---
title: Reduced Abelian Complexity Function
url: https://www.emergentmind.com/topics/reduced-abelian-complexity-function
type: topic
---

# Reduced Abelian Complexity Function

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 [2509.16034].

## 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
\[
\operatorname{red}(w) = c_1 c_2 \cdots c_m,
\]
that is, each maximal block of identical characters is replaced by a single letter [2509.16034].

For an infinite word \(\mathbf{s}\) over \(\Sigma\), let \(\operatorname{Fact}_{\mathbf{s}}(n)\) denote the set of its factors of length \(n\). Two factors \(v,w \in \operatorname{Fact}_{\mathbf{s}}(n)\) are said to be abelian-reduced-equivalent, written
\[
v \equiv_{ab,red} w,
\]
if
1. \(|\operatorname{red}(v)| = |\operatorname{red}(w)|\), and  
2. \(\operatorname{red}(v)\) and \(\operatorname{red}(w)\) are abelian-equivalent, meaning that their Parikh vectors coincide.

The reduced abelian complexity function is therefore
\[
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{s}}(n)
= \bigl|\operatorname{Fact}_{\mathbf{s}}(n)\big/\!\equiv_{ab,red}\bigr|.
\]
It counts the number of abelian-reduced-equivalence classes represented among the length-\(n\) factors of \(\mathbf{s}\) [2509.16034].

A structural feature of the definition is that reduction is performed factorwise. What is counted are the abelian types of the words \(\operatorname{red}(v)\) arising from length-\(n\) factors \(v\), 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 \(\Sigma=\{a_1,\dots,a_k\}\) and \(\psi(w)=(|w|_{a_1},\dots,|w|_{a_k})\), then for an infinite word \(\mathbf{s}\),
\[
\rho^{ab}_{\mathbf{s}}(n)=\bigl|\{\psi(\mathbf{s}[i..i+n-1]) : i \ge 0\}\bigr|,
\]
so \(\rho^{ab}_{\mathbf{s}}(n)\) counts the number of distinct Parikh vectors among length-\(n\) factors [2011.00453].

The reduced abelian complexity differs from this in a precise way. Classical abelian complexity counts length-\(n\) factors up to abelian equivalence of the factors themselves, whereas reduced abelian complexity first applies \(\operatorname{red}(\cdot)\) to each factor and then counts the resulting shorter words up to abelian equivalence [2509.16034]. In practice one finds
\[
\rho^{ab,red}_{\mathbf{s}}(n)\le \rho^{ab}_{\mathbf{s}}(n),
\]
and many sequences whose classical abelian complexity grows unboundedly turn out to have a much simpler reduced abelian complexity, often eventually periodic [2509.16034].

This places the notion alongside the broader hierarchy of \(k\)-Abelian complexities. In that framework, \(k=1\) gives ordinary abelian equivalence and the associated complexity function counts factors modulo Parikh-vector equality [1302.3783]. The run-reduced variant is therefore not merely the \(k=1\) 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 \(\mathbf{f}\) over \(\Sigma=\{0,1\}\), Campbell–Currie–Rampersad prove the following closed form:
\[
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{f}}(n)=
\begin{cases}
3, & n \text{ even},\\[6pt]
4, & n>1,\; n\equiv 1 \pmod{4},\\[4pt]
5, & n\equiv 3 \pmod{4}.
\end{cases}
\]
This holds for every \(n \ge 1\) [2509.16034].

| Length condition on \(n\) | \(\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{f}}(n)\) |
|---|---:|
| \(n\) even | 3 |
| \(n>1,\; n\equiv 1 \pmod{4}\) | 4 |
| \(n\equiv 3 \pmod{4}\) | 5 |

In particular, the map \(n \mapsto \rho^{ab,red}_{\mathbf f}(n)\) is eventually periodic of period \(4\). The significance of this statement lies in its contrast with the classical abelian complexity of \(\mathbf f\), which grows logarithmically [2509.16034]. 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 \(\mathbf t\), the same work establishes a clean \(2\)-regular recurrence for the reduced factor complexity, but it does not provide a full recursion for
\[
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{t}}(n)
\]
[2509.16034].

The initial values observed empirically are
\[
(2,3,3,4,3,5,4,5,3,4,5,6,4,6,5,4,3,5,4,\dots),
\]
and one partial relation is
\[
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{t}}(2n+1)
=
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{t}}(n+1).
\]
These data indicate nontrivial regularity, but the full structure remains unresolved [2509.16034].

The main open problems stated for \(\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf t}(n)\) are threefold. First, one asks for an explicit recursion, for instance a \(2\)-regular scheme, determining the function for all \(n\). Second, one asks to prove or disprove the conjectured relation
\[
\bigl|
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{t}}(4n+2)
-
\rho^{\mathrm{ab},\mathrm{red}}_{\mathbf{t}}(4n)
\bigr|
=
\begin{cases}
0, & \mathbf{t}_{n+1}=\mathbf{t}_{3n+1},\\
1, & \text{otherwise},
\end{cases}
\]
and to determine the sign in the nonzero case. Third, one asks to show that the sequence \(n \mapsto \rho^{\mathrm{ab},\mathrm{red}}_{\mathbf t}(n)\) is not \(k\)-automatic for any \(k\) [2509.16034].

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 \(\mathbf s\), if \(\rho^{ab}_{\mathbf s}(n)\) is bounded and the prefix Parikh-vector sequence \(n \mapsto \psi(\mathbf s[0..n-1])\) is synchronized, then \(n \mapsto \rho^{ab}_{\mathbf s}(n)\) is automatic, and one can effectively build a DFAO for it [2011.00453]. The construction proceeds by forming automata for Parikh vectors of arbitrary factors, computing the finite range of relative Parikh vectors
\[
f(i,n):=\psi(\mathbf s[i..i+n-1])-\psi(\mathbf s[0..n-1]),
\]
enumerating the subsets \(A_n\) of realized relative vectors, and taking a direct product whose output is \(|A_n|\) [2011.00453].

A related but more general finite-state framework appears for \(c\)-balanced Parry words. If \(\mathbf u\) is such a word, then its abelian complexity is \(U\)-automatic, and the same DFAO mechanism applies to any function \(F(n)\) determined by the set of relative Parikh vectors
\[
\mathcal R_{\mathbf u}(n)
=
\{\Psi_{\rm rel}(w): w \text{ is a factor of } \mathbf u,\ |w|=n\},
\qquad
\Psi_{\rm rel}(w)=\Psi(w)-\Psi(u_0u_1\cdots u_{n-1}).
\]
The exposition explicitly lists “the so-called ‘reduced abelian complexity’” among examples of such functions [1406.1876].

The finite-state mechanism in that setting is based on a finite family of sets \(S(n)\subset \mathbb Z^m\times A\times A^{2L+1}\), where each element stores a relative Parikh vector, a first letter, and a bounded context window. Because \(\mathbf u\) is \(c\)-balanced, only finitely many distinct sets \(S(n)\) occur, yielding a DFAO whose output recovers the chosen function of \(\mathcal R_{\mathbf u}(n)\) [1406.1876]. 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 \(\operatorname{red}(\cdot)\) and the equivalence relation \(\equiv_{ab,red}\) [2509.16034].

In expository discussions of \(k\)-Abelian complexity, however, the phrase may be used differently. One summary of Karhumäki–Saarela–Zamboni describes the \(k=1\) case itself as the “reduced Abelian complexity,” namely the ordinary abelian complexity obtained from Parikh vectors [1302.3783]. A separate exposition, based on a general treatment of \(k\)-Abelian equivalence, proposes “reduced Abelian complexity” in the sense of a reduction by the Sturmian baseline, for example through the additive excess
\[
\Delta_{\omega}(n)
=
\mathcal P^{(1)}_{\omega}(n)-(n+1)
\]
or the normalized quantity
\[
R_{\omega}(n)
=
\frac{\mathcal P^{(1)}_{\omega}(n)}{n+1},
\]
with \(\Delta_{\omega}(n)\equiv 0\) or \(R_{\omega}(n)\equiv 1\) corresponding to Sturmian behavior among aperiodic words [1301.5104].

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” [1907.08247]. The terminological landscape therefore contains at least three distinct uses: the run-reduced function of Campbell–Currie–Rampersad, the ordinary \(k=1\) 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 \(k=1\) abelian level, or by subtraction or normalization relative to \(n+1\).

Source: https://www.emergentmind.com/topics/reduced-abelian-complexity-function