A Ladder-Based Traversal Notation for Repeated Symbolic Sequences

1. Introduction

Repeated structures arise throughout symbolic systems, including language, genomic data, compressed sequences, and numerical expansions. Existing approaches to repetition analysis include factor automata and compact directed acyclic word graphs [1], hidden subsequence pattern analysis [2], overlap statistics [3], runs and maximal repetitions [4], and grammar-based compression systems such as SEQUITUR and Re-Pair [5,6]. These frameworks primarily represent repetition through intervals, factors, graphs, or recursive substitutions. In contrast, the present framework represents repeated symbolic structures as traversal systems defined by ordered progression between repetition states.

The present work introduces a preliminary traversal-based formalism for analysing repeated symbolic structures. Rather than characterising repetition solely through substring occurrence, periodicity, or grammatical substitution, the framework interprets repeated structures as ordered traversal relationships between repetition states. This perspective motivates a traversal-oriented representation of symbolic repetition and suggests descriptive quantities such as ladder density and traversal asymmetry.

Existing repetition frameworks typically characterise repeated symbolic structure through maximal periodic intervals [4], recursive grammatical substitutions [5,6], factor-transition graphs [1], or overlap statistics [3]. In contrast, the proposed framework represents repeated symbolic structures as traversal systems defined by ordered progression between adjacent rung states. Rather than compressing, factorising, or replacing repeated substrings, the framework preserves the original symbolic sequence while introducing a traversal-oriented representation of hierarchical repetition organisation.

The work is exploratory and primarily conceptual. Its aim is not to replace established methods in stringology, compression, or symbolic sequence analysis, but to provide a complementary representation that may operate alongside existing frameworks. Future work will determine whether the proposed traversal measures provide useful structural information beyond conventional repetition and overlap statistics.

2. Ladder Interpretation

The ladder framework models repeated symbolic structures as ordered traversal systems embedded within a finite symbolic sequence. Table 1 summarises the notation used throughout the ladder traversal framework.

Let

$$S=(s_1,s_2,\dots ,s_n)$$

(1)

denote a sequence over an alphabet $\mathsf{\Sigma }$.

For a chosen target symbol $\sigma \in \mathsf{\Sigma }$, a ladder rung (or rung state) of order $k$ is defined as a contiguous repetition block

$$R_k={\sigma }^k$$

(2)

consisting of $k$ consecutive occurrences of $\sigma$. The integer $k$ denotes the rung order of rung state $R_k$.

Traversal within the framework occurs horizontally through the sequence and vertically between repetition orders. A traversal path is therefore interpreted as an ordered progression between ladder rungs of varying repetition size.

Ascending traversal corresponds to incremental expansion through successive repetition states:

$$R_1\to R_2\to R_3\to \cdots$$

while descending traversal corresponds to incremental contraction:

$$R_m\to R_{m-1}\to \cdots \to R_1$$

A valid ladder traversal is constrained to adjacent repetition orders only. Consequently, direct transitions between non-adjacent rung states are excluded. Valid ascent paths therefore satisfy

$$R_k\to R_{k+1}$$

(3)

while valid descending paths satisfy

$$R_k\to R_{k-1}$$

(4)

for consecutive repetition orders, where two rung orders differ by one.

Traversal paths may consist of pairwise transitions such as

$$R_1\to R_2$$

or chained traversals such as

$$R_1\to R_2\to R_3$$

where each successive rung satisfies the incremental traversal constraint.

Within a sequence, traversal between rungs occurs across intermediary subsequences termed traversal gaps. If rung $R_i$ occupies positions $a_i$ to $b_i$, and rung $R_j$ begins at position $a_j$, then the associated traversal-gap length is defined as

$$g_{ij}=a_j-b_i-1$$

(5)

A traversal path containing multiple rung transitions therefore possesses an ordered gap vector

$$G=\left(g_1, g_2,\dots g_{m-1}\right)$$

(6)

describing the spacing between consecutive rung states.

The framework additionally permits overlapping traversal windows. Individual occurrences of the target symbol may therefore participate simultaneously in multiple traversal paths. For example, within the binary sequence

10110111

examples of ascending traversal embeddings may include

[1]0[11]0111

10[1]10[11]1

101[1]01[11]

In each case, a rung state of order 1 participates in an ascending traversal to a rung state of order 2. The same occurrence may therefore contribute to multiple valid traversal embeddings within the sequence.

The framework additionally permits chained traversals, such as

[1]0[11]0[111]

which represents the progression

$$R_1\to R_2\to R_3$$

Consequently, the ladder traversal is not interpreted as a unique decomposition of a sequence but rather as the complete set of valid incremental traversal paths satisfying the rung-transition constraints.

Traversal direction additionally introduces climb-state asymmetry. Reading a sequence left-to-right and right-to-left may generate distinct ascent and descent structures because traversal relationships depend upon ordered positional arrangement within the sequence. The framework therefore treats traversal direction as an intrinsic component of rung-state definition rather than a purely representational convention. Figure 1 provides a visual representation of ladder traversal behaviour within the binary sequence 10110111

3. Core Concepts

Traversal states within the ladder framework are represented by the tuple

$$T=(\sigma ,d,G,C,r)$$

(7)

where:

• $\sigma$ denotes the repeated target symbol,
• $d\in \{L\to R,\hspace{0.25em} R\to L\}$ denotes traversal direction,
• $G=\left(g_1, g_2,\dots ,g_{m-1}\right)$denotes the ordered traversal-gap vector between consecutive ladder rungs,
• $C\in \{A,D\}$ denotes climb behaviour, where $A$ represents ascending traversal and $D$ represents descending traversal,
• $r=(k_1,k_2,\dots ,k_m)$ denotes the ordered traversal path through rung states

Ascending and descending traversal follow the adjacent-rung transition rules defined in Eqs. (3) and (4). The ladder length is defined as

$$\left|r\right|=m$$

(8)

where $m$ denotes the number of rung states within the traversal path.

Consecutive rung states in a traversal path

$$r=(R_{k_1},R_{k_2},\dots ,R_{k_m})$$

must satisfy the adjacency constraint

$$\mid k_{i+1}-k_i\mid =1, i=1,\dots , m-1$$

(9)

This restates the adjacent-rung requirement for arbitrary traversal paths.

4. Traversal Notation

The traversal-state notation may be illustrated through simple and chained traversal examples extracted from binary symbolic sequences. The following examples demonstrate how traversal direction, rung progression, traversal gaps, and climb state are represented within the ladder framework.

4.1. Simple Traversal Example

Consider the binary sequence

$$S=1011$$

with repeated symbol

$$\sigma =1$$

Reading the sequence from left to right gives traversal direction

$$d=L\to R$$

Within the sequence, the traversal

$$\mathrm{r}=\left(1, 2\right)$$

progresses from repetition rung state $R_1$ to rung state $R_2$, corresponding to the repeated structures

$$1\to 11$$

The intermediary traversal gap consists of a single symbol $0$, yielding traversal-gap vector

$$G=\left(1\right)$$

Since traversal proceeds through increasing repetition order, the climb state is ascending:

$$C=A$$

The associated rung-order sequence is therefore

$$\left(\mathrm{1,2}\right)$$

Corresponding to the traversal path

$$\mathrm{r}=\left(R_1, R_2\right)$$

Hence the traversal state is represented by

$$T=(1,L\to R,(1),A,(\mathrm{1,2}\left)\right)$$

4.2. Chained Traversal Example

For a more complex example, consider the binary sequence

$$S=10110111$$

With repeated target symbol

$$\sigma =1$$

One valid chained traversal within the sequence is

$$1\to 11\to 111$$

Corresponding to the rung progression

$$\left(R_1, R_2, R_3\right)$$

or equivalently the rung-order sequence

$$\left(\mathrm{1,2},3\right)$$

In this traversal, the intermediary traversal gaps consist of one symbol between $R_1$ and $R_2$, and one symbol between $R_2$ and $R_3$, yielding traversal-gap vector

$$G=\left(\mathrm{1,1}\right)$$

Since traversal proceeds through increasing repetition order, the climb state is ascending:

$$C=A$$

Reading the sequence from left to right therefore yields the traversal-state tuple

$$T=(1,L\to R,(\mathrm{1,1}),A,(\mathrm{1,2},3\left)\right)$$

5. Rung-Jump and Skipped-Repetition Rule

Direct traversal between non-adjacent rung orders is excluded by the adjacency constraint defined in Eq. (9). For example

$$R_1\to R_3$$

does not constitute a valid single-step traversal because the intermediary rung state

$$R_2$$

is absent from the traversal chain. A valid ascent traversal must instead proceed incrementally:

$$R_1\to R_2\to R_3$$

Similarly, a valid descending traversal must also proceed incrementally:

$$R_3\to R_2\to R_1$$

This constraint defines the skipped-repetition rule and prevents arbitrary rung-jump transitions between non-adjacent rung states.

Although non-adjacent repetition jumps are excluded, ladder states may appear in multiple valid embeddings within a larger repetition structure. For example, within the repeated structure

$$1111$$

the rung state

$$R_2=11$$

may appear as:

$$\left[11\right]11$$

$$1\left[11\right]1$$

$$11\left[11\right]$$

Similarly for the lower-order rung

$$R_1=1$$

may appear as

$$\left[1\right]111$$

$$1\left[1\right]11$$

$$11\left[1\right]1$$

$$111\left[1\right]$$

These embedded traversal positions do not represent skipped-repetition transitions, but rather alternative contextual embeddings of the same valid rung order within a larger repeated structure.

More generally, repeated symbolic structures may contain recursively overlapping lower-order ladder states. For example,

$$111\supset 11\supset 1$$

indicates that the repetition structure $111$ contains overlapping embedded occurrences of $11$, which themselves contain embedded occurrences of $1$. Consequently, higher-order rung states may recursively embed multiple lower-order traversal configurations, generating nested and overlapping traversal paths within the same symbolic sequence.

The ladder framework therefore distinguishes between:

• Valid incremental traversal between adjacent rung orders; and
• Invalid skipped-repetition transitions between non-adjacent rung orders.

This distinction ensures that ladder traversal represents continuous hierarchical progression through repetition space rather than arbitrary transitions between repeated symbolic structures.

6. Potential Applications

The ladder traversal framework provides a traversal-based method for analysing repeated structures within symbolic and numerical sequences. Rather than representing repetition solely through substring occurrence or frequency, the framework interprets repeated structures as collections of measurable traversal paths characterised by traversal direction, rung progression, traversal gaps, climb behaviour, and overlapping embeddings.

One potential application of the framework is symbolic sequence fingerprinting. Under this interpretation, symbolic sequences such as binary streams, genomic strings, and symbolically encoded information sources may be analysed through their traversal distributions. Traversal organisation may therefore function as a structural descriptor of repeated symbolic behaviour within a sequence, potentially revealing similarities and differences in hierarchical repetition structure across diverse symbolic domains.

The framework permits measurement of traversal properties including ladder density, traversal length, overlap density, directional asymmetry, and the occurrence of chained traversal structures.

The traversal representation naturally suggests statistical extensions in which symbolic sequences are characterised through traversal profiles, overlap structure, and rung-transition behaviour. Such quantities may provide complementary structural information beyond conventional repetition counts.

A natural extension of the framework is Monte-Carlo simulation over randomly generated symbolic sequences. Traversal profiles extracted from observed symbolic systems may then be compared against random baselines in order to investigate whether traversal organisation exhibits statistically detectable structural regularities.

Potential exploratory applications additionally include traversal analysis of irrational-number expansions such as $\pi$, where digit-specific traversal distributions and overlap structures may serve as structural fingerprints. Such fingerprints could facilitate comparison between numerical expansions and between numerical and other symbolic sequences.

7. Discussion

The ladder traversal framework recasts repeated symbolic structures as ordered progression through repetition states rather than as isolated substring occurrences. By imposing adjacency between rung orders and permitting overlapping embeddings, the framework provides a hierarchical representation of repetition structure within symbolic sequences.

A key consequence of this interpretation is that repeated structures may be analysed in terms of traversal organisation rather than solely through occurrence counts. This perspective naturally suggests traversal-derived quantities such as overlap density, traversal-gap distributions, directional asymmetry, and hierarchical traversal organisation, which may provide complementary information about sequence organisation.

At present, the framework should be regarded as a preliminary symbolic and computational model rather than a complete mathematical theory. Future work may include traversal-extraction algorithms, probabilistic traversal models, graph-theoretic representations, entropy-based measures, and statistical comparison of traversal behaviour within random and non-random symbolic systems.

The framework presently provides a descriptive formalism rather than a predictive or algorithmically optimised repetition theory. Its practical utility, computational properties, and statistical behaviour therefore remain subjects for future investigation.

8. Conclusion

This paper introduces a ladder-based traversal framework for representing repeated symbolic structures within symbolic and numerical sequences. The framework provides a simple formalism in which repetition is viewed as progression through ordered repetition states rather than solely as collections of repeated substrings.

By representing repeated structures as traversal paths, the framework offers an alternative perspective on symbolic organisation that incorporates hierarchy, overlap, and directional behaviour within a unified representation.

Although preliminary, the framework provides a foundation for further exploration of traversal-based symbolic analysis and the development of quantitative measures derived from traversal structure.