The Presentation of Evolutionary Concepts

Abstract

The paper considers an approach to solving the problem of supporting the semantic stability of information system (IS) objects. A set of IS objects is addressed as a semantic network consisting of concepts and frames. The interpretation that assigns intensional (meaning) and extensional (value) characteristics to network designs is connected to the constructions of the semantic network. The interpretation in the general case depends on the interpreting subject, time, context, which can be considered as parameters. The possibility to preset a consistent interpretation for a given semantic network is regarded as a semantic integrity, and the possibility to control changes in interpretation when the parameter is changed is regarded as semantic stability. Among the tasks related to supporting semantic stability, the problem of modelling evolutionary concepts (EC) is highlighted. It is proposed to construct a computational model of EC based on the theory of categories with a significant use of the concept of variable domain. The model is constructed as a category of functors, and it is shown that the Cartesian closure of the basic category implies Cartesian closure of the category of models. The structure of the exponential object of the category of models has been studied, and it is shown that its correct construction requires taking into account the evolution of concepts. The testing of the model’s constructions was carried out when lining the means of semantic support for the implementation of the best available technologies (BAT).

Appendix

The appendix contains proofs of the properties of the defined constructions.

The Cartesian Product in the Category Set. Category Set of sets is cartesian. Cartesian products in it are built as usual as sets of pairs: if A and B are two sets (objects of category Set), their cartesian product is a set $\{(a, b)| a \in A, b \in B\}$. Projections are defined by the natural way:

$$\begin{aligned} p (a, b) = a, \qquad q (a, b) = b. \end{aligned}$$

The arrow of pair evaluation $\langle {f, g}\rangle$ for $f : C \rightarrow A$, $g : C \rightarrow B$ is defined so:

$$\begin{aligned} \langle {f, g}\rangle (c) = (fc, gc), \end{aligned}$$

where $c \in C$. It is easy to check that the product of arrows is evaluated so:

$$\begin{aligned} (f \times g)(a, b) = (fa, gb). \end{aligned}$$

The Properties of the Cartesian Product. 1. $U \times V$ preserves the composition. Really,

$$\begin{aligned} \begin{array}{c} (U \times V)(f \circ g) = U(f \circ g) \times V(f \circ g) = (U(f) \circ U(g)) \times (V(f) \circ V(g)) \\ = \langle {U(f) \circ U(g) \circ p, V(f) \circ V(g) \circ q}\rangle = \\ = (Uf \times Vf) \circ (Ug \times Vg) = (U \times V)f \circ (U \times V)g. \end{array} \end{aligned}$$

2. $U \times V$ preserves the unit arrows. Really,

$$\begin{aligned} (U \times V)1 = U1 \times V1 = 1 \times 1 = \langle {1 \circ p, 1 \circ q}\rangle = \langle {p, q}\rangle = 1. \end{aligned}$$

So $U \times V$ is really a functor. We verify now that p and q are natural transformations. So we check the condition of naturality

$$\begin{aligned} pB \circ (U \times V)f = p \circ (Uf \times Vf) = p \circ \langle {Uf \circ p, Vf \circ q}\rangle = Uf \circ p = Uf \circ pA. \end{aligned}$$

The condition satisfies so p is really a natural transformation. The check for q is analogous. This check shows that projections in Set are defined really naturally.

We can make the evaluations above also in terms of restriction mappings. The indexes of objects can be reconstructed in a unique way, and so they can be omitted because of natural property of the projection.

The Properties of the Pairing. We check that defined arrow is a natural transformation, i.e. check the naturality condition:

$$\begin{aligned} \begin{array}{c} \langle {\mu , \nu }\rangle B \circ Wf = \langle {\mu B \circ Wf, \nu B \circ Wf}\rangle = \langle {Uf \circ \mu A, Vf \circ \nu A}\rangle \\ = \langle {Uf \circ p, Vf \circ q}\rangle \circ \langle {\mu A, \nu A}\rangle = ((U \times V)f) \circ \langle {\mu , \nu }\rangle A. \end{array} \end{aligned}$$

The condition is satisfied, so the natural transformation $\langle {\mu , \nu }\rangle$ is defined correctly.

Now we have to check the characteristic properties of the projection and pairing. We show now that $p \circ \langle {\mu , \nu }\rangle = \mu$. We calculate a component $(p \circ \langle {\mu , \nu }\rangle )A$ and see that it is equal to ?A. We have

$$\begin{aligned} (p \circ \langle {\mu , \nu }\rangle )A = pA \circ \langle {\mu , \nu }\rangle A = p \circ \langle {\mu , \nu }\rangle A = \mu A. \end{aligned}$$

So components of $p \circ \langle {\mu , \nu }\rangle$ are really identical with components of $\mu$ and so $p \circ \langle {\mu , \nu }\rangle = \mu$. We show now that $\langle {p \circ \eta , q \circ \eta }\rangle = \eta$, where $\eta : W \rightarrow (U \times V)$. We have

$$\begin{aligned} \langle {p \circ \eta , q \circ \eta }\rangle A = \langle {pA \circ \eta A, qA \circ \eta A}\rangle = \langle {p \circ \eta A, q \circ \eta A}\rangle = \eta A. \end{aligned}$$

So this characteristic property is also satisfied.

The Properties of the Exponential. We have defined the object and arrow mappings for the functor. Now we check that this is really a functor. We must check the composition preservation property

$$\begin{aligned} (U \rightarrow V)(f \circ g) = (U \rightarrow V)f \circ (U \rightarrow V)g \end{aligned}$$

and the unit preservation property

$$\begin{aligned} (U \rightarrow V)(1_A) = 1_{(U \rightarrow V)A}. \end{aligned}$$

We begin with the composition. Let $f: B \rightarrow A$ and $g: C \rightarrow B$. Then

$$\begin{aligned} (U \rightarrow V)(f \circ g): (U \rightarrow V)A \rightarrow (U \rightarrow V)C. \end{aligned}$$

The elements of $(U \rightarrow V)A$ are the families $\varphi$ of mappings. For the comparison of these families we apply them to the arrow $h: D \rightarrow C$. Then

$$\begin{aligned} (U \rightarrow V)(f \circ g)(\varphi ) = \psi , \psi h = \varphi (f ? g) ? h \end{aligned}$$

and

$$\begin{aligned} \begin{array}{c} ((U \rightarrow V)f \circ (U \rightarrow V)g)(\varphi )h = (U \rightarrow V)f ((U \rightarrow V)g)(\varphi )h) \\ = (U \rightarrow V)f (\varphi )(g \circ h) = \varphi (f \circ (g \circ h)). \end{array} \end{aligned}$$

The property of the unit preservation can be checked in the similar way.

The Properties of the Currying. We check the functor character of the defined mapping. We have

$$\begin{aligned} \begin{array}{c} (\varLambda \psi )B \circ Uf = (V \rightarrow W)f \circ (\varLambda \psi )A \\ ((\varLambda \psi )((Uf)a)) g (b) = \psi C (Ufa ( g), b) = \psi C (a (f \circ g), b) \\ ((V \rightarrow W)f \circ (\varLambda \psi )A a) g (b) = (\varLambda \psi )A a) {f \circ g} (b) = \psi C (a (f \circ g), b). \end{array} \end{aligned}$$