Evolutionary Domains for Varying Individuals - Science Direct

Procedia Computer Science Volume 88, 2016, Pages 347–352 7th Annual International Conference on Biologically Inspired Cognitive Architectures, BICA 2016

Evolutionary Domains for Varying Individuals Viacheslav E. Wolfengagen1 , Larisa Yu. Ismailova1 , Sergey V. Kosikov2 , Viacheslav V. Navrotskiy2 , Sergey I. Kukalev2 , Alexander A. Zuev2 , and Polina V. Belyatskaya2 1

National Research Nuclear University “MEPhI” (Moscow Engineering Physics Institute), Moscow, 115409 RF, {[email protected]; [email protected]} 2

Institute for Contemporary Education “JurInfoR-MGU”, Moscow, 119435 RF, [email protected]

Abstract The domains ranged by the variables using Web-resources can vary with a time. This is possible even in a runtime of Web-application giving rise to various vulnerabilities and bugs. This paper focuses at the problem mentioned as the individual migration in a problem domain. There is a lack of computational models which operate in an environment of variable domains and the contribution is to develop such a model. The advance is in establishing the mechanism for driving the dynamics of the sets and individuals. As a consequence, the behavior of the variables in query logical expression becomes predictable suppressing the possible semantic instability. Keywords: computational model, variable domains, individual migration, tangled individuals

1

Introduction

It is well known that modern Web incorporates the commonly used data space which is mainly open to many kinds of access, traversing manyfolds of Web-resources. This obviously needs to increase greatly the flexibility of software tools on hand. The recent attempts to bring in more flexibility to the data models in an area of information systems [6], [5] are pursued using a category theory. Modern considerations of flexible types and associated domains is based on the advance in homotopy type theory and unification of foundations for mathematics and computer science [7], [1]. This leads to increase of type flexibility based on homotopy. The notion of variable objects, types and domains, in fact, was discussed earlier in the works of this paper authors [11], [2], [3]. The indexed families of objects were computationally constructed in [10] and analyzed for their semantic stability. Corresponding computational Selection and peer-review under responsibility of the Scientific Programme Committee of BICA 2016 c The Authors. Published by Elsevier B.V.

doi:10.1016/j.procs.2016.07.447

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Evolutionary Domains for Varying Individuals . . .

Wolfengagen, Ismailova, Kosikov et al.

environments were studied in [9] and their applicative models were studied in more details in [8]. The generalized computational models were introduced in [4] using “combinatory glue” and their equational characterization was given. Unfortunately, most of the available approaches ignore such a fact that Internet and Web give as a new kind of reality, where variables range over the varying domains. The commonly used methods and computational models assume that the ranges for variables are fixed enabling usage of “ordinary” mathematics. As the ranges of a variable of Web Information Systems (WIS) are varying with a time, this gives rise to a variety of vulnerabilities, semantic instability and other bugs. Among them there is an effect of tangled Web and, in particular, the most tricky task to detect bugs initiated by tangled individuals when they migrate over disparate Web-pages and domains. This paper covers some questions on the migration of individuals over Web. Section 2 covers the introduction of variable sets to model the variability of ranges for variables in Webapplications and WIS. Section 3 covers the possible vulnerabilities arising during individual migration. Section 4 briefly depicts the future roads to build the advanced computational models to drive the individual migration and detection the possible semantic bugs. Nevertheless, we hope that the method been applied is wider and can become the tool for analysis the dynamics of a problem domain.

2 2.1

Variable sets Generalized elements

In many cases we need the extended notion of element. Following this way we refuse the notion of the constant element in favor of idea of a variable element. Using phrase element of T , often indicates the mapping to T from any A (not obviously, that carrying out A = 1, where 1 is a terminal element). To indicate namely this circumstance the expression generalized element of T is used. This is an arbitrary map for which a codomain is selected out as T . This map has some domain but not obviously terminal element 1. The more exact content of this description looks as variable element of T , ranging over A. In reality, this is somewhat of widely used idea of an element. Assuming the way of discussing the inhabitants of a hotel, we say ‘inhabitant’, that, as occurred, is an element of the set of possible inhabitants, but we mean the actual inhabitant. On the other hand, yesterday this was one value but today this became other value. The values are changing, but stay the definite ones. Let A is a set of the past year days, for any of which there is the element in a constant sense. 348



The actual inhabitant in such a day is a value of mapping and the mapping itself represents the inhabitant, which is mentioned by the hotel administrator. He is changeable yet assumed as some unit, one “element”. In practice this is a usual way to discuss many of the situations. To discuss this appearance using such a notion of variability, rather general kind of models is used involving as one of the counterparts that should not be alone: – an abstract set, playing the role of “temporal” parameters; – some other set, playing the role of values; – special map describing the process of evolving. This process can possess its own cause or prerequisite, which should be taken into account as well, but the effect of this cause/prerequisite is the sequence of values, or listing. Variability or evolving is often described as a sequence or listing. But the persistence for such an evolution is given by indicating their concreteness or certainty, representing a certainty of the element under consideration. An element of the inhabitant set could surely be (1) single and certain inhabitant (single unchangeable value of inhabitants); (2) inhabitant of the Hotel “Hilton” during the last week. In the last case this is still yet single, but a generalized element of the inhabitant set, and the domain of corresponding map is A. The variable sets are often occurred and used in practice. In particular, a set of persons, inhabiting the hotel, is varying with time, say, by days. For mathematical modeling of this effect, the category of variable sets can be fruitfully applied to. In use will be both the categories of constant sets and categories of variable sets.

2.2

Continuity

Take into account only “continuous” operations over classes. They are objects C such that C(X) is determined for any class X ⊆ V and C(X), where V is a class of all the sets, which is also a class. Besides that, C should conform the following conditions: (1) X ⊆ Y always implies C(X) ⊆ C(Y ); (2) if C ⊆ C(X) and C is a set, then C ⊆ C(B ) for some set B ⊆ X. Conditions (1) and (2) are not so restrictive, as could appear. Any usual mapping c : V → V determines continuous operator over class according the definition: C(X) = {c(x) | x ∈ X}. In addition, C determines c, because we have: y = c(x) iff {y} = C({x}) for all x, y ∈ V . As appeared, in some sense nothing has been lost. For the diagram in Figure 1, this is a case that X ≡ B and C ⊆ C(B ). Then we can take B ⊆ B and, as follows, can take C ≡ Cf . As far as B ⊆ B, then C(B ) ⊆ C(B). Now, because C ⊆ C(B ) ⊆ C(B) and C ≡ Cf , then Cf ⊆ C(B), as required. Show that the properties of operations on classes above lead to some certainty of a kind of controlling their elements. This can be expanded up to the replacement some elements by other ones, without violation of some predetermined feature/property. 349


Wolfengagen, Ismailova, Kosikov et al. f

World A

World B

Evolvent A

B

⊆

T

Populating

C

h C(A) Inhabitants of the world A

1

⊇

⊆

C

Populating

S

¯ h= f h

¯ h

Cf

⊆

C(f )

Transformation

C(B) Inhabitants of the world B

Transformers of the world (f, B)

Figure 1: Evolving the domains for individuals.

3

Driving the bugs

Show, as follows, how, figuratively speaking, some T -distinguished part of “entity” of B (B¯ occurred as been replaced by (f, B)-transformer hf with some distinct origin inhabitant h) and, generally speaking, with other T -undistinguishable features/propertties. Assume that A is fixed, and f , B are varying as parameters.

3.1

Evolving the events

A “solution” for the individuals from Figure 1 is outlooked as follows. ¯ inhabits in, or lives in C(B), and h(b) ¯ The individual h ∈ T for b ∈ B. The individual h inhabits in C(A), and h(a) ∈ T for a ∈ A. Assuming that b, a are linked by the evolvent f so, that f (b) = a, it follows, that hf became an inhabitant in Cf and is an image of h, because of hf (b) = h(a). certainly, hf infiltrates C(B), because of Cf ⊆ C(B). For pure mathematical ¯ are ‘merged’ reasons, this is due to continuity of operations in use. If, in addition, hf (b) and h(b) ¯ ¯ ¯ by equivalence hf (b) = h(b) for hf (b), h(b) ∈ T , then hf = h also turn, for some condition (for some “time”), into indistinguishable individuals in C(B).

3.2

Coincidence, and not an identity

¯ threaded his way to C(B), Purely meaningfully, hf of Cf in the guise of (in the likeness of) h ¯ and this can be unwanted. As it turned out, both hf (b) and h(b) of T are merging, i.e. by predetermined feature/property they are alike/indistinguishable. 350



But in this case inverse image of hf for the evolvent f presented h and was an inhabitant ¯ of C(A). And in T , as it turns out, hf (b) simply replaces h(b). However, for other properties they can be rather distinguishable individuals, i.e. it may turns out, that ¯ hf = h, but the same time ¯ hf ≡ h. ¯ MeaningIt turned out, that (f, B)-transformer hf replaced legal, indigenous B-inhabitant h. fully, here we faced to the coincidence of individuals, but not to their identity. Coincidence occurs by the property T , but not by all other properties, which can be well distinguished. Substantively, this can be treated as the manageable bugging, which within known boundaries, without violation the property T , can be A-controlled. (Select out the A-points, which, along the evolvent f , lead to B-points, generating definitely (f, B)-transformers of Cf . They can be resettled to C(B), hiding for a while the fact of their transformation.)

4

Future Work

– First, one would obtain and verify the formal properties of the variable domains that contain the individuals like HT (I) = {h|h : I → T }, where types T of the objects are obtained from the definitions like T = {h : D|Ψ} for the domain D. In general, the object HT (I) is a variable domain indeed, because of the parameter I and T occurrence. Also its elements are the functions h. It is viable to show that the set of T -sorted potential individuals is included in D while the set of actual in i individuals HT ({i}) is included in T . – Second, for the event driven model the studying of the evaluation map · · : Objects × Assignments −→ Values can fill the gap between logical constructs and variable domains. For instance, for variables x ¯, y¯ the following reasons are applicable. The events evolve from A to B. The inhabitants of the world A evolve as well, so they inhabit the world B. The world B contains the clones of A-inhabitants, and also some other inhabitants if any. A

f

←−−−− H(f )

B ⊆

y B ¯ xA ∈HT (A) −−−−→ (HT )f −−−−→ HT (B) ¯ ¯ xA = ¯ y B = ¯ xf B

(1)

And so on. This is especially important to analysis of Web-application dynamics, using the script-driven computations. – At last, there are other direction of the method application to studying the tangled individuals. 351



Conclusions To solve the problem of variable ranges for the variables in query expressions we determined and applied the constructions of variable sets. This naturally leads to the variable domains in semantic considerations. The variance of the sets in use in analogous to homotopy models in modern mathematica. The authors have their own vision of application the variable sets using Web-application. The established f –dynamics of individuals reflects their behavior with a migration. There is an ability of merging the migrant from some domain with pre-existed individual in other domain. The unwanted merging can be detected by f -threading the “suspicious” individuals. The further analysis using the described method can be applied to solve the tricky problem of the tangled individuals in the tangled Web. The authors hope that this work will stimulate further research.

Acknowledgements This research is supported in part by the RFBR grants 16-07-00914, 16-07-00892, 16-07-00912, 16-07-00909; 15-07-06933, 15-07-06898; 14-07-00119; 14-07-00087, 14-07-00107, 14-07-00072, 14-07-00054.

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