Reduced Abelian Complexity Function

• Reduced abelian complexity function is defined by applying run reduction to factors of infinite words and then counting them up to abelian equivalence.
• For the regular paperfolding sequence, the function admits a closed form that is eventually periodic, contrasting with its logarithmically growing classical counterpart.
• The Thue–Morse case presents open problems, prompting further research into explicit recurrences and automata-theoretic methods for complexity analysis.

The reduced abelian complexity function is a complexity function for infinite words over finite alphabets in which each finite factor is first simplified by collapsing every maximal run of identical letters to a single letter and is then counted up to abelian equivalence. In the formulation introduced by Campbell–Currie–Rampersad, this produces a variant of classical abelian complexity that can be substantially simpler while retaining a factor-based combinatorial structure; for the regular paperfolding sequence it admits a closed form that is eventually periodic of period $4$, whereas for the Thue–Morse sequence the full recursion remains open (Campbell et al., 19 Sep 2025).

1. Definition through run reduction

Let $\Sigma$ be a finite alphabet and let $w \in \Sigma^{+}$ be a nonempty finite word. Write $w$ in its unique run-decomposition

$w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$

where each $\ell_i \ge 1$, each $c_i \in \Sigma$, and $c_i \neq c_{i+1}$. Each block $c_i^{\ell_i}$ is a maximal run of the letter $c_i$. The run-reduction operator is then

$\Sigma$0

that is, each maximal block of identical characters is replaced by a single letter (Campbell et al., 19 Sep 2025).

For an infinite word $\Sigma$1 over $\Sigma$2, let $\Sigma$3 denote the set of its factors of length $\Sigma$4. Two factors $\Sigma$5 are said to be abelian-reduced-equivalent, written

$\Sigma$6

if

1. $\Sigma$7, and
2. $\Sigma$8 and $\Sigma$9 are abelian-equivalent, meaning that their Parikh vectors coincide.

The reduced abelian complexity function is therefore

$w \in \Sigma^{+}$0

It counts the number of abelian-reduced-equivalence classes represented among the length-$w \in \Sigma^{+}$1 factors of $w \in \Sigma^{+}$2 (Campbell et al., 19 Sep 2025).

A structural feature of the definition is that reduction is performed factorwise. What is counted are the abelian types of the words $w \in \Sigma^{+}$3 arising from length-$w \in \Sigma^{+}$4 factors $w \in \Sigma^{+}$5, not the abelian types of factors of a globally reduced infinite word.

2. Relation to classical abelian complexity

Classical abelian complexity is defined from Parikh vectors without any run reduction. If $w \in \Sigma^{+}$6 and $w \in \Sigma^{+}$7, then for an infinite word $w \in \Sigma^{+}$8,

$w \in \Sigma^{+}$9

so $w$0 counts the number of distinct Parikh vectors among length-$w$1 factors (Shallit, 2020).

The reduced abelian complexity differs from this in a precise way. Classical abelian complexity counts length-$w$2 factors up to abelian equivalence of the factors themselves, whereas reduced abelian complexity first applies $w$3 to each factor and then counts the resulting shorter words up to abelian equivalence (Campbell et al., 19 Sep 2025). In practice one finds

$w$4

and many sequences whose classical abelian complexity grows unboundedly turn out to have a much simpler reduced abelian complexity, often eventually periodic (Campbell et al., 19 Sep 2025).

This places the notion alongside the broader hierarchy of $w$5-Abelian complexities. In that framework, $w$6 gives ordinary abelian equivalence and the associated complexity function counts factors modulo Parikh-vector equality (Karhumäki et al., 2013). The run-reduced variant is therefore not merely the $w$7 case; it is a separate modification obtained by composing abelianization with the run-reduction operator.

3. Explicit evaluation for the regular paperfolding sequence

For the regular paperfolding sequence $w$8 over $w$9, Campbell–Currie–Rampersad prove the following closed form: $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$0 This holds for every $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$1 (Campbell et al., 19 Sep 2025).

Length condition on $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$2	$w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$3
$w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$4 even	3
$w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$5	4
$w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$6	5

In particular, the map $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$7 is eventually periodic of period $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$8. The significance of this statement lies in its contrast with the classical abelian complexity of $w = c_1^{\ell_1} c_2^{\ell_2} \cdots c_m^{\ell_m},$9, which grows logarithmically (Campbell et al., 19 Sep 2025). The paperfolding word thus provides a canonical example in which run reduction suppresses enough local multiplicity that the resulting abelian complexity becomes periodic.

This example also shows that reduced abelian complexity can exhibit low-amplitude oscillation among a small set of values even when the unreduced abelian complexity is not constant.

4. The Thue–Morse case and open problems

For the Thue–Morse sequence $\ell_i \ge 1$0, the same work establishes a clean $\ell_i \ge 1$1-regular recurrence for the reduced factor complexity, but it does not provide a full recursion for

$\ell_i \ge 1$2

(Campbell et al., 19 Sep 2025).

The initial values observed empirically are

$\ell_i \ge 1$3

and one partial relation is

$\ell_i \ge 1$4

These data indicate nontrivial regularity, but the full structure remains unresolved (Campbell et al., 19 Sep 2025).

The main open problems stated for $\ell_i \ge 1$5 are threefold. First, one asks for an explicit recursion, for instance a $\ell_i \ge 1$6-regular scheme, determining the function for all $\ell_i \ge 1$7. Second, one asks to prove or disprove the conjectured relation

$\ell_i \ge 1$8

and to determine the sign in the nonzero case. Third, one asks to show that the sequence $\ell_i \ge 1$9 is not $c_i \in \Sigma$0-automatic for any $c_i \in \Sigma$1 (Campbell et al., 19 Sep 2025).

The Thue–Morse case is therefore the principal unresolved example in the current theory of the run-reduced abelian complexity function.

5. Automata-theoretic and relative-Parikh approaches

Automata-theoretic methods for abelian complexity provide an important methodological backdrop. For an automatic sequence $c_i \in \Sigma$2, if $c_i \in \Sigma$3 is bounded and the prefix Parikh-vector sequence $c_i \in \Sigma$4 is synchronized, then $c_i \in \Sigma$5 is automatic, and one can effectively build a DFAO for it (Shallit, 2020). The construction proceeds by forming automata for Parikh vectors of arbitrary factors, computing the finite range of relative Parikh vectors

$c_i \in \Sigma$6

enumerating the subsets $c_i \in \Sigma$7 of realized relative vectors, and taking a direct product whose output is $c_i \in \Sigma$8 (Shallit, 2020).

A related but more general finite-state framework appears for $c_i \in \Sigma$9-balanced Parry words. If $c_i \neq c_{i+1}$0 is such a word, then its abelian complexity is $c_i \neq c_{i+1}$1-automatic, and the same DFAO mechanism applies to any function $c_i \neq c_{i+1}$2 determined by the set of relative Parikh vectors

$c_i \neq c_{i+1}$3

The exposition explicitly lists “the so-called ‘reduced abelian complexity’” among examples of such functions (Turek, 2014).

The finite-state mechanism in that setting is based on a finite family of sets $c_i \neq c_{i+1}$4, where each element stores a relative Parikh vector, a first letter, and a bounded context window. Because $c_i \neq c_{i+1}$5 is $c_i \neq c_{i+1}$6-balanced, only finitely many distinct sets $c_i \neq c_{i+1}$7 occur, yielding a DFAO whose output recovers the chosen function of $c_i \neq c_{i+1}$8 (Turek, 2014). This suggests a broad automata-theoretic strategy for reduced complexity variants whenever the relevant relative-Parikh data are finite-state representable.

6. Terminological scope and common ambiguities

The expression “reduced abelian complexity” is not uniform across the literature. In the Campbell–Currie–Rampersad usage, it refers specifically to the run-reduced function defined through $c_i \neq c_{i+1}$9 and the equivalence relation $c_i^{\ell_i}$0 (Campbell et al., 19 Sep 2025).

In expository discussions of $c_i^{\ell_i}$1-Abelian complexity, however, the phrase may be used differently. One summary of Karhumäki–Saarela–Zamboni describes the $c_i^{\ell_i}$2 case itself as the “reduced Abelian complexity,” namely the ordinary abelian complexity obtained from Parikh vectors (Karhumäki et al., 2013). A separate exposition, based on a general treatment of $c_i^{\ell_i}$3-Abelian equivalence, proposes “reduced Abelian complexity” in the sense of a reduction by the Sturmian baseline, for example through the additive excess

$c_i^{\ell_i}$4

or the normalized quantity

$c_i^{\ell_i}$5

with $c_i^{\ell_i}$6 or $c_i^{\ell_i}$7 corresponding to Sturmian behavior among aperiodic words (Karhumaki et al., 2013).

By contrast, Kaye–Rampersad explicitly state that their work on abelian complexity and the Frobenius problem does not introduce any separate notion of “reduced abelian complexity” (Kaye et al., 2019). The terminological landscape therefore contains at least three distinct uses: the run-reduced function of Campbell–Currie–Rampersad, the ordinary $c_i^{\ell_i}$8 abelian complexity, and normalized or excess-type reductions against a Sturmian benchmark.

For research usage, this ambiguity makes the defining operation essential. When the term is used without qualification, the crucial question is whether the reduction is by run collapse, by passage to the $c_i^{\ell_i}$9 abelian level, or by subtraction or normalization relative to $c_i$0.