# Ladderpath Approach: How Tinkering and Reuse Increase Complexity and Information

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## Abstract

**:**

## 1. Introduction

## 2. Ladderpath Approach: Starting from Scratch

#### 2.1. A Thought Experiment

- ACXLGICXGOXEMZBRCNKXACXLPICXEMZBRCNKX.

#### 2.2. Definition of Generation-Operation

**basic set**, denoted as ${\mathbf{S}}_{0}=\{$A(0), B(0), C(0), D(0), E(0), F(0)} which is defined to be a special type of partially ordered multiset, i.e., its elements are partially ordered (the ordered parts are separated by double slash “⫽”, between which we can call “one

**level**”; and the unordered parts are separated by comma “,”), and the number of instances (called multiplicity) of each element is written in the brackets behind the element. Note that the basic set ${\mathbf{S}}_{0}$ should be predefined, according to the specific research problem currently at hand, after which the analysis can be conducted (see Section 4.1 for more discussions). In this specific case we defined the basic set to be constituted from single letters.

**building block**(block for short). The elements in the basic set are called

**basic (building) blocks**. In addition, the string we want to obtain, $\mathcal{X}$, is called the

**target (building) block**. Note that, in ${\mathbf{S}}_{0}$, as shown above, the basic blocks are all unordered and their multiplicities are all zeros. Then, we define an operation on this type of partially ordered multiset:

**Generation-operation**is defined as: take any number of blocks in the partially ordered multiset (the multiplicity decreases accordingly, but note that any block can be taken even if its multiplicity is zero or negative) and combine them in a certain way (note that the newly-generated blocks must not be present in the set), and then put the combined one back into the set at the level that is one level higher than the highest level of the constituted blocks. After this operation, a new partially ordered multiset of this type is obtained.

- Ex1.
- 1st generation-operation, ${\mathbf{S}}_{0}$:C + D = CD $\to {\mathbf{S}}_{1}$ = {A(0), B(0), C(−1), D(−1), E(0), F(0) ⫽ CD(1)}
- .
- 2nd, ${\mathbf{S}}_{1}$:B + CD = BCD $\to {\mathbf{S}}_{2}$ = {A(0), B(−1), C(−1), D(−1), E(0), F(0) ⫽ CD(0) ⫽ BCD(1)}
- .
- 3rd, ${\mathbf{S}}_{2}$:E + F = EF $\to {\mathbf{S}}_{3}$ = {A(0), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽ CD(0), EF(1) ⫽ BCD(1)}
- .
- 4th, ${\mathbf{S}}_{3}$:A + BCD + BCD + BCD + CD + EF + EF = $\mathcal{X}$$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{S}}_{4}$ = {A(−1), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽ CD(−1), EF(−1) ⫽ BCD(−2) ⫽$\mathcal{X}$(1)}
- .
- Lastly, take one $\mathcal{X}$ out from ${\mathbf{S}}_{4}$, and then we achieve our goal: obtained one target block $\mathcal{X}$. The last step can be considered one special generation-operation, that is, only take out but do not put back, which can be denoted as ${\mathbf{S}}_{4}$:$\mathcal{X}$(−1) $\to {\mathbf{S}}_{5}$ = {A(−1), B(−1), C(−1), D(−1), E(−1), F(−1)⫽CD(−1), EF(−1) ⫽ BCD(−2) ⫽$\mathcal{X}$(0)}

- Ex2.
- 1st generation-operation, ${\mathbf{S}}_{0}$:D + B + C = DBC $\to {\mathbf{S}}_{1}^{\prime}$ = {A(0), B(−1), C(−1), D(−1), E(0), F(0) ⫽ DBC(1)}
- .
- 2nd, ${\mathbf{S}}_{1}^{\prime}$:E + F = EF $\to {\mathbf{S}}_{2}^{\prime}$ = {A(0), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽ DBC(1), EF(1)}
- .
- 3rd, ${\mathbf{S}}_{2}^{\prime}$:A + B + C = ABC $\to {\mathbf{S}}_{3}^{\prime}$ = {A(−1), B(−2), C(−2), D(−1), E(−1), F(−1) ⫽ DBC(1), EF(1), ABC(1)}
- .
- 4th, ${\mathbf{S}}_{3}^{\prime}$:DBC + DBC = DBCDBC $\to {\mathbf{S}}_{4}^{\prime}$ = {A(−1), B(−2), C(−2), D(−1), E(−1), F(−1) ⫽ DBC(−1), EF(1), ABC(1) ⫽ DBCDBC(1)}
- .
- 5th, ${\mathbf{S}}_{4}^{\prime}$:C + D = CD $\to {\mathbf{S}}_{5}^{\prime}$ = {A(−1), B(−2), C(−3), D(−2), E(−1), F(−1) ⫽ DBC(−1), EF(1), ABC(1), CD(1) ⫽ DBCDBC(1)}
- .
- 6th, ${\mathbf{S}}_{5}^{\prime}$:ABC + DBCDBC + D + CD + EF + EF = $\mathcal{X}$$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{S}}_{6}^{\prime}$ = {A(−1), B(−2), C(−3), D(−3), E(−1), F(−1) ⫽ DBC(−1), EF(−1), ABC(0), CD(0) ⫽ DBCDBC(0) ⫽$\mathcal{X}$(1)}
- .
- Lastly, ${\mathbf{S}}_{6}^{\prime}$:$\mathcal{X}$(−1)$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{S}}_{7}^{\prime}$ = {A(−1), B(−2), C(−3), D(−3), E(−1), F(−1) ⫽ DBC(−1), EF(−1), ABC(0), CD(0) ⫽ DBCDBC(0) ⫽$\mathcal{X}$(0)}

#### 2.3. Definition of Ladderpath

- For a path ${\mathbf{S}}_{0}\to {\mathbf{S}}_{1}\to {\mathbf{S}}_{2}\to \cdots \to {\mathbf{S}}_{n}\Rightarrow \mathcal{X}$ through which we obtain the target block $\mathcal{X}$, we construct another partially ordered multiset in the following way: Take the final set ${\mathbf{S}}_{n}$, delete all of the blocks with zero multiplicity, and then set all other multiplicities to be the absolute value of the corresponding multiplicities (with the partial orders preserved). This procedure generates a new partially ordered multiset J, which we call the
**ladderpath**of $\mathcal{X}$ that corresponds to this particular path. - Any block in the ladderpath is called a
**ladderon**.

- Ex3.
- 1st generation-operation, ${\mathbf{S}}_{0}$:D + B + C = DBC $\to {\mathbf{S}}_{1}^{\prime}$ = {A(0), B(−1), C(−1), D(−1), E(0), F(0) ⫽ DBC(1)}
- .
- 2nd, ${\mathbf{S}}_{1}^{\prime}$:E + F = EF $\to {\mathbf{S}}_{2}^{\prime}$ = {A(0), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽ DBC(1), EF(1)}
- .
- 3rd, ${\mathbf{S}}_{2}^{\prime}$:A + B + C + DBC + DBC + D + C + D + EF + EF = $\mathcal{X}$$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{S}}_{3}^{\u2033}$ = {A(−1), B(−2), C(−3), D(−3), E(−1), F(−1) ⫽ DBC(−1), EF(−1) ⫽$\mathcal{X}$(1)}
- .
- Lastly, ${\mathbf{S}}_{3}^{\u2033}$:$\mathcal{X}$(−1) $\to {\mathbf{S}}_{4}^{\u2033}$ = {A(−1), B(−2), C(−3), D(−3), E(−1), F(−1) ⫽ DBC(−1), EF(−1) ⫽$\mathcal{X}$(0)}

- Any target sequence has at least one
**trivial ladderpath**in which all the ladderons are basic blocks.

- One ladderon is a block that has been reused (i.e., been part in a generation-operation at least twice). The reuse is the ultimate reason why we can simplify the process of making the target block (later we shall see that this type of simplification is closely related to the “information” contained in the target block).
- For any ladderpath ${J}_{\mathcal{X}}$,$${N}_{a}=\sum _{i\in {J}_{\mathcal{X}}}({m}_{i}\times {n}_{i,a})$$Take ${J}_{\mathcal{X},1}$ as an example. The letter B appears 3 times in the target sequence $\mathcal{X}$, i.e., ${N}_{\mathrm{B}}=3$. On the right hand side of Equation (3), the contribution of ladderon BCD is $2\times 1=2$ (as its multiplicity is 2 and the letter B appears once in BCD); the contribution of ladderon B is $1\times 1=1$ (as its multiplicity is 1 and the letter B appears once in B); and the contribution of other ladderons is zero. So the right hand side is also $2+1=3$. It is straightforward to see that Equation (3) holds for all other letters A, C, D, E and F, too. Likewise, we can also verify ladderpath ${J}_{\mathcal{X},2}$ and ${J}_{\mathcal{X},0}$.

**laddergraph**. An example is shown in Figure 1a, where the levels represent the order relationships in the ladderpath, and the links among blocks represent generation-operations (e.g., E and F link up to EF, representing that EF is generated from E and F).

- 1.
- Dim all the lines that are linked to the basic blocks.
- 2.
- If a block i at a lower level can be linked to a block j at a higher level via other blocks, then even if i and j are directly linked, we do not draw the lines between i and j. For example, in Figure 1a, the block CD at a lower level is directly linked to the target block $\mathcal{X}$ at the higher lower since in the ladderpath ${J}_{\mathcal{X},1}$, CD is directly involved when generating $\mathcal{X}$; but because CD is linked to BCD, and BCD is linked to $\mathcal{X}$, then we should not draw a line between CD and $\mathcal{X}$.
- 3.
- The multiplicities of blocks can either be written down explicitly (as in Figure 1) or not.

#### 2.4. Definition of Ladderpath-Index ($\lambda $), Order-Index ($\omega $), and Size-Index (S)

- For the string examples this paper describes, we define the
**length unit of a ladderpath**as follows: for each operation, that concatenates any two strings (namely, blocks), we associate a unit called a “**lift**” (note that since how building blocks are combined together might be different in different systems, the length unit of a ladderpath could be defined differently, yet we always term the unit as a “lift”).

- The
**length of a ladderpath**J, denoted as $\left|J\right|$, is the sum of the lengths of all generation-operations along the ladderpath J. Note that, the length of any path is thus naturally defined as the sum of the lengths of all generation-operations along this path.

- The
**ladderpath-index**of target block $\mathcal{X}$, denoted $\lambda \left(\mathcal{X}\right)$, is the length of the**shortest ladderpath(s)**of $\mathcal{X}$ (there may be one or several shortest ladderpaths). Thus, the length unit of ladderpath-index is also “lift”.

- The
**size-index**$S\left(i\right)$ of any block i (e.g., the target block or a ladderon) is the length of its shortest trivial ladderpath (there could be one or several shortest trivial ladderpaths, but they all have the same length).

- The
**order-index**$\omega \left(\mathcal{X}\right)$ of the target block $\mathcal{X}$ is defined to be:$$\omega \left(\mathcal{X}\right):=S\left(\mathcal{X}\right)-\lambda \left(\mathcal{X}\right),$$

- The length of a ladderpath ${J}_{\mathcal{X}}$ can be directly calculated from its partially ordered multiset representation, with no need to convert the ladderpath into one of its corresponding path. The length can be calculated as:$$|{J}_{\mathcal{X}}|=S\left(\mathcal{X}\right)-\sum _{i\in {J}_{\mathcal{X}}}{m}_{i}\xb7(S\left(i\right)-1),$$

- In fact, based on Equation (5) and the definition of the order-index, the order-index of the target block $\mathcal{X}$ can be readily calculated from the following formula:$$\omega \left(\mathcal{X}\right)=\sum _{i\in {J}_{\mathcal{X}}}{m}_{i}\xb7(S\left(i\right)-1),$$

#### 2.5. Ladderpath-Index and Order-Index Are Two Axes of “Complexity”

- The ladderpath-index $\lambda \left(\mathcal{X}\right)$ describes the amount of “information” that $\mathcal{X}$ carries, that is, how many extra steps/“costs” or how much extra “information” the external agent needs to input in order to generate $\mathcal{X}$, equivalent to the difficulties to reproduce $\mathcal{X}$ (which is distinct from the “information” that Shannon entropy or thermodynamic entropy refers to);
- The order-index $\omega \left(\mathcal{X}\right)$ describes how much “information” can be saved, i.e., the amount of redundant “information” (equivalently, the difference between the trivial ladderpath and the shortest ladderpath). This is consistent with the intuition, as the more steps/“costs” (namely, how many lifts) it saves, the more ordered the target block $\mathcal{X}$ is;
- Now we can see that “complexity” that we often intuitively talk about has two aspects: One is described by the ladderpath-index $\lambda $ which focuses more on the difficulties and costs of constructing the target; While the other aspect is described by the order-index $\omega $ which focuses more on how the target is built in an organized and hierarchical manner;
- Lastly, for a particular target (or target system), the sum of its ladderpath-index and its order-index is always equal to its size-index, which automatically solves the “normalization” problem that may occur when comparing different sized targets.

- 1.
- For patterns with the same size-index (e.g., [i], [iii] and [vi]), as $\omega $ increases, the pattern becomes more and more ordered, which is consistent with our intuitions. On the other hand, as the ladderpath-index $\lambda $ increases, the pattern becomes more and more difficult to reproduce (e.g., to reproduce [vi], we need to memorize the position of almost every stone, but to reproduce [i], we only need to memorize the description of the ladderpath, rather than the position of each stone), which is also consistent with our intuitions;
- 2.
- There are some counter-intuitive points indeed, yet they are also the most important: For example, [i] is more ordered than [ii], but the difficulty to generate either of them is the same, as their $\lambda $’s are identical. [i] is more ordered than [iii] and [vi], but it is less difficult and takes fewer lifts to generate [i]. [i] is more ordered than [v], yet not only is it less difficult to generate [i], the size of [i] is also larger than [v] (the same argument applies to [ii], compared with [iv], [v] and [vi]; and so on). We shall see later that this point is the very key to explaining why the emergence of life is not as difficult as imagined before.

#### 2.6. Extension of the Concept: The Ladderpath of a Whole System

**target system**for convenience (but note that these strings are not connected, which is distinct from the single string AABAABBCBCBCDDF). It deserves to mention that the ladderpath of a target system is different from the concept of the “molecular assembly tree” which only handles a group of distinct types of molecules [33].

- Ex4.
- 1st generation-operation, ${\mathbf{U}}_{0}$:A + B = AB $\to {\mathbf{U}}_{1}$ = {A(−1), B(−1), C(0), D(0), E(0) ⫽ AB(1)}
- .
- 2nd, ${\mathbf{U}}_{1}$:C + AB = CAB $\to {\mathbf{U}}_{2}$ = {A(−1), B(−1), C(−1), D(0), E(0) ⫽ AB(0) ⫽ CAB(1)}
- .
- 3rd, ${\mathbf{U}}_{2}$:AB + D = ABD $\to {\mathbf{U}}_{3}$ = {A(−1), B(−1), C(−1), D(−1), E(0) ⫽ AB(−1) ⫽ CAB(1), ABD(1)}
- .
- 4th, ${\mathbf{U}}_{3}$:ABD + ABD = ABDABD $\to {\mathbf{U}}_{4}$ = {A(−1), B(−1), C(−1), D(−1), E(0) ⫽ AB(−1) ⫽ CAB(1), ABD(−1) ⫽ ABDABD(1)}
- .
- 5th, ${\mathbf{U}}_{4}$:E + D = ED $\to {\mathbf{U}}_{5}$ = {A(−1), B(−1), C(−1), D(−2), E(−1) ⫽ AB(−1), ED(1) ⫽ CAB(1), ABD(−1) ⫽ ABDABD(1)}
- .
- 6th, ${\mathbf{U}}_{5}$:ABD + ED = ABDED$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{U}}_{6}$ = {A(−1), B(−1), C(−1), D(−2), E(−1) ⫽ AB(−1), ED(0) ⫽ CAB(1), ABD(−2) ⫽ ABDABD(1), ABDED(1)}
- .
- 7th, ${\mathbf{U}}_{6}$:ABDED + B + ED = ABDEDBED$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\to {\mathbf{U}}_{7}$ = {A(−1), B(−2), C(−1), D(−2), E(−1) ⫽ AB(−1), ED(−1) ⫽ CAB(1), ABD(−2) ⫽ ABDABD(1), ABDED(0)$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\u2afd$ ABDEDBED(1)}
- .
- Lastly, take all the blocks included in the target system $\mathcal{Q}$ from ${\mathbf{U}}_{7}$, and then the goal is achieved. The last step can be considered one special generation-operation, i.e., take out but do not put back, denoted as ${\mathbf{U}}_{7}$:$\mathcal{Q}$(−1) $\to {\mathbf{U}}_{8}$ = {A(−1), B(−2), C(−1), D(−2), E(−1) ⫽ AB(−1), ED(−4) ⫽ CAB(−1), ABD(−2) ⫽ ABDABD(0), ABDED(−1) ⫽ ABDEDBED(−1)}

#### 2.7. Algorithm to Compute the Shortest Ladderpaths

- 1.
- First, create an empty multiset $\mathscr{H}$ to store blocks in. Later the ladderpath can be readily computed from $\mathscr{H}$;
- 2.
- Starting from the target system $\mathcal{Q}$, we preserve only one instance of each type of distinct blocks in $\mathcal{Q}$, and put all other repetitions into $\mathscr{H}$;
- 3.
- Keep slicing the blocks in $\mathcal{Q}$ (i.e., slicing strings into multiple substrings or letters) in a pre-determined systematic manner, until in $\mathcal{Q}$ there are at least two substrings (or letters) that are identical. Preserve only one of such identical substrings in $\mathcal{Q}$ and put all other repetitions into $\mathscr{H}$. Note that there could be many systematic manners to slice the string, referring to Appendix D for an example;
- 4.
- Repeat from step 3, until no repetitive substrings or letters can be found in $\mathcal{Q}$. Then, cut all of the remaining substrings into basic blocks (i.e., single letters in this case), and put them all into $\mathscr{H}$. Now, $\mathscr{H}$ records one ladderpath;
- 5.
- The final step is to align the (sub)strings in $\mathscr{H}$ in the right hierarchical order (that is, based on how they are sliced: which is the parent and which are the children), which results in the partially ordered multiset representation of this ladderpath.

- In step 3, we bias the algorithm to search for repetitive substrings of maximum length.

## 3. Ladderpath’s Significance in Evolution: Ladderpath-Systems

- we call a system as a
**ladderpath-system**if it satisfies the two conditions: (i) the ability to generate new blocks, and (ii) some or all blocks can replicate;

## 4. Discussions

**isolated**and

**non-isolated /united**systems.

#### 4.1. On Information: Alien Signals (Isolated System)

**isolated system**, because there are no other signals or sequences that can be considered together with $\mathcal{Y}$.

#### 4.2. On Information: Human Language (Non-Isolated/United System)

**non-isolated /united system**) where even though a word did not repeat in the target sentence, it repeats in this non-isolated system, and we thus must consider these words as basic blocks.

#### 4.3. On Origins of Life

- The generation-operation for molecules is defined as follows: combine several molecular structures or fragments (namely, the blocks) into one, whereas the combination means that chemical bonds between atoms are formed;
- The length unit of a ladderpath is defined to be one chemical bond formed. That is, if n chemical bonds are formed in one generation-operation, the length of this generation-operation is n lifts.

#### 4.4. On Why Life Is Ordered

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Glossary

Name: | Refer to: | Description or Definition: | |
---|---|---|---|

Basic set | Section 2.2 | The partially ordered multiset that contains all the basic elements that constitute an object, denoted as ${\mathbf{S}}_{0}$. | |

Level | Section 2.2 | It refers to the partial order in the partially ordered multiset. Each level is separated by double slash “⫽”. | |

⋆ | (Building) block | Section 2.2 | Any element in the partially ordered multiset. |

Basic (building) block | Section 2.2 | Any element in the basic set ${\mathbf{S}}_{0}$. | |

Target (building) block | Section 2.2 | The object to generate in the end, denoted as $\mathcal{X}$. | |

⋆ | Generation-operation | Section 2.2 | A specific operation applied on the partially ordered multiset. Details in the main text. |

⋆ | Ladderpath | Section 2.3 | A sequence of generation-operations that generate the target block $\mathcal{X}$ in the end, which can be represented by a partially ordered multiset, denoted as ${J}_{\mathcal{X}}$. Details in the main text. |

⋆ | Ladderon | Section 2.3 | Any block in the ladderpath is called a ladderon. |

Trivial ladderpath | Section 2.3 | The particular ladderpath(s) in which all of the ladderons are the basic blocks. | |

⋆ | Laddergraph | Section 2.3 | A ladderpath can also be represented by a graph or network (besides a partially ordered multiset), called a laddergraph. |

⋆ | Ladderpath-index ($\lambda $) | Section 2.4 | The length of the shortest ladderpath(s) of an object. |

⋆ | Size-index (S) | Section 2.4 | The length of the shortest trivial ladderpath(s) of an object. |

⋆ | Order-index ($\omega $) | Section 2.4 | Defined as $(S-\lambda )$, referring to Equation (4). |

Length unit of a ladderpath | Section 2.4 | A system-dependent quantity. In the string examples in this paper, it is defined as carrying out the action that concatenates any two strings, once. | |

Length of a ladderpath | Section 2.4 | The sum of the lengths of all generation-operations along the ladderpath. | |

⋆ | lift | Section 2.4 | The name of the length unit of the ladderpath. |

Shortest ladderpath | Section 2.4 | The ladderpath(s) that has the minimum length. | |

⋆ | Target system | Section 2.6 | A group of target blocks that need to be generated altogether in the end. |

⋆ | Ladderpath-system | Section 3 | A system that satisfies the two conditions: (i) the ability to generate new blocks, and (ii) some or all blocks can replicate. |

Isolated system | Section 4.1 | We call a system an isolated system if we force it to have nothing to do with any other systems. | |

Non-isolated (/united) system | Section 4.2 | It may not be possible to isolate a system. For example, when considering one single sentence, we should consider it in the context of the language it belongs to, so, this sentence and the language altogether form a united system. |

## Appendix B. Examples of the Ladderpath of Strings

- Ex.B1.
- 1st generation-operation, ${\mathbf{R}}_{0}$: A + B + C + D + E + F = $\mathcal{A}$$\to {\mathbf{R}}_{1}$ = {A(−1), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽$\mathcal{A}$(1)}
- .
- The last generation-operation, ${\mathbf{R}}_{1}$: $\mathcal{A}$(−1) $\to {\mathbf{R}}_{2}$ = {A(−1), B(−1), C(−1), D(−1), E(−1), F(−1) ⫽$\mathcal{A}$(0)}

- Ex.B2.
- 1st generation-operation, ${\mathbf{R}}_{0}$: A + A = AA $\to {\mathbf{R}}_{1}$ = {A(−2) ⫽ AA(1)}
- .
- 2nd, ${\mathbf{R}}_{1}$: AA + AA = AAAA $\to {\mathbf{R}}_{2}$ = {A(−2) ⫽ AA(−1) ⫽ AAAA(1)}
- .
- 3rd, ${\mathbf{R}}_{2}$: AAAA + AAAA = $\mathcal{B}$$\to {\mathbf{R}}_{3}$ = {A(−2) ⫽ AA(−1) ⫽ AAAA(−1) ⫽$\mathcal{B}$(1)}
- .
- Lastly, ${\mathbf{R}}_{3}$: $\mathcal{B}$(−1) $\to {\mathbf{R}}_{4}$ = {A(−2) ⫽ AA(−1) ⫽ AAAA(−1) ⫽$\mathcal{B}$(0)}

- Ex.B3.
- 1st generation-operation, ${\mathbf{R}}_{0}$: A + A + A = AAA $\to {\mathbf{R}}_{1}$ = {A(−3) ⫽ AAA(1)}
- .
- 2nd, ${\mathbf{R}}_{1}$: AAA + AAA = AAAAAA $\to {\mathbf{R}}_{2}$ = {A(−3) ⫽ AAA(−1) ⫽ AAAAAA(1)}
- .
- 3rd, ${\mathbf{R}}_{2}$: A + A + AAAAAA = $\mathcal{B}$$\to {\mathbf{R}}_{3}$ = {A(−5) ⫽ AAA(−1) ⫽ AAAAAA(0) ⫽$\mathcal{B}$(1)}
- .
- Lastly, ${\mathbf{R}}_{3}$: $\mathcal{B}$(−1) $\to {\mathbf{R}}_{4}$ = {A(−5) ⫽ AAA(−1) ⫽ AAAAAA(0) ⫽$\mathcal{B}$(0)}

- Ex.B4.
- 1st generation-operation, ${\mathbf{R}}_{0}$: T + G = TG $\to {\mathbf{R}}_{1}$ = {A(0), T(−1), G(−1), C(0) ⫽ TG(1)}
- .
- 2nd, ${\mathbf{R}}_{1}$: A + TG = ATG $\to {\mathbf{R}}_{2}$ = {A(−1), T(−1), G(−1), C(0) ⫽ TG(0) ⫽ ATG(1)}
- .
- 3rd, ${\mathbf{R}}_{2}$: ATG + TG + C + ATG = $\mathcal{B}$$\to {\mathbf{R}}_{3}$ = {A(−1), T(−1), G(−1), C(−1) ⫽ TG(−1) ⫽ ATG(−1) ⫽$\mathcal{C}$(1)}
- .
- Lastly, ${\mathbf{R}}_{3}$: $\mathcal{C}$(−1) $\to {\mathbf{R}}_{4}$ = {A(−1), T(−1), G(−1), C(−1) ⫽ TG(−1) ⫽ ATG(−1) ⫽$\mathcal{C}$(0)}

- Ex.B5.
- 1st generation-operation, ${\mathbf{R}}_{0}$: X + Y = XY $\to {\mathbf{R}}_{1}$ = {X(−1), Y(−1), Z(0) ⫽ XY(1)}
- .
- 2nd, ${\mathbf{R}}_{1}$: XY + XY + XY + Z + XY + Y = $\mathcal{K}$$\to {\mathbf{R}}_{2}$ = {X(−1), Y(−2), Z(−1) ⫽ XY(−3) ⫽$\mathcal{K}$(1)}
- .
- Lastly, ${\mathbf{R}}_{2}$: $\mathcal{K}$(−1) $\to {\mathbf{R}}_{3}$ = {X(−1), Y(−2), Z(−1) ⫽ XY(−3) ⫽$\mathcal{K}$(0)}

- Ex.B6.
- 1st generation-operation, ${\mathbf{R}}_{0}$: A + U = AU $\to {\mathbf{R}}_{1}$ = {A(−1), U(−1), G(0), C(0) ⫽ AU(1)}
- .
- 2nd, ${\mathbf{R}}_{1}$: C + AU = CAU $\to {\mathbf{R}}_{2}$ = {A(−1), U(−1), G(0), C(−1) ⫽ AU(0) ⫽ CAU(1)}
- .
- 3rd, ${\mathbf{R}}_{2}$: G + G + G + CAU + CAU + AU + AU + AU + G = $\mathcal{L}$$\to {\mathbf{R}}_{3}$ = {A(−1), U(−1), G(−4), C(−1) ⫽ AU(−3) ⫽ CAU(−1) ⫽$\mathcal{L}$(1)}
- .
- Lastly, ${\mathbf{R}}_{3}$: $\mathcal{L}$(−1) $\to {\mathbf{R}}_{4}$ = {A(−1), U(−1), G(−4), C(−1) ⫽ AU(−3) ⫽ CAU(−1) ⫽$\mathcal{L}$(0)}

## Appendix C. Compute the Ladderpaths of Stone Patterns

**Figure A1.**(

**a**) The sequence of generation-operations that generate the target stone pattern [i] in the end. Each arrow on the left represents one generation-operation. (

**b**) Likewise, the sequence of generation-operations that generate the target stone pattern [ii]. (

**c**) It shows the six stone patterns, with the irregular part highlighted. The red color means that these stones are positioned in an irregular manner, and thus, those stones have to be added one by one. For example, for stone pattern [iii], after the black part has been generated, we need six more generation-operations each of which adds one “red” stone.

## Appendix D. Example to Illustrate the Algorithm for the Shortest Ladderpaths

- i.
- First create an empty multiset $\mathscr{H}$ to store blocks, and later the ladderpath can be readily computed from $\mathscr{H}$;
- ii.
- Starting from the target system $\mathcal{Q}$, we preserve only one instance of each type of distinct blocks in $\mathcal{Q}$, and put all other repetitions into $\mathscr{H}$. So we have $\mathcal{Q}=\{ABDEDBED,ABDED,ABDABD,CAB,ED\}$ and $\mathscr{H}$ = {ABDEDBED, CAB, ED(2)} (if there is no bracket behind a block, it means the multiplicity of this block is 1);
- iii.
- Then, in the 3rd step, we keep slicing the blocks in $\mathcal{Q}$ in a pre-determined systematic manner, until in $\mathcal{Q}$ there are at least two substrings that are identical. There could be many systematic manners to slice strings, one of which is to treat the positions where slicing could happen as binary numbers and consecutively count. For example, for string $abcd$, there are three positions where slicing could happen, so the consecutive counting of binary numbers is 001, 010, 011, 100, 101, 110, 111, which can be considered a sequence of an inclusive slicing scheme where 1 denotes for slicing and 0 denotes otherwise. Thus, the sequence of a slicing scheme for $abcd$ is as follows: $abc|d$, $ab|cd$, $ab\left|c\right|d$, $a|bcd$, $a\left|bc\right|d$, $a\left|b\right|cd$, $a\left|b\right|c|d$.In this case, when we count until 0000100, namely slicing ABDEDBED into ABDED and BED, we find that there are two ABDED in $\mathcal{Q}$. So, we put one ABDED into $\mathscr{H}$, and then we have $\mathcal{Q}$ = {BED, ABDED, ABDABD, CAB, ED} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(2), ABDED};
- iv.
- Repeat step 3. We can slice BED into B and ED, and also ABDED into ABD and ED. Then we find that there are three ED in $\mathcal{Q}$. So, we put two ED into $\mathscr{H}$, and then we have $\mathcal{Q}$ = {B, ABD, ABDABD, CAB, ED} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED};
- v.
- Repeat step 3. We can slice ABDABD into ABD and ABD. Then we find that there are three ABD in $\mathcal{Q}$. So, we put two ABD into $\mathscr{H}$, and then we have $\mathcal{Q}$ = {B, ABD, CAB, ED} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED, ABD(2)};
- vi.
- Repeat step 3. We can slice ABD into AB and D, and also CAB into C and AB. Then we find that there are two AB in $\mathcal{Q}$. So, we put one AB into $\mathscr{H}$ and then we have $\mathcal{Q}$ = {B, D, C, AB, ED} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED, ABD(2), AB};
- vii.
- Repeat step 3. We can slice AB into A and B. Then we find that there are two B in $\mathcal{Q}$. So, we put one B into $\mathscr{H}$, and then we have $\mathcal{Q}$ = {B, D, C, A, ED} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED, ABD(2), AB, B};
- viii.
- Repeat step 3. We can slice ED into E and D. Then we find that there are two D in $\mathcal{Q}$. So, we put one D into $\mathscr{H}$, and then we have $\mathcal{Q}$ = {B, D, C, A, E} while $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED, ABD(2), AB, B, D};
- ix.
- Then we cannot find repetitive letters or strings anymore. Now we put all of the remaining letters into $\mathscr{H}$, and then we have $\mathscr{H}$ = {ABDEDBED, CAB, ED(4), ABDED, ABD(2), AB, B(2), D(2), C, A, E}, while $\mathcal{Q}$ becomes empty.

## Appendix E. Looking for the Evidence of Intelligent Life

- 1.
- The order-index $\omega $ of life is very high; intelligent life including the things created by intelligent life has an even higher order-index; and the more intelligent, the higher the order-index (although intelligent life can also create things with low order-indices, it is the upper limit that we are considering here). The reason is that $\omega $ describes how different the system is from the systems that are completely generated by random processes, that is, the larger $\omega $ is, the more different the system is from random systems; and the involvement of life and intelligent life must deviate the system from randomness;
- 2.
- However, the system with a high order-index $\omega $ may not always be life, intelligent life, or things created by intelligent life (e.g., crystals produced by simple chemical/physical processes are very much ordered but not life);
- 3.
- The ladderpath-index $\lambda $ (equivalent to the costs or the difficulties to generate/reproduce the system) of life is relatively high; intelligent life including the things created by intelligent life has an even higher ladderpath-index; and the more intelligent, the higher the ladderpath-index (likewise, we only consider the upper limit here). The reason is that, as discussed in the last section, ladderpath-systems automatically evolve more and more “complex” (in both aspects, i.e., increasing order-index and ladderpath-index);
- 4.
- However, the system with a high ladderpath-index $\lambda $ may not always be life, intelligent life, or things created by intelligent life (e.g., the ladderpath-index of a random system is high, but it is not life);
- 5.
- The four points above infer that if a system is life, intelligent life, or created by intelligent life, its order-index $\omega $ and ladderpath-index $\lambda $ should be both high, and vice versa (but as for how high can be considered high, that is another question).

**Figure A2.**The imagined five photos of stone patterns, on the surfaces of five extraterrestrial planets. Each black dot represents a stone.

**Table A2.**The three indices of the shortest ladderpaths of the five corresponding stone patterns in the figure above.

$\left(a\right)$ | $\left(b\right)$ | $\left(c\right)$ | $\left(d\right)$ | $\left(e\right)$ | |

S | 8 | 48 | 8 | 48 | 48 |

$\lambda $ | 8 | 13 | 4 | 9 | 7 |

$\omega $ | 0 | 35 | 4 | 39 | 41 |

- $\left(f\right)$
- We found an airplane (denoted $\mathcal{F}$) on another planet, which is exactly the same as our airplane on Earth, even the positions of the windows are the same;
- $\left(g\right)$
- We found a UFO-like thing, denoted $\mathcal{G}$, but we have absolutely no idea whether it is an aircraft. It looks like a disc, with a similar size to an airplane, with some patterns on the surface, with a door-like thing but no clue how to get in;
- $\left(h\right)$
- We found two exact $\mathcal{G}$ on another planet.

_{rep}} (where $\mathcal{F}$

_{rep}is the repetitive part of $\mathcal{F}$), so its order-index is ${\omega}_{f}^{\prime}=S\left({\mathcal{F}}_{rep}\right)-1\doteq S\left({\mathcal{F}}_{rep}\right)$ (according to Equation (6)). However, we should consider $\mathcal{F}$ and airplanes on Earth as a united system since $\mathcal{F}$ is repetitive in our experiences, indeed. So, $\mathcal{F}$ itself is a ladderon and its ladderpath is thus {basic blocks ⫽ $\mathcal{F}$

_{rep}⫽ $\mathcal{F}$} (similar to the scenario where we find two identical $\mathcal{F}$), and (f)’s order-index is thus ${\omega}_{f}\doteq S\left({\mathcal{F}}_{rep}\right)+S(\mathcal{F}).$ However, in case (g), we can only consider $\mathcal{G}$ as an isolated system, since we do not know what it is (thus we cannot consider it and the airplanes on Earth as a united system). So, its ladderpath can only be written as {basic blockswe ⫽ $\mathcal{G}$

_{rep}} (where $\mathcal{G}$

_{rep}is the repetitive parts of $\mathcal{G}$), and its order-index is ${\omega}_{g}\doteq S\left({\mathcal{G}}_{rep}\right).$ In case (h), we do not know what $\mathcal{G}$ is neither, but as $\mathcal{G}$ is repetitive in the system itself, its ladderpath is thus {basic blocks⫽ $\mathcal{G}$

_{rep}⫽ $\mathcal{G}$}, and its order-index is ${\omega}_{h}\doteq S({\mathcal{G}}_{rep})+S(\mathcal{G})$.

_{rep}) = S($\mathcal{G}$

_{rep}), then we have ${\omega}_{f}={\omega}_{h}>{\omega}_{g}$ and ${\lambda}_{f}={\lambda}_{h}={\lambda}_{g}$. Then, (f) and (h) are very similar cases, which would be more likely to be caused by intelligent life than (g). Nevertheless, note that if ω

_{g}and λ

_{g}are already very large, that is, the order-index and ladderpath-index of $\mathcal{G}$ are already larger than the “thresholds” that might be caused by intelligent life, inferred from other sources, it is still possible that (g) is also caused by intelligent life.

- 1.
- On another planet, we find a whole set of processes (or traces of processes) that follow the ladderpath mechanism, while the processes are completely different from what happened on Earth (e.g., silicon-based life, immersed in liquid methane);
- 2.
- We try to look for high-$\lambda $ objects (namely, difficult to make) that look like things in our own world (such as man-made airplanes, architectures with complex shapes but similar to ours, complex metabolic activities but similar to ours, complex machines but similar with ours), which is analogous to case $\left(f\right)$. As long as we find a high-$\lambda $ object similar to the counterpart in our own world (even if we only find one such object), we can treat it as a ladderon (since it is repetitive in this united system). Therefore, both the order-index and the ladderpath-index of this united system are high, which implies that it is the result of a civilization (However, if we and they are independent civilizations, this scenario is very unlikely because the probability that two independent civilizations emerged from literally an infinite number of possible civilizations happen to be identical is practically zero);
- 3.
- We find only one single machine (or the relic of one machine), yet it has many repetitive and hierarchical structures inside, resulting in both its order-index and ladderpath-index being very high. In this case, it is also very likely that this machine was made by an alien civilization;
- 4.
- Of course, we cannot exclude the last possibility: we find a large number of identical high-$\lambda $ objects that have, however, no repetitive structure inside (i.e., each object has a very low order-index). This case is also very likely to be an alien civilization, because on one hand, as the number of objects is very large (the object is thus ladderon), the order-index of the whole system is very large, while on the other hand, as the object has a very high $\lambda $, the ladderpath-index $\lambda $ of the whole system will be also very high. However, the probability that this type of civilization emerges is almost zero (which does not follow the ladderpath mechanism), just as the metaphor of the junkyard tornado, as discussed in Section 4.3.

## Appendix F. Connections between the Ladderpath and Shannon Entropy

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**Figure 1.**(

**a**) The laddergraph that corresponds to ladderpath ${J}_{\mathcal{X},1}$. (

**b**) The laddergraph that corresponds to ladderpath ${J}_{\mathcal{X},2}$. Grey blocks represent target blocks. “[$\times 1$]” in front of the target block represents that “we need to obtain 1 of such target block in the end”. If $\left(n\right)$ is added behind a block, it means the multiplicity (in the partially ordered multiset representation) of this block is n, while if there is no $\left(n\right)$ behind, it means the multiplicity is 1. Finally, $\left(0\right)$ behind the target block means that its multiplicity in this ladderpath is 0 (in principle, we should not draw blocks if their multiplicities are 0’s, but here we explicitly drew the target block, just in order to show the readers the hierarchical relationships between it and other blocks).

**Figure 2.**(

**a**) The relationships among the ladderpath-index $\lambda $, the order-index $\omega $ and the size-index S. The blue diagonals are the contour lines of S. The points [i]–[vi] correspond to the patterns in (

**b**) (note that one coordinate could correspond to an infinite number of patterns), and the coordinates are $(6,12),(6,7),(11,7),(6,2),(11,2),(16,2)$, respectively. (

**b**) The patterns corresponding to the six coordinates in (

**a**). (

**c**) The ladderpaths of the six patterns.

**Figure 3.**The laddergraph representation of one ladderpath of the target system $\mathcal{Q}$ (this ladderpath ${J}_{\mathcal{Q}}$ is actually the shortest one for $\mathcal{Q}$). All of the grey blocks constitute the target system $\mathcal{Q}$. “[$\times n$]” in front of the grey blocks represents that there are n such blocks included in the target system $\mathcal{Q}$. If $\left(n\right)$ is added behind a block, it means the multiplicity (in the partially ordered multiset representation) of this block is n, while if there is no $\left(n\right)$ behind, it means the multiplicity is 1. Finally, $\left(0\right)$ means that its multiplicity in this ladderpath is 0 (in principle, we should not draw blocks if their multiplicities are 0’s, but here we explicitly drew them, just in order to show the readers the hierarchical relationships among important blocks).

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Liu, Y.; Di, Z.; Gerlee, P. Ladderpath Approach: How Tinkering and Reuse Increase Complexity and Information. *Entropy* **2022**, *24*, 1082.
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**AMA Style**

Liu Y, Di Z, Gerlee P. Ladderpath Approach: How Tinkering and Reuse Increase Complexity and Information. *Entropy*. 2022; 24(8):1082.
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**Chicago/Turabian Style**

Liu, Yu, Zengru Di, and Philip Gerlee. 2022. "Ladderpath Approach: How Tinkering and Reuse Increase Complexity and Information" *Entropy* 24, no. 8: 1082.
https://doi.org/10.3390/e24081082