Automata on Linear Orderings

• Automata on linear orderings are automata-theoretic frameworks where words, seen as functions over arbitrary linear orders, incorporate cuts and limits to capture order-specific phenomena.
• They use both successor and limit transitions to systematically handle diverse ordering features such as gaps, dense segments, and scattered structures.
• Key results include structural rank bounds, decidability frontiers, and finite-basis decompositions that underline the intricate interplay between order theory and automata.

Automata on linear orderings form a family of automata-theoretic frameworks in which linear order is not merely an ambient combinatorial notion but part of the input model, the presentation mechanism, or the structural invariant under study. In the most direct sense, due to Bruyère and Carton, a word is a function $w:J\to\Sigma$ whose domain $J$ is a linear ordering, and automata run on the cuts of $J$ rather than on positions. Closely related lines of work study $L$-automatic structures, word-automatic and tree-automatic linear orders, and several adjacent order-sensitive models. Across these settings, the subject has developed around three recurring themes: semantic uniformity over arbitrary order types, structural rank bounds for representable linear orders, and sharp decidability or undecidability frontiers for equivalence, rigidity, and logical definability (Braipson et al., 2 Jul 2026, Kartzow et al., 2013, Huschenbett, 2012, Cristau, 2011).

1. Core automaton models over linearly ordered domains

In the Bruyère–Carton framework, a word indexed by a linear ordering is a function

$w:J\to \Sigma,$

where $J$ is the “length” of the word. This subsumes finite words, $\omega$-words, bi-infinite words, and ordinal or transfinite words in a single definition. The central technical device is the set of cuts $\widehat J$: a cut is a partition $(K,L)$ of $J$ such that every $J$0 is $J$1 every $J$2. Runs are defined on cuts rather than positions, which makes the model uniform over orders with limit phenomena, dense regions, and gaps (Braipson et al., 2 Jul 2026).

An automaton on linear orderings has the form

$J$3

where $J$4 is finite, $J$5, and $J$6 contains successor transitions $J$7, right-limit transitions $J$8, and left-limit transitions $J$9. A run on $J$0 is a map $J$1 satisfying the endpoint conditions, the successor condition at consecutive cuts, and the two limit conditions at cuts without predecessor or successor. The relevant limit sets are

$J$2

and symmetrically for $J$3. The accepted languages are called rational (Braipson et al., 2 Jul 2026).

For arbitrary linear time, the cut-based viewpoint is indispensable. If $J$4 is finite, the model collapses to ordinary finite automata. If $J$5, it behaves like a transfinite or Büchi-style limit mechanism at the final cut. If $J$6 is arbitrary, gaps and non-successor cuts require the full left-limit and right-limit transition apparatus (Cristau, 2011).

A distinct but closely connected presentation-theoretic line studies $J$7-words and $J$8-automatic structures. Fix a linear order $J$9 and a padding symbol $L$0. An $L$1-word is a map $L$2 with finite support

$L$3

Bruyère–Carton automata on such words yield $L$4-automatic structures; restricting to finite-support $L$5-words gives finite word $L$6-automaticity, and allowing scattered $L$7 leads to the class of scattered-automatic structures (Kartzow et al., 2013).

2. Temporal logic over arbitrary linear time

A major logical motivation for automata on linear orderings is linear temporal logic over arbitrary linear time. Over words in $L$8, the relevant operators are Until, Since, and the future and past Stavi connectives. For strict Until,

$L$9

iff there exists $w:J\to \Sigma,$0 such that $w:J\to \Sigma,$1 and every $w:J\to \Sigma,$2 with $w:J\to \Sigma,$3 satisfies $w:J\to \Sigma,$4. Since is defined by reversal. The future Stavi connective is gap-sensitive: $w:J\to \Sigma,$5 iff there exists a gap $w:J\to \Sigma,$6 such that $w:J\to \Sigma,$7 holds everywhere between $w:J\to \Sigma,$8 and $w:J\to \Sigma,$9, there is no interval starting at $J$0 on which $J$1 always holds, and $J$2 holds on some interval starting at $J$3 (Cristau, 2011).

The role of gaps is decisive. On Dedekind-complete orderings, Until and Since suffice for first-order completeness, but on arbitrary linear orderings they do not; the Stavi operators are needed to express properties involving gaps. This is mirrored exactly by the automaton model, whose runs live on cuts and whose transition system explicitly handles limit behavior on both sides of a cut (Cristau, 2011).

For every LTL formula with Until, Since, and the Stavi connectives, there is an automaton $J$4 that computes its truth word pointwise. If $J$5 is defined by

$J$6

then $J$7 outputs $J$8 on input $J$9. The construction is compositional, using product and composition of transducers together with elementary automata for atomic propositions, Boolean connectives, Until, Since, and the Stavi operators. The paper gives a doubly exponential procedure to compute such an automaton from the formula, and since emptiness for the automaton model is decidable, this yields a decision procedure for satisfiability (Cristau, 2011).

A notable boundary result is that the automaton model is stronger than the temporal fragment used in the translation. The paper exhibits an automaton that outputs $\omega$0 iff there is a gap somewhere in the future and notes that this property is not first-order definable. This shows that automata on arbitrary linear orderings are not merely a presentation device for a fixed logic; they have independent expressive power (Cristau, 2011).

3. Structural bounds for automatic linear orders

A central strand of the subject concerns how complicated a linear order can be while still admitting an automatic presentation. For tree-automatic structures, the relevant invariant is finite-condensation rank. Given a linear ordering $\omega$1, the equivalence relations $\omega$2 are defined by iterating finite-distance condensation, and

$\omega$3

For scattered orderings this coincides with Hausdorff’s scattered-order rank $\omega$4, up to the finite-sum variant $\omega$5 used in proofs (Huschenbett, 2012).

The structural mechanism behind the major upper bounds is Delhommé’s decomposition theorem. For a tree-automatic structure $\omega$6 and an $\omega$7-formula $\omega$8, there is a finite family $\omega$9 such that every induced substructure on $\widehat J$0 is a sum augmentation of tame box augmentations built from members of that family. Sum augmentations partition the domain into finitely many pieces inducing prescribed structures; box augmentations use a bijection

$\widehat J$1

such that each coordinate map is an embedding when the others are fixed. Tameness means that basic relations are determined from finitely many coordinatewise colors, in Feferman–Vaught style (Huschenbett, 2012).

Using this decomposition, the paper proves the main theorem: $\widehat J$2 More sharply, if $\widehat J$3 admits a $\widehat J$4-free tree-automatic presentation of branching complexity rank $\widehat J$5, then

$\widehat J$6

At the ordinal level, this yields

$\widehat J$7

and unrestricted tree-automatic ordinals are exactly those below $\widehat J$8 (Huschenbett, 2012).

For $\widehat J$9-automaticity over a fixed scattered index order $(K,L)$0, the governing rank is $(K,L)$1. If $(K,L)$2 is scattered of $(K,L)$3-rank $(K,L)$4, then every finite word $(K,L)$5-oracle-automatic scattered linear order $(K,L)$6 satisfies

$(K,L)$7

In particular, every $(K,L)$8-automatic ordinal is below

$(K,L)$9

The same paper proves bounds on well-founded forests: if $J$0 is an ordinal or a scattered order with $J$1, every $J$2-oracle-automatic forest has rank strictly below $J$3; for arbitrary scattered $J$4, the general upper bound is

$J$5

It also separates tree-automaticity from $J$6-automaticity for ordinal $J$7 by showing that the countable atomless Boolean algebra, known to be tree-automatic, is not $J$8-automatic for any ordinal $J$9 (Kartzow et al., 2013).

These results collectively show that automaticity over linear orderings is rank-sensitive. Word automata, tree automata, and $J$00-automatic presentations do not merely differ in convenience of coding; they occupy sharply distinct zones in the ordinal and scattered-order landscape (Huschenbett, 2012, Kartzow et al., 2013).

4. Decidability, normal forms, and undecidability frontiers

For scattered tree-automatic linear orders, the question whether a given tree-automatic presentation is already word automatic is decidable. The key presentation-theoretic criterion is slimness of the domain tree language. If the thickness of a tree $J$01 is

$J$02

then a regular tree language is slim if all accepted trees have uniformly bounded thickness, and fat otherwise. Slimness of a regular tree language is decidable; if a tree automatic structure has slim domain, it is effectively word automatic; and if a tree automatic scattered linear ordering has fat domain, then it cannot be word automatic. Hence, given a tree automatic presentation of a scattered linear ordering, one can decide whether it is word automatic, and in the positive case compute a word automatic presentation (Huschenbett, 2012).

At the symmetry level, strong negative results appear even inside the scattered fragment. The isomorphism problem for scattered tree automatic linear orders is $J$03-hard. For word automatic scattered linear orders, existence of a non-trivial automorphism is undecidable. More precisely, the set of word automatic presentations of automatically rigid linear orders is $J$04-complete, so the existence of an automatic non-trivial automorphism of a word automatic linear order is $J$05-complete. For tree automatic scattered orders, rigidity is even harder: the set of tree automatic presentations of rigid linear orders is $J$06-hard (Kuske, 2012).

These hardness results coexist with positive normal-form phenomena in special subclasses. For DFAs accepting lexicographically well-ordered regular languages, the represented orders are exactly the ordinals below $J$07. There is a polynomial-time algorithm that computes the Cantor normal form of the ordinal represented by such an automaton, and therefore a polynomial-time algorithm deciding whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages (Ésik, 2010).

The contrast is substantial. Rigidity of $J$08 for regular $J$09 is decidable, yet rigidity becomes $J$10-hard for automatic linear orders presented via the automatic order $J$11, even when the resulting orders are scattered (Kuske, 2012). A plausible implication is that automatic linear orders become algorithmically difficult not simply because they are linear orders, but because the presentation formalism permits highly nontrivial decompositions and codings that are absent in the plain lexicographic regular-language setting.

5. Rational languages over scattered linear orderings

A recent development addresses a longstanding difficulty of Bruyère–Carton automata: words indexed by arbitrary scattered orders are very general objects, often with countable or uncountable length, and are hard to reason about directly. The new concept of a constructible word isolates a finitely describable subclass that suffices to characterize rational languages over scattered linear orderings (Braipson et al., 2 Jul 2026).

Over a finite alphabet $J$12, the set $J$13 of constructible words is the least set containing $J$14 and each one-letter word, and closed under concatenation, countable one-sided repetition $J$15 and $J$16, and uncountable one-sided repetition $J$17 and $J$18. Thus constructible words are generated by

$J$19

They may have uncountable length, but they admit finite syntax (Braipson et al., 2 Jul 2026).

For a fixed automaton $J$20, the paper defines an automaton-induced equivalence $J$21 on scattered words: $J$22 iff for every triple $J$23, there exists a path

$J$24

reading $J$25 iff there exists such a path reading $J$26. The quotient is finite and carries a semigroup structure. One major lemma shows that for regular uncountable ordinals $J$27,

$J$28

so automata on scattered linear orderings cannot distinguish one uncountable regular cofinality from another. This is why $J$29 suffices as a canonical representative of uncountable one-sided repetition (Braipson et al., 2 Jul 2026).

The main representative theorem states that every non-empty $J$30-class contains a constructible word. The proof uses Colcombet’s theorem on Ramseyan splits for multiplicative labelings on linear orderings. Applied to the cut ordering $J$31, this yields a bounded-height decomposition whose highest-level classes determine repetitions by an idempotent semigroup element. Scatteredness is essential here because the proof needs consecutive elements in certain subsets of cuts, a property that fails in dense contexts (Braipson et al., 2 Jul 2026).

From the representative theorem the paper derives its main characterization: $J$32 for rational languages $J$33 of scattered words. Equivalently, two automata on scattered linear orderings accept the same language iff they accept the same constructible words. The paper is careful not to claim closure under complement in the unrestricted scattered setting; on the contrary, it emphasizes that complementability fails in general and that semigroup recognizability and rationality separate in the uncountable scattered case (Braipson et al., 2 Jul 2026).

This result is best viewed as a finite-basis theorem for reasoning about scattered-word automata. It does not collapse arbitrary scattered words to a single automaton-independent normal form, but it does show that rational language equality is controlled by a finitely generated test family (Braipson et al., 2 Jul 2026).

6. Related meanings of “ordered automata” and adjacent boundary results

The phrase “automata on linear orderings” is also used, sometimes loosely, for several neighboring but technically distinct paradigms. This is a source of confusion in the literature.

One nearby line studies synchronous binary relations over unary words, hence over $J$34 via unary length encoding. In that setting, synchronous linear orders are exactly the poor orders, namely finite sums

$J$35

and equivalence of such synchronous orders is decidable (Choffrut, 2023). This is not the Bruyère–Carton model of arbitrary order-indexed words; it is ordinary finite-word automaticity in a unary encoding.

Another line studies finite automata whose state sets admit an order preserved by every transition. A deterministic automaton is monotonic if there exists a linear order $J$36 on states such that

$J$37

Recognizing monotonic automata is NP-complete, even for binary alphabets, and the same holds for oriented automata preserving a cyclic order (Szykuła, 2014). Here the order lives on states, not on input positions or domains.

Co-lexicographically ordered automata form a further adjacent theory. Every automaton admits a co-lexicographic partial order on states compatible with the co-lex order of reaching prefixes, and the minimum width of such an order is a structural complexity parameter. The later development identifies a canonical minimum-width DFA for each language, the Hasse automaton $J$38, and proves

$J$39

for $J$40 the minimum DFA of $J$41 (Cotumaccio et al., 2022, D'Agostino et al., 2021). This again concerns an order on states induced by words, not words indexed by arbitrary linear orders.

Two additional nearby models bring linear order in through data rather than position order. One-sided Church games over $J$42 and $J$43 with specifications given by deterministic register automata are decidable in time polynomial in $J$44 and exponential in the number of priorities and registers; unrestricted Church games over $J$45 are undecidable even for deterministic register automata (Exibard et al., 2020). Tree register automata with suprema and infima constraints over a complete dense linear order with endpoints have decidable emptiness, and this yields decidability of countable satisfiability for two-variable logic with a tree order and a linear order, and in particular with two interpreted linear orders (Toruńczyk et al., 2021).

Finally, Presburger-definable and computable-structure perspectives supply complementary limits. Every $J$46-dimensionally interpretable linear order in $J$47 has $J$48-rank at most $J$49, and in dimension $J$50 there is a complete criterion: each $J$51-definable ordering is a restriction of a lexicographic order on $J$52 to a definable set (Zapryagaev, 2019). Conversely, arbitrary graph information cannot in general be uniformly recovered from linear-order codings: there is a graph not effectively interpretable in any linear ordering, there is a graph not interpretable in any linear ordering using computable $J$53 formulas, every graph is interpretable in some linear ordering using computable $J$54 formulas, and there is no fixed tuple of $J$55-formulas that uniformly interprets every input graph $J$56 in the Friedman–Stanley coding $J$57 (Knight et al., 2019).

A common misconception is therefore that all “ordered automata” results concern the same mathematical object. They do not. Some papers study automata reading words indexed by arbitrary linear orders; some study automatic presentations of linear orders; some study automata whose states carry a preserved order; and some study automata over ordered data domains. The technical overlap is substantial, but the semantic role of linear order differs sharply across these settings (Choffrut, 2023, Szykuła, 2014, D'Agostino et al., 2021, Exibard et al., 2020).