automaton, or \(2\)-HRFA, operating on rectangular picture languages. Here two heads scan the picture simultaneously from opposite sides: the first starts at the top row and scans active rows from left to right, while the second starts at the bottom row and scans active rows from right to left. A \(2\)-HRFA is
\[
M=(Q,\Sigma,\delta,s,F,\#,\square),
\]
with
\[
\delta: Q\times ((\Sigma\cup \{\varepsilon,\#\})\times (\Sigma\cup \{\varepsilon,\#\})\setminus\{(\varepsilon,\varepsilon)\})\rightarrow 2^Q.
\]
Configurations are written
\[
(q,P_\square,\mu,\mu'),
\]
where \(P_\square\) is the partially erased picture and \(1\le \mu\le \mu'\le m\) identify the current active rows. Acceptance occurs when the whole picture has been processed, all interior symbols have been replaced by \(\square\), the two heads have met, and the state is final [2606.26680].

The synchronized restriction B2-HRFA requires both heads to move in lockstep, except for a final single-head step when only one unread symbol remains. These models generate a strict hierarchy:
\[
\mathcal L(RFA)\subsetneq \mathcal L(B2\mbox{-}HRFA)\subsetneq \mathcal L(2\mbox{-}HRFA)\subsetneq \mathcal L(RPDA).
\]
The first strictness is witnessed by
\[
L_{rev}=\{\,w\ominus w^R\mid w\in\{a,b\}^+ \,\},
\]
which is accepted by a deterministic \(2\)-HRFA and by a B2-HRFA but not by a standard one-head RFA. The second strictness is witnessed, via one-dimensional restriction, by the language \(\{a^{3k}b^k\mid k\ge 1\}\), which lies in \(2\)-HRFA but not in B2-HRFA. The class \(\mathcal L(2\mbox{-}HRFA)\) is incomparable with the class of picture languages generated by context-free matrix grammars.

The closure theory reflects the difference between unrestricted and synchronized motion. \(\mathcal L(2\mbox{-}HRFA)\) is closed under union, but not under intersection, complement, or row and column concatenation. By contrast, B2-HRFA are determinizable, satisfy
\[
\mathcal L(B2\mbox{-}HRFA)=\mathcal L(B2\mbox{-}HRDFA),
\]
and are closed under union, intersection, complement, vertical reflection, horizontal reflection, and \(180^\circ\) rotation, though not under transpose or rotation by \(\pm 90^\circ\). The paper also notes that decision problems for unrestricted \(2\)-HRFA are widely open [2606.26680].

## 6. Returning as acceptance discipline and communication reset

In graph-walking automata, “returning” no longer refers to rescanning an input word or picture. A deterministic graph-walking automaton is
\[
A=(Q,q_0,F,\delta)
\]
over a graph signature \(S=(D,-,\Sigma,\Sigma_0,(D_a)_{a\in\Sigma})\). It is returning precisely when
\[
F\subseteq Q\times \Sigma_0,
\]
which means that acceptance is possible only in the initial node. A returning automaton may still reject away from the initial node and may still loop forever; returning is therefore independent of halting and weaker than reversibility. The normalization cost is quantified sharply: making an \(n\)-state GWA over a signature with \(k\) directions returning requires at most
\[
2nk+n
\]
states and at least
\[
2(n-1)(k-3)
\]
states in the worst case [2011.14856].

In deterministic returning centralized parallel communicating finite automata systems, “returning” refers to communication semantics. A system
\[
A=\langle \Sigma, A_1,\dots,A_k,Q,\triangleleft\rangle
\]
consists of deterministic component automata that read the same input in parallel. If component \(A_i\) enters query state \(q_j\), then it requests the current state of \(A_j\). In the returning mode, the requester receives the sender’s current state,
\[
p_i=s_j,
\]
and simultaneously the sender is reset to its initial state,
\[
p_j=s_{0,j}.
\]
This reset-based notion of returning yields a rich communication-complexity theory. There is a strict hierarchy
\[
L(\mathrm{O}(\log n)\mbox{-}DRCPCFA(k)) \subset L(\mathrm{O}(\sqrt n)\mbox{-}DRCPCFA(k)) \subset L(\mathrm{O}(n)\mbox{-}DRCPCFA(k)),
\]
and, for every \(r\ge 1\),
\[
L(\mathrm{O}(\log(n)^r)\mbox{-}DRCPCFA(k)) \subset L(\mathrm{O}(\log(n)^{r+2})\mbox{-}DRCPCFA(k))
\]
for every \(k\ge 4\). At the same time, for \(\mathrm{O}(\log n)\)-bounded returning centralized systems of degree \(k\ge 4\), emptiness, finiteness, infiniteness, universality, inclusion, equivalence, regularity, and context-freeness are not semi-decidable [1405.5600].

## 7. Structural themes and open directions

Across these literatures, returning acts as a control discipline that restricts where progress may resume. In \(DFAwtl\) and \(NFAwtl\), the return is a restart from the left end of a shrinking word. In picture RFAs and GHRFAs, it is a restart from the beginning of the next row or frontier. In graph-walking automata, it is a restriction on where acceptance may occur. In returning communicating systems, it is a reset of the sender to its initial state. This suggests that “returning finite automata” is best understood not as one formalism but as a family of reset-based finite-state architectures.

Theoretical behavior depends strongly on the surrounding model. Deterministic and nondeterministic returning translucent-letter automata diverge once repetitive endmarker behavior is introduced, but ordinary rectangular picture RFAs do not separate deterministic from nondeterministic power. Returning can be weaker than synchronized two-head returning motion, as in
\[
\mathcal L(RFA)\subsetneq \mathcal L(B2\mbox{-}HRFA),
\]
yet strong enough to preserve nontrivial structural properties such as the letter-equivalent regular sublanguage property used in inverse-homomorphism lower bounds for \(NFAwtl\) [2606.26683] [2606.26680].

Several boundary questions remain explicit. For returning finite automata with translucent letters, the closure of \(L(NFAwtl)\) under reversal remains open. For unrestricted \(2\)-HRFA, the corresponding decision problems are described as widely open. These unresolved points indicate that the return discipline, although finite-state in control, continues to interact in subtle ways with nondeterminism, multidimensional access, and restart semantics.

Source: https://www.emergentmind.com/topics/returning-finite-automata