# A Discrete-Event Simulation of Claude Lévi-Strauss’ Structural Analysis of Myths Based on Symmetry and Double Twist Transformations

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## Abstract

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## 1. Introduction

## 2. Transformational Analysis of Myths Following Structural Methodology

#### 2.1. Introduction to Structural Analysis

#### 2.2. Understanding Claude Lévi-Strauss Method of Myth Analysis

- The ogre (named Orcu) knows the secret of the fabrication of Corsican cheese using milk.
- The shepherds are jealous of the Orcu because of this secret.
- The ogre is captured because of an ingenious trap.
- The shepherds ask for the secret of the fabrication of the cheese.

- 1st mythem: $a=orcu$; $x=secret$
- 2nd mythem: $a=shepherd$; $x=jealous$
- 3rd mythem: $a=orcu$; $x=trapped$
- 4th mythem: $a=shepherd$; $x=secret$

- A homology between two terms a and b. The result of a homology between a and b consists in replacing term a by term b in each mythem in order to obtain a new generated mythical transformation. For example, considering the four mythems previously introduced, we are able to generate mythems of a new myth using a homology between the ogre and the Sybille. The result consists in replacing the term ogre by the term Sybille in all the mythems containing the ogre term: first and third mythems.
- Inversion. The result of an inversion of a term a consists in replacing the term a by term $1/a$ in each mythem of the given myth in order to obtain a new generated mythical transformation. For example, in the Corsican culture, the devil is often considered as having an inverse character to the ogre. So, we are able to generate a mythem of a new myth using an inversion of the terms ogre and devil. The result consists of replacing the term ogre by the term devil in all the mythems containing the ogre term: first and third mythems.
- Opposition. The result of an opposition of a term a consists in replacing term a by term ${a}^{-1}$ in each mythem of the given myth in order to obtain a new generated mythical transformation. For example, we can point to usual oppositions introduced by Claude Lévi-Strauss in the Mythologiques Series: fire = $wate{r}^{-1}$, Jaguar = $anteate{r}^{-1}$, vegetable = $anima{l}^{-1}$.
- Symmetry. The result of a symmetry of a term consists in replacing term a by term $-a$ in each mythem in order to obtain a new generated mythical transformation. For example, in some myths studied by Claude Lévi-Strauss, a menstruating woman is sterile. She becomes a non-woman (symmetry operation), but without being a man (opposition operation).
- The canonical formula transformation (also double twist transformation) leans on the algebraic expression given in Equation (1).

- Adding a mythem: this is a very simple transformation which consists in adding a mythem to a given myth. The new mythem is built with a term a and a function x.
- Removing a mythem: once again a very simple transformation which consists in removing a given mythem from a given myth.

#### 2.3. Applying Claude Lévi-Strauss Method of Myth Analysis

- To allow a user to define an initial myth of reference, defined the Myth M1 by Claude Lévi-Strauss.
- To allow the modeling of a given myth through a set of mythems. The software will allow a user to define a myth as an interconnection of mythems which is represented using two variables: a term usually noted a and a function usually noted x.
- To allow a user to generate a new myth from a given myth by performing a transformation selected through the following set of basic operations which have been presented and detailed previously (homology, inversion, symmetry, opposition, addition of a mythem, removal of a mythem) as well as the Canonical formula.
- To allow a user to visualize the set of transformations already performed using a graph representation. The graph should present the transformation relations between the myths. The nodes of the graph will represent the set of myths labeled by their associated number.

## 3. DEVS Formalism Based Software Framework

#### 3.1. Basic Concepts of the DEVS Formalism

- $X:\left\{(port,value)\right|(port\in InputPorts,\phantom{\rule{0.166667em}{0ex}}value\in {X}_{port})\}$: the set of input ports and values.
- $Y:\left\{(port,value)\right|(port\in OutputPorts,\phantom{\rule{0.166667em}{0ex}}value\in {Y}_{port})\}$:the set of output ports and values.
- S is the set of states.
- ${\delta}_{int}:S\to S$: the internal transition function that allows the system to go into the next state after the time returned by the time advance function.
- ${\delta}_{ext}:Q\times X\to S$: the external transition function that allows to schedule the states changes according to an external input event.
- $\lambda :S\to Y$: the output function that allows to generate external events on the outputs before the internal transition will be executed.
- ${t}_{a}:S\to {\mathbb{R}}_{0}^{+}$: the time advance function, that gives the life time of the current state.

- $Q=\left\{(state,elapsed)\right|state\in S,0<elapsed<{t}_{a}\left(s\right)\}$ is the total state set,
- $elapsed$ is the elapsed time since last transition, and $state$ the partial set of states for the duration of ${t}_{a}\left(state\right)$ if no external event occur.
- ${\delta}_{int}:$ the model being in a $state$ at ${t}_{i}$, it will move into $stat{e}^{\prime}$, ${s}^{\prime}={\delta}_{int}\left(state\right)$, if no external events occurs before ${t}_{i}+{t}_{a}\left(s\right)$.
- ${\delta}_{ext}:$ when an external event occurs, the model being in the state $state$ since the elapsed time $elapsed$ moves in ${s}^{\prime}$, The next state depends on the elapsed time in the present state. At every state change, $elapsed$ is reset to 0.
- $\lambda :$ the output function is executed before an internal transition and before emitting an output event, the model remains in a transient state.
- A state with an infinite life time is a passive state otherwise it is an active state (transient state). If the $state$ is passive, the model can evolve only when an input event occurs.

- X: the set of input ports for the reception of external events.
- Y: the set of output ports for the emission of external events.
- $COMP$: the set of components (coupled or basic models).
- ${M}_{d}$: the DEVS model for each $d\in COMP$.
- $EIC$: the set of input links, that connects the inputs of the coupled model to inputs of the components that it contains.
- $EOC$: the set of output links, that connects the outputs of contained components to the output of the coupled model.
- $IC$: the set of internal links, that connects the output ports of components to the input ports of components in the coupled models.

#### 3.2. The DEVSimPy Framework

## 4. Implementation of DEVS M+S of Claude Lévi-Strauss Myth Analysis

- Modeling of a given myth through a set of mythems. The software will allow a user to define a myth as an interconnection of mythems.
- Definition of an initial myth as descried by Claude Lévi-Strauss.
- Generation of a new myth from a given myth by performing a set of transformations belonging to several types of basic transformation (Section 2.2).
- Visualization of the set of transformations already performed using a graph representation.

#### 4.1. Myth Modeling Based on the Notion of Mythems

#### 4.2. Definition of an Initial Myth

#### 4.3. Generation of a New Myth from a Given Myth

#### 4.4. Visualization of the Set of Transformations Already Performed

## 5. Validation

- Lib_to_copy_to_Domain_dir which gives the directory containing all the python code dedicated to myths transformation. It must be extracted in the DEVSimPy Domain directory and added to the $\_\_all\_\_$ variable in the $\_\_init\_\_.py$
- M1-Bororo.dat which is the file containing the initial myth of the South America myths transformations. It must be used to initialized the $filename$ attribute of the M1-Bororo DEVS model
- MC1-U-Lurcu.dat which is the file containing the initial myth for the Corsica myths transformations. It must by used as the M1-Bororo model
- MythCLSv7.dsp which is the file to be selected in order to perform the South America myths transformations
- MythsCorsi2020.dsp which is the file to be selected to perform the Corsican myths transformations
- MythsCorsi2020.mp4 which gives a video retracing how to perform myths transformation.

## 6. Discussion

## 7. Future Applications

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Santucci, J.-F.; Doja, A.; Capocchi, L.
A Discrete-Event Simulation of Claude Lévi-Strauss’ Structural Analysis of Myths Based on Symmetry and Double Twist Transformations. *Symmetry* **2020**, *12*, 1706.
https://doi.org/10.3390/sym12101706

**AMA Style**

Santucci J-F, Doja A, Capocchi L.
A Discrete-Event Simulation of Claude Lévi-Strauss’ Structural Analysis of Myths Based on Symmetry and Double Twist Transformations. *Symmetry*. 2020; 12(10):1706.
https://doi.org/10.3390/sym12101706

**Chicago/Turabian Style**

Santucci, Jean-Francois, Albert Doja, and Laurent Capocchi.
2020. "A Discrete-Event Simulation of Claude Lévi-Strauss’ Structural Analysis of Myths Based on Symmetry and Double Twist Transformations" *Symmetry* 12, no. 10: 1706.
https://doi.org/10.3390/sym12101706