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Syntactic and Semantic Relationships in Models of Complex Systems: An Ecological Case

José Luis Usó-Doménech, Josué-Antonio Nescolarde-Selva*, Miguel Lloret-Climent

Department of Applied Mathematics, University of Alicante, Alicante, Spain

Abstract: In this paper, the authors extend and generalize the methodology based on the dynamics of systems with the use of differential equations as equations of state, allowing that first order transformed functions not only apply to the primitive or original variables, but also doing so to more complex expressions derived from them, and extending the rules that determine the generation of transformed superior to zero order (variable or primitive). Also, it is demonstrated that for all models of complex reality, there exists a complex model from the syntactic and semantic point of view. The theory is exemplified with a concrete model: MARIOLA model.

Keywords: complex system, flow equations, mathematical model, seme, sememe, state equations, transformed function

Received June 17, 2015; Revised July 15, 2015; Accepted August 26, 2015

Citation: José Luis Usó-Doménech, Josué-Antonio Nescolarde-Selva, Miguel Lloret-Climent, Syntactic and Semantic Relationships in Models of Complex Systems: An Ecological Case. American Journal of Systems and Software, 3 (4). 73-82. January 2015. doi:10.12691/ajss-3-4-1

1. Introduction

We assume [11] that the dynamics of the system can be modelled starting off with a set of ordinary non-lineal differential equations as follows:

[picture] | (1)

x and y being functions of in , where is the phase space and t the time and y the variable of state. Therefore there will be formulated a system of differential ordinary not linear equations:

[picture] | (2)

Being the flow variable which produces the variable of state .

Each of the variables of flow can depend simultaneously on the variables of entry or on the variables of state. We will call "w" to the set formed by the variables of entry and those of condition and will identify it as a subset opened of Rn. It is possible to write it of the following way:

[picture] | (3)

A particular modeling methodology has been exposed by the authors in previous works [11] and successfully used in specific ecological models [10]. A linguistic theory of this methodology has also been developed by the authors [6,7,13,14,15]. In this methodology, the flow equation is expressed by a linear combination of transformed functions [10]. In this methodology, the flow equation It is expressed as a linear combination of transformed functions, which are generated from the generative grammar [6], where the functions of the same order transformed or greater than two, are generated from transformed function of the first order applied to variables or primitive, belonging to the model. These functions depend on the modeller, and the flow equation takes the form, according to the criteria of recognizability, adopted by the model builder.

[picture] | (4)

Syntax refers to relationships between the sign and words of formal system, while semantics refers to relationships between expressions of the said formal system and the objects expressed.

2. Monoadic and Polyadic Symbols

In the first place, we will point out that not all attributes of Ontological Reality can be represented by numerical values, only some. An attribute can be measurable if we find some way of deciding the equality and addition of being, so that said rules are applied. To do that, we must assume specific laws that have to be true so that the attributes are measurable. The process to find out if an attribute is measurable and to establish a procedure to measure it is based entirely on experimental investigation. To all measurable attribute of an entity belonging to an Ontological system, it will be called variable, or primitive monoad.

Let be an open connected space in the complex plane [2]: we will call H the set of all analytical functions over which have a ring structure with the operations of addition and product. For , A(F) will denote the subring generated by F and A*(F) will be the set of analytical functions which depend algebraically on some subset of F.

1.-

2.- If then f', Ef, Pf Lf E being f' the derivative, Ef the exponential of f, Pf a primitive and Lf the logarithm of f.

3.- If f, g E, then f + g, f/g, (g ≠ 0), fg (that is exp(g log f)), are elements of E.

4.- If f, g E y f(Ω) Ω then g f E.

Definition 1: Given F H we will call a transformed function [11] of order n to any function as the following one : f1o f2 o......o fn with fi F , i=1,2,..,n.

The set with w F, will be the set of primitive monads and will denote in symbolic representation as . The application of an analytical function over a primitive monads define the transformed of order 1 or first order monoad: fi i = 0 1,...,n and will denote in symbolic representation as . The composition or recursion between two first order monoad will be called the transformed of order 2 or second order monoad: ; i,j = 1,2,...,n. and will denote in symbolic representation as and so on.

In general, a transformed of order n or n-order monoad which is a composition of n elements of, j=1,2,..,n will be expressed as: and will denote in symbolic representation as .

Definition 2: We define as associative field of a primitive monoad x and we called , the set constituted by all possible symbols of said measurable attribute:

The set will be a number set. In the practical tool, it will be a requisite to define one subset whose cardinal will be an integer number. will be called M-Vocabulary of first order of primitive monoad x and will be designed as .

Let r be the number of first order monads . Said number m is arbitrary, that is to say, it depends on the modeller. The cardinal of vocabulary will be being r being the number of first order monads and n the order of the monads.

Definition 3: The expression

where is the product, quottion or composition and m the maximun order of monoad, will be called a dyad. We define M-vocabulary of order two as formed by dyads.

Definition 3: The expression

will be called a tryad. We define M-vocabulary of order three as formed by tryads.

Definition 4: The expression

will be

The analytical expression of polyadic symbols of zero order is:

[picture] | (5)

Where v is the number of elementary variables, aij are real coefficients, hj is the maximum exponent admitted with step h, established by the modeller.

The set i=1,2....v; j=1,2....p with F, will be the set of composed symbols of zero order (complex variables) and will denote in symbolic representation as .

Consequence 1: An elementary variable is a special case of complex variable if and only if the multiplicative coefficients and the exponents of all elementary variables that compose it are zero except a term whose multiplicative coefficient and its exponent is equal to unity.

Definition 5: Given F Hwe will call a complex variable or polyad of zero order to any function as the following one : ω₁₁ ω₁₂ ...... with F , i=1,2,..,v and j = 1,2,3,......p , where is the product, quottion or composition.

3. The Syntactic System

Definition 6: We shall call the set L of all simple vocabularies of any order Simple Lexicon and we shall denote as L.

Definition 7: The set of variables is a subset of simple Lexicon L.

This set will be the set of elementary variables and each element of this set is a elementary variable. A elementary variable is a symbol belonging to a simple Lexicon with the property of being measurable.

For linguistic simple vocabulary let . We say that is related linguistically to and we will call it if and only if and .

Simple linguistic relationships among monoadic symbols of different simple vocabularies can be defined. Let be two monoadic symbols. We say that is related linguistically to in a second order simple relationship and we call it if and only if and .

Let be three monoadic symbols. We say that ,, are related linguistically in an third order simple relationship, and we call it , if and only if

and .

Let . We say that are related linguistically in an n-order simple relationship and we call it if and only if

and .

We will call the whole of all simple linguistic relationships

In general, a simple linguistic relationship between different order simple vocabularies can be defined.

Let be simple vocabularies of n, m,...,l orders, respectively.

We say that are related linguistically and we will call it

if and only if simple vocabulary exists so that

where

Let be the whole of all simple relationships between simple vocabularies

Let S =(T, R) be a system and a model system of said system. Let L be the simple lexicon of and in it a group of simple relationships is defined. At the same time let be the simple lexicon system associated with the model system or simply the Lexicon-Model System (LMS). Since the whole of simple linguistic relationships in L, is formed by mathematical relationships between monoadic symbols belonging to L, we shall call the Elementary Mathematical Model of S =(T, R).

Definition 8: We define as associative field of a composed variable and we called the set constituted by all possible composed symbols of said composed variable:

[picture] | (6)

[picture] | (7)

The value of m is subjective, depending on the modeller.

Each compound symbol is formed from four sums with terms, or each may have terms, being v the number of simple zero-order symbols (variables), and n is the order of the transformed function.

V 'and C' are variations and combinations with repetition respectively.

The set will be a number set. In the practical toll, it will be a requisite to define one subset whose cardinal will be an integer number.

The cardinal of is

[picture] | (8)

We call composed vocabulary of first order. Each element of that set is called a composed symbol.

Definition 9: We define sign vocabulary as the one formed by signs. and it will be written a element of by .

Definition 10: We defined composed vocabulary of order two

[picture] | (9)

If we want a short notation, it could be denoted by =ij .

Definition 11: We define simple vocabulary of order three the one formed by:

[picture] | (10)

It will be denoted the operation of three elements of simple vocabulary order one, in a short notation, by .

Definition 12: We define simple vocabulary of order n the one formed by:

[picture] | (11)

A short notation would be

4. The Semantic System

Definition 13: A semantic field [9] is a part of the vocabulary closely associated, where each particular sphere is divided, classified and organised so that the elements contribute to define their surroundings.

Each one of the symbols of Vocabulary Vx, can be considered as a sememe from the point of view of meaning. The seme is the smallest unit of meaning recognized in semantics, refers to a single characteristic of a sememe

Definition 14: The seme is meaning’s distinctive feature [8]. It is the primary unit or quantum of significance, not susceptible to independent fulfilment, and aggregated into a semantic configuration or sememe.

Definition 15: A sememe is defined [8] as a set of semes. The lexeme’s significance contents would be its sememe. The symbol will be a sememe from the Semantic point of view.

For example, the transformed third order function exp(sin(x)) will be a sememe exp(sin(x)), with x, sin, exp semes.

Definition 16: All symbols of a syntactic vocabulary Vx, can be considered the Semantic Field of a measurable symbol x.

Let Sm = (Tm, Rm) be a system and x Tm one primitive symbol. The symbol of zero order will have one semantic mean or seme only, that we denote . A symbol of first one will have two semes and. . For example sin x has two semes , etc.

Let m be the cardinal set of first order symbols . The number m is arbitrary, that is to say, it depends on the Modeller. In Table 1 the number of sememes and semes are specified.

Table 1. Set of semes of a symbol x

We can approach a semantic system from two points of view: the quantum level and the level of semes and the atomic level or the level of sememes.

For example (see in section 5) in sememe PORDT = 2.8949log(DBVFS)- 5.0052, DBVS is a seme of first order and log(DBVFS) a seme of second order formed for two seme: DBVFS and log. If coslogDBVFS would be a seme of third order.

4.1. The Quantum Level. The Q-System

The Q-system formed has as elements the quantum unity of meaning: the seme. The associative field of semes for a primitive symbol x is ( see Table 1).

We define the set {sxj} as (i = 0,...,n; j = 1,..., mn+mn-1+...+1) the semes or quantum semantic units of semantic and n the order of the symbol.

Definition 17: The Semantic first order Q-Vocabulary for symbol x, defined as is the set formed by all semes of the semantic field of this symbol x. Card .

Consider one subset such that whose cardinal value will be an integer number. The cardinal of will be

Consequence 2: Each syntactic vocabulary of the first order Vx¹ of a primitive symbol x, has associated one semantic Q-vocabulary of the first order .

In Linguistic Theory, an operator is a linguistic element that is used to constitute a phrasic physical structure. This operator while universal is an “immanent datum” and empty of sense, that acquires meaning in a particular context. It is tied in short, to a conceptual analysis. For us, S will be a semantic operator, being the particular semantic sense of one elementary mathematical operation or seme of addition, product, division or logic connections. From point of view of the Modeller “...it is added to..”, “..it is multiplied by..”, etc.. We can considerer S as an operation that it is not commutative, is not associative and it has necessarily a neutral function.

Definition 18: The Semantic Product denoted as, is the semantic relationships between all elements of a first semantic vocabulary of primitive symbol x and all elements of a first semantic vocabulary of other primitive symbol y through an operator. works as a Cartesian product but it contains all semantic operators S.

The Semantic product between two semantic sets and established a binary semantic relationship between all elements of and :

being S the semantic operator. In the same way are defined the following semantic Q-vocabularies of with orders superior to 1.

Definition 19: The Semantic Q-Vocabulary of order two is the formed by:

Definition 20: The Semantic Q-Vocabulary of order three is the formed by:

Definition 21: The Semantic Q-Vocabulary of order n is the formed by:

Definition 22: The Semantic Q-Lexicon is the set of all semantic Q-vocabularies of any order and denoted as.

Definition 23: The Semantic primary Q-Lexicon and denoted as is the set of all semantic Q-vocabularies of first order

The complement of shall be denoted as such as .

The complement set defines the semantic Q-relations between elements of Semantic primary Q-Lexicon.

Let be the semantic Q-lexicon and we shall make a partition such as

We shall make in another partition such as

Definition 23: The Semantic Q-system is an ordered pair of and sets, With being the set object in the Semantic primary Q-Lexicon and a set of semantic relationships such as

Is Χ an operation product of semantic relationships i between Q-vocabularies of the first order and that allow the existence of Q-vocabularies of order higher than one. This operation is equivalent to a Generalized Cartesian Product. The relational set is not formed by binary relations as a classic concept of system but n-tuplets. The concept of system is enlarged when semantic concepts haves been introduced in theory.

An elementary semantic Q-vocabulary is formed by one quantum semantic unity, that is to say

Let be a set such as

Definition 24: The Q-Semantic Base Model denoted as SS = (, ) is the system determined by the elementary semantic Q-lexicon of all primitive symbols and the relational set formed by = .

Definition 25: The Q-Supreme Semantic Model and denoted as = ( , ), being the Semantic primary Q-Lexicon of all primitive symbols and is the set formed for semantic Q-vocabularies of order higher than one.

Consequence 3: The Supreme Semiotic Model has associated a Q-Supreme Semantic Model that serves as superstructure.

4.2. The Atomic Level. The A-System

From the moment that an accurate meaning is conferred to lexemes of any syntactic system, putting them in correspondence with the entities of a functional mathematical universe, we obtain a representation of the said system. The lexeme is converted therefore into a sememe. That is to say . This operation of representation is not something that returns adding to the lexemic symbols something which had been abstracted in its presentation. The formal semiotic system can be considered as an abstraction of its representations and of its presentations. But in such abstraction there is a dialectical unit in how much the syntactic system lacks sense without the existence of semantic associated and per se, this leads to the absurd. A semantic system constitutes the superstructural unit of a syntactic system, constituting an object called Semiotic System. From the point of view of syntax it constitutes a lexeme. Each lexeme has associated one sememe according to which it is understood what is referred at a higher level. Previously the sememe has been defined as the set of semes. We are obliged to introduce a new definition, one which would consider at sememe as a set of semes related by the logical operation of the conjunction. A sememe Si (i = 0,...,m) of order i will be being m the arbitrational number that depends on the modeller.

The A-system formed has as elements the atomic unity of meaning: the sememe. The associative semantic field of sememes for a primitive symbol x is (Table 2):

Table 2. Set of sememes of a symbol x

Definition 26: The Semantic A-Vocabulary of order one of primitive symbol x, denoted as is the set formed by all sememes of the semantic field of said symbol x. Card = .

We consider one subset .The cardinal of A-Vocabulary of first order VSx will be the same syntactic vocabulary of lexemes, is to say . For the same reason, upon equating any lexeme with a sememe at the symbolic level or the level of presentation, we will be able to establish the second, third,..., n order A-vocabularies in the same way that was established syntactically. The operator has the semantic sense defined in paragraph 3.1, being denoted also as S.

Definition 27: The A–vocabulary of order two is formed by:

Definition 28: The A–vocabulary of order three is formed by:

Definition 29: The A-vocabulary of order n is formed by:

Definition 30: The Semantic primary A-Lexicon and denoted as is the set of all semantic A-vocabularies of first order

Definition 31: The Semantic A-Lexicon is the set of all semantic A-vocabularies of any order and is denoted as LS.

An elementary semantic A-vocabulary is formed by one quantum semantic unity, it is to say

Definition 32: The A-Semantic Base Model, denoted as is the system determined by the elementary semantic A-lexicon of all primitive symbols and the relational set formed by

Definition 33: The A-Supreme Semantic Model, denoted as , being the Semantic primary A-Lexicon of all primitive symbols and , is the set formed by

Consequence 4: The Supreme Semiotic Model has associated one A-Supreme Semantic Model that serves it as superstructure and the super-superstructure is A-Supreme Semantic Model.

5. An Ecological Case: The Model Mariola

The MARIOLA model [10,11], so called for having taken as the base the mountainous terrestrial ecosystem of the Sierra de Mariola in Alicante, Spain (Figure 1), is a simulation of the behaviour and development of a typical bush ecosystem of the Mediterranean area (Figure 2).

Figure 1. Sierra de Mariola

Figure 2. Ecosystem of the Sierra de Mariola

In these shrub lands we find the representative bushes: Bupleurus fruticescens L., U/ex parviflorus Pourret, Helychrysum stoechas (L.) Moench, Rosmarinus officinalis L., Lavandula latifolia Medicus, Sedum sediforme (Jacq.) Pau, Genista scorpius (L.) OC. in Lam. and OC., Marrubium vulgaris L., Thymus vulgaris L, Cistus albidus L. They are common plants which play an important role in the shrub communities of the western Mediterranean region, especially during the first ten years after a fire. lt is interspersed with areas of artificial reforestation of Pinus halepensis.

The MARIOLA can be characterized as flow lows. (1) It is a compartmental but not necessarily linear model. (2) The input and output flows of each compartment or level are calculated by means of nonlinear regression equations. (3) The fauna is considered indirectly through a process of defoliation or destruction of the biomass by action of invertebrate predators and herbivore mammals. (4) Human action is not explicit. (5) The temporal unit for the measurements and simulation is one month for the reproductive submodel; the temporal resolution is one week. (6) The spatial extent is of 100 m2. (7) The basic magnitude is biomass, with as unit grams of dry living material. (8) The model simulates the individual development of each bush species and the process of decomposition, in the space limited by the canopy of the plant. (9) The model does not take into consideration problems of competition. (l0) The disaggregation is intermediary, that is, it is not sufficiently disaggregated to study behaviour in the morpho or ecophysiological scales. (11) Processes of decomposition are considered as "black box"; that is, the existence of decomposers causing the decomposition is not taken into account. Nor are biochemical processes of degradation of cellulose and lignin considered. (12) The processes of decomposition of humus are referred to the O horizon of the soil. (13) In the actual state, the MARIOLA model has been validated with one shrub species, the Cistus albidus (Figura 3).

Figure 3. Cistus albidus (white rockrose)

Nothing impedes its validation in any other species, arbutus or herbaceous, provided that the equations of growth are known. (14) The model simulates the behaviour of the evolution of the plant biomass on short and medium terms and establishes an objective to observe the development in normal and limited (desertification) conditions. The MARIOLA model consists of the following submodels (Figure 4):

Figure 4. Simplified causal diagram of MARIOLA model

1. Submodel of growth:

- growth,

- defoliation,

- destruction of the biomass.

2. Submodel on the decomposition of fallen biomass.

3. Submodel on reproduction:

- formation of florist buds,

- flowering,

- fructification.

a. State variables

Y:   Description (unit)

BL:   woody biomass (g)

BV:   green biomass (g)

MOTS:   total organic soil material (%)

NRO:   organic material of animal origin on the ground (g)

RBL:   litter of woody biomass on the ground (g)

RBV:   litter of green biomass on the ground (g)

b. Flow variables

X:   Description (unit)

ARRS:   rate of loss of the organic soil material through dragging and washing (%)

CRBL:   rate of production by growth of the woody biomass (g)

CRBV:    rate of production by growth of the green biomass (g)

DBLAR:   rate of destruction of the woody biomass through the action of arthropods (g)

DBLPL:   rate of destruction of the woody biomass through the action of phytoplagues (g)

DBVFS:   rate of destruction of the green biomass through the action of mammals (g)

DBVI:   rate of destruction of the green biomass through the action of insects (g)

DBVPL:   rate of destruction of the green biomass through the action of phytoplasgues (g)

DCBL:   rate of catastrophic destruction of the woody biomass (g)

DCBV:   rate of catastrophic destruction of the green biomass (g)

DF:   rate of defoliation (g)

DMOTS:   rate of decomposition of the total organic soil material (%)

DRBL:   rate of decomposition of the litter of the woody biomass on the soil (g)

DRBV:   rate of decomposition of the litter of the green biomass on the soil (g)

DRO:   rate of decomposition of the detritus of an animal narure (g)

MOFD:   rate of finely divided organic material (%)

PMOTS:   rate of production of organic soil material (humus) (%)

PRO2:   rate of production of organic detritus of animal origin (g)

VMN:   rate of destruction of the woody biomass (g)

c. Exogenous variables [semes of first level]

e:   Description (unit)

H:   environmental humidity (%)

IFAP:   maximum intensity of precipitation (max.l/h)

PLU:   precipitation(l)

POBHV:   population of mammals (Oryctolagus cuniculus) (number of individuals)

T:   environmental temperature (ºC)

VEVI:   wind speed (km/h max)

d.-Auxiliary variables and parameters

a:   Description (unit)

BT:   total biomass (g)

CRO2:   parameter of residual production of the rodents (g)

PORDT:   the herbivore diet (%)

Flow and auxiliary equations for MARIOLA (Cistus albidus) [SEMEMES]

CRBV = BT(0.0011T + 0.0028 H- 0.0271) + 0.012 PLU- 0.1436

CRBL = 0.6773 BT- 0.0079 BT H + 0.0004 BT PLU- 3.1864

BT=BV+BL

DF = 1.2382BV 2 - 0.0025BV- 0.0063BV T- 0.0081 BV H + 17.47630/PLU)- 0.9696

DBVFS = 0.000428BV 2 + 0.087560BVPOBHV- 0.184747

DCBV = 0.0020 BV IFAP + 0.0007 BV VEVI + 0.0007 exp(0.1 IFAP) + 0.0020

DBVPL = -0.00064BV 2 + 0.0066BV T- 0.3142cos H- 1.0665

VMN = 0.0187 BL + 0.0001 BL PLU - 0.5732

DCBL = 0.7023 cos BL + 0.0005 BLIFAP + 0.0003 BL VEVI - 0.4707

DBLPL = 0.0022BL T + 259.9959exp( -0.1 BL)- 1.4981cosBL- 3.59

CR02 = 1900

PORDT = 2.8949log(DBVFS)- 5.0052

PR02 = POBHV CR02 x (PORDT/100)

DRBV = 0.0007 T 2 - 0.0041T RBV + 0.0021 H RBV + 0.00002exp(0.1 H)- 0.3774

DRBL = -0.0030 T 2 + 0.0005 TH + 0.121 exp(0.1T) + 0.0170cos H- 0.3125

DRO = 0.0538NRO T- 0.0016 T 2 + 1.1457cosNRO- 0.8088

PMOTS = (0.0045T 2 - 0.0013 TH- 0.1623 T DBL + 0.3111T DBV + 0.5191 cos T+ 1.1102DRO + 1.0542)/100

MOFD = ( -0.0287 T 2 + 0.0058 TH + 1.0304exp(0.1T)- 0.0002exp(0.1 H)- 2.3152)/100

DMOTS = [MOTS(-0.0509 MOTS + 0.0133 T + 0.0012 H + 0.0014 PLU) + 0.0018 T 2 - 0.0509)/100

ARRS = (-0.0065 T 2 + 0.0024

6. Discussion

1.- Generally, it is easy to confuse meaning with the interpretation or decoding of the received message. Semantic units (semes and sememes), have associated a meaning and a decoding possibility. But if the order of sememe increases, and therefore, the number of semes, the interpretation or decoding leaves making more and more difficult. We are stopped here with limits imposed by knowledge and human psychology. The binary character of language (informative-expression of the transmitter) forces reopening the problem of meaning as a dual structure construed by the significance and the signifier. [4,5]. The signifier is something which possesses a prior process to that of the concept, which defines. It is the semantic component of the information emitted by the process indicating its source. Then it is independent of the observer. Significance is what appears when the process concept is identified and it is united to a certain context. It exists, when the process appears as a syntagmatic set element, this being considered as the cognitive structure set of the processes. It depends of the observer. It is equivalent to interpretation in Pierce [1] and may be defined as its transformation into a new sign being itself a sign. We can distinguish between having a signifier as a process, as an inherent propriety, and to have significance when it is related with the rest of the reality processes considered as system. So, the significance is ontological system propriety, whereas the significant will be of the semiotic systems or meaning system property.

2.- The adjustment of the lexical units to experimental data (explanation) versus its interpretation, is a semantic problem of decoding accomplished in accordance with a Semantic Principle of Uncertainty [12].

3.- Another inherent problem in working with any language, included the language LMT, is the duality synchrony-diachrony. A language can be studied according its double perspective: the synchrony (static, the axis of simultaneities of the system) and the diachrony (the axis of successions, evolution, and history). We have outlined as much the syntactic as the semantic vision from a synchronous or restricted diachronic point of view. Our first supposition is the stability of the Ontological System, that is to say, the conservation of variables and relationships among these, during a certain period of time. This is a rough idealization of the reality. In such a way that the Modeler proposes a specific replacement function, so it will be able to move from a metasystem to another that includes it, that is to say, from a Metatext to another of which the first one is not more than a simple subtext.

4.- The complete diachronic perspective implies the existence of changes, adaptations, modifications of the structure, phenomenon very well-known to ecologists. Language has all the possibilities in abstract in its vocabularies, as long as any lexical or semantic unit modeling, and provided it is not assigned a certain meaning. That is to say, when for example exp(atan(x)) modeling, x can be any variable, past, present or future, that is to say, it has existed or it can exist. For this reason it makes sense to speak of a Supreme Text in the current situation. Any modification of the structure will force us to build another text with a corresponding series of Metatexts, with all the possibilities offered by the language. To give a previous meaning to the lexical units would suppose accepting the neoplatonic idea of the Text’s existence before the reality that describes for it [3]. The Text and their Metatexts are related dynamically in turn, as dynamic as the reality that they seek to describe.

References

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[2]   | Markushevich, A. 1978. Teoría de las funciones Analíticas. Vol I. Editorial Mir. Moscow. (In Spanish, ranslated from Russian).
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[11]   | Usó-Domènech, J. L., Mateu, J and J.A. Lopez. 1997. Mathematical and Statistical formulation of an ecological model with applications ECOLOGICAL MODELLING. 101, 27-40.
[12]   | Usó-Domènech, J.L., Villacampa, Y., Mateu, J., and Sastre-Vazquez, P. 2000. “Uncertainty and Complementary Principles in Flow Equations of Ecological Models”. Cybernetics and Systems: an International Journal, 31(2), p 137-160.
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[14]   | Usó-Doménech, J. L., Nescolarde-Selva, J. 2014. Dissipation Functions of Flow Equations in Models of Complex Systems. American Journal of Systems and Software. 2 (4), pp. 101-107
[15]   | Usó-Doménech, J. L.; Nescolarde-Selva, J.; Lloret-Climent, M.; González-Franco, L. 2014. Diversity for Texts Builds in Language L(MT): Indexes Based in Theory of Information. American Journal of Systems and Software. 2(5). pp. 113-120.