Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings

Abstract: Automata on linear orderings are finite-state automata introduced by Bruyère and Carton as a broad generalization of finite, infinite and transfinite-word automata. In this context, a word is defined as a function from a linear ordering to a finite alphabet. This general definition can make automata on linear orderings difficult to reason about. In this work, we introduce constructible words as an intuitive way of tackling this difficulty. These words can be obtained by a finite number of applications of simple operators and thus admit a finite notation. We show that a rational language of words indexed by scattered (countable and uncountable) linear orderings is characterized by its constructible words. Our proof of this result relies on an interesting theorem of semigroup theory due to Colcombet. We expect this property to be useful in future theoretical developments about automata on scattered linear orderings.

• The paper establishes that constructible words precisely characterize rational languages over scattered linear orderings.
• It employs automata theory and algebraic semigroup techniques to manage uncountable complexities and define equivalence classes.
• The authors leverage generalized Ramsey theory to reduce language equivalence to a finite set of constructible witnesses, enhancing algorithmic analysis.

Constructible Words and the Characterization of Rational Languages over Scattered Linear Orderings

Introduction

The study of automata operating on linear orderings provides a canonical generalization encompassing finite, infinite, bi-infinite, and transfinite word automata. In this context, a word is formalized as a function from a linear ordering to a finite alphabet. The resulting framework is intractable with traditional intuition, especially when orderings are allowed to be scattered and uncountable. This work introduces and uses the notion of constructible words—objects generated by a finite sequence of canonical operators—to resolve the expressive gap between the complexity of words indexed by scattered orderings and the tractability of automata-theoretic properties over such domains.

The paper establishes that rational languages—those languages recognizable by automata on scattered linear orderings—are entirely characterized by their constructible words. This extends the classical result on $\omega$-regular languages (characterized by ultimately periodic words) to this broader class, with significant theoretical and practical consequences (2607.01858).

Automata on Linear Orderings and Scattered Words

Automata on linear orderings, as defined by Bruyère and Carton, generalize word automata to domains indexed by arbitrary linear orderings. Words become functions from a (possibly uncountable, scattered) linear ordering to a finite alphabet. The acceptance semantics necessarily generalize classical runs by using cut-based state assignments, requiring an automaton to respect successor and limit behaviors depending on the topology of the underlying ordering.

Automata are defined by tuples $(Q, \Sigma, \Delta, I, F)$, partitioning transitions into successor, right-limit, and left-limit moves, thus capturing not only successor dynamics but also limiting (infinite or transfinite) behaviors. For words indexed by scattered orderings, classical closure properties degrade, particularly the failure of closure under complementation, making language equivalence and structural characterizations non-trivial [BC07, Ris05].

Constructible Words: Formal Definition and Properties

Constructible words are defined inductively:

• Base cases: The empty word and words of length one.
• Operations:
  • Concatenation: $u, v$ $\in$ $\mathscr{C} \implies u \cdot v \in \mathscr{C}$
  • Countable and uncountable "omega-powers": If $u \in \mathscr{C}$, then $$u^\omega, u^{-\omega}, u^{\omega_1}, u^{-\omega_1} \in \mathscr{C}$$

This set generalizes ultimately periodic words ($uv^\omega$) to allow for cofinalities beyond $\omega$, and forms a syntactic fragment that can finitely represent automata-relevant phenomena occurring in both the countable and uncountable regimes.

Of key import is that constructible words precisely capture all observable distinctions made by automata on scattered orderings. As such, the equivalence of two automata is determined by their constructible words, not by the uncountably many words in the full class $\Sigma^{\textsf{scat}}$.

Equivalence Classes and Semigroup Structure

A central technical device is the equivalence relation $(Q, \Sigma, \Delta, I, F)$0, defined w.r.t. an automaton $(Q, \Sigma, \Delta, I, F)$1: $(Q, \Sigma, \Delta, I, F)$2 and $(Q, \Sigma, \Delta, I, F)$3 are indistinguishable by $(Q, \Sigma, \Delta, I, F)$4 in that they induce identical sets of transition triples $(Q, \Sigma, \Delta, I, F)$5. This equivalence relation naturally partitions $(Q, \Sigma, \Delta, I, F)$6 into finitely many equivalence classes, forming a finite semigroup under concatenation.

The theoretical leverage is obtained by:

• Showing that the product of equivalence classes is well-defined and associative.
• Demonstrating via semigroup tools (notably Colcombet's factorization forests generalized to infinite, scattered orderings) that each non-empty equivalence class contains a constructible word.
• Proving that automata cannot distinguish between words such as $(Q, \Sigma, \Delta, I, F)$7 and $(Q, \Sigma, \Delta, I, F)$8 when $(Q, \Sigma, \Delta, I, F)$9 are regular uncountable ordinals—clarifying automaton limits on cofinality distinctions.

The semigroup structure ties together automata theory with algebraic recognizability techniques [Ris05, Colcombet-ramseyan-splits].

Ramseyan Factorizations and the Core Theorem

Applying Colcombet's generalized Ramsey theorem for semigroups, the authors establish the existence of finite-height Ramseyan splittings for any multiplicative labeling of the cuts of a scattered word. Inductively, any word in a given automaton equivalence class can be reduced to a constructible word through this finite splitting process, leveraging the fact that automata can only distinguish words up to certain canonical powers (e.g., $u, v$0, $u, v$1, $u, v$2, $u, v$3) [Colcombet-ramseyan-splits].

The main result (stated as Theorem~\ref{thm:constructible-words-scattered}) is that two automata accept the same rational language (over scattered orderings) if and only if they accept the same set of constructible words.

This result is nontrivial due to the lack of closure under complementation and, for uncountable domains, the absence of a direct model-theoretic or effective description akin to the MSO or algebraic characterizations in the countable case [CCP, automata-MSO].

Implications and Future Directions

The result systematizes the tractable core of rational languages of scattered words, providing a reduction to finitely representable objects. Practically, it suggests that language equivalence, containment, and other questions may be reduced to checking on (abounded number of) constructible witnesses, thereby facilitating algorithmic analysis and providing a means to sidestep uncountable complexity where automata distinctions collapse.

Theoretically, the result clarifies the expressive boundaries of automata on scattered orderings, especially their inability to distinguish uncountable cofinalities. This aligns with the constraints on definability in fragments of logic (e.g., MSO or MFO), and opens questions about the exact relationship between rational languages and definability in the monadic theory of scattered domains [Shelah_1975, Lauchli-Leonard].

It also suggests a broader potential: the method of characterizing rationality through constructible fragments could extend to larger classes of orderings or other automata-theoretic contexts.

Conclusion

By rigorously linking automata on scattered linear orderings to constructible words, the paper provides a potent combinatorial and algebraic tool for language-theoretic analysis in the transfinite and uncountable regime. This framework exposes the necessity and sufficiency of a finitely representable core, shaped by semigroup theory and structural Ramsey arguments, for the rationality of languages indexed by scattered orderings.

Further research may clarify the connection to logical definability and effective decision procedures, and perhaps achieve similar characterizations in broader settings where scatteredness is either relaxed or different automaton models are considered.

Reference:

"Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings" (2607.01858).