Comparing Behavior Patterns of Swarms that Learn

Comparing Behavior Patterns of Swarms that Learn Using Tolerance Perceptual Near Sets K.S.Patnaik1 1

G.Sahoo1 1

Computer Science and Engineering Birla Institute of Technology, Ranchi, India-835215 1

Computer Science and Engineering, Birla Institute of Technology Ranchi, India-835215 1

[email protected]

[email protected]

J.F.Peters2 2

Electrical & Computer Engineering, University of Manitoba, 75A Chancellor's Circle, E1-526, Winnipeg, Manitoba R3T 5V6, Canada 2

[email protected]

Abstract- The problem considered in this paper is how to measure the degree of nearness of behaviours of swarms that learn. The solution to this problem is set forth a methodology for discovering perceptual granules (i.e., sets of perceptual objects) that are, in some sense, close to each other. A perceptual object is something presented to the senses or knowable by the mind. The basic approach to comparing perceptual objects is inspired by the early 1980s work by Zdzis law Pawlak on the classification of objects. Objects are classified by comparing descriptions of objects stored in information tables. In this work, descriptions of swarm behavior are stored in tables called rough ethograms. A swarm behavior description is defined by means of probe functions (sensor readings) that represent behaviour features. The proposed approach to comparing bahaviours and extracting pattern information in different ethograms takes advantage of recent studies of the nearness of objects and near sets. Behavior patterns are near each other if they have similar descriptions. The contribution of this paper is a framework for determining the nearness of behaviours represented in two or more ethograms. Keywords: tolerance

perceptual

object,

near

set,

ethogram,

I INTRODUCTION The problem considered in this paper is how to measure the degree of nearness of behaviours of swarms that learn. The solution to this problem is set forth in a methodology for discovering perceptual

granules (i.e., sets of perceptual objects) that are, in some sense, close to each other. Perceptual granules can be gleaned from what are known as perceptual information systems (or, more concisely, perceptual system). A perceptual system is a real-valued, total, deterministic information system. Briefly, such an information system consists of a set of perceptual objects and a set of probe functions representing object features. Specifically, a perceptual object is something presented to the senses or knowable by the mind [6]. This form of an information system represents a perceptual interpretation of an information system introduced in [20] and elaborated in [23]. Nondeterministic information systems were introduced by Lipski [5], while deterministic information systems independently by Pawlak [12] (see, also, [10, 9]). The notion of a perceputal system admits a wide variety of different interpretations that result from the selection of sample perceptual objects contained in a particular sample space O. Perceptual objects are known by their descriptions. Examples of perceptual objects include observable organism behaviour, growth rates, soil erosion, events containing the outcomes of experiments such as energizing a network, microscope images, MRI scans, and the results of searches for relevant web pages. A perceptual object is something that has its origin in the physical world. Perceptual objects that have the same measured appearance are considered perceptually near each other, i.e., objects with matching or, at least, similar descriptions. A description is a tuple of values of functions

representing features of a perceptual object. The basic approach to comparing perceptual information systems is inspired by the early 1980s work by Zdzislaw Pawlak on the classification of objects by means of attributes [12]. Objects are classified by comparing descriptions of objects stored in information tables. In this work, descriptions of swarm behavior are stored in decision tables called rough ethograms introduced in [15,21]. In general, an ethogram is a tabular representation of observed behaviours of organisms [4]. A rough ethogram is a specialized form of decision table first introduced in [12]. A swarm behavior description is defined by means of probe functions (sensor readings) that represent behaviour features. In this article, the focus is on perceptually near behaviours. The phrase perceptually near is close to the usual understanding of the adjective similar [6]. Insight into the perception of near objects comes from Zdzislaw Pawlak's work on classification of objects [14,12,13] and from Ewa Orlowska's observation about approximation spaces as formal counterparts of perception [8]. An understanding of perception either by humans or imitated by thinking machines entails a consideration of the appearances of objects characterized by functions representing object features [19,17]. Perceptual objects are considered near each other if the objects have similar descriptions to some degree. For example, two peas in a pod are considered near each other if they have approximately the same colour or shape or weight independent of the relative position of the peas. A set of perceptual objects is a near set if it contains objects that are near each other. Hence, any equivalence class containing perceptual objects with matching descriptions is a near set. Again, for example, any set of buildings in the same village is a near set if one considers location as part of the description of the buildings. Any non-empty rough set contains one or more equivalence classes, i.e. sets containing objects with matching descriptions. Hence, any rough set is a nearset. Originally, the notion of nearness between sets (proximity relation between sets) was derived from a spatial meaning of distance, Efremovic [1]. Later, it would seem that the notion of proximity between sets was more abstract, i.e., proximity not limiting to spatial interpretation (see, also, [2, 7]). This later form of proximity relation permits either a quantitative (spatial) or qualitative (non-spatial) interpretation. The proposed approach to comparing behaviours and extracting pattern information in different ethograms takes advantage of recent studies of the nearness of objects and near sets.

Behavior patterns are near each other if they have similar descriptions. The contribution of this paper is a framework for determining the nearness of behaviours represented in two or more ethograms. II PERCEPTUAL INFORMATION SYSTEMS Definition 1: A perceptual information system or, more concisely, perceptual system, is a real valued total deterministic information system where O is a non-empty set of perceptual objects, while F a countable set of probe functions. A Perceptual Object Description Perceptual objects are known by their descriptions. An object description is defined by means of a tuple of function values φ(x) associated with an object x ∈ X (see Table 1). The important thing to notice is the choice of functions φi∈B used to describe an object of interest. Assume that B ⊆ F (see Table 1) is a given set of functions representing features of sample objects X ⊆ O and F is finite. Let φi∈B , where φi :O → R. In combination, the functions representing object features provide a basis for an object description φi :O → RL , a vector containing measurements (returned values) associated with each functional value φ(x) for x ∈ X, where | φ|= L, i.e. the description length is L. Object Description: φ(x) = (φ1(x), φ2(x), φ3(x),…, φi(x) ,…, φL(x)): The intuition underlying a description φ(x) is a recording of measurements from sensors, where each sensor is modeled by a function φi(x). TABLE 1 DESCRIPTION SYMBOLS Symbol R O X x F

Interpretation Set of real numbers Set of perceptual objects X⊂O, Set of sample objects x∈X, sample object. A set of functions representing object features

B φ

B⊆F, φ :O → RL, object description

L i φi

Description length i≤L φi∈B,where φ :O → R,probe function,

φ(x)



φ(x) =(φ1(x), φ2(x), φ3(x),…, φi(x) ,…, φL(x)),description φ1(x),…,φ(x|X|), Perceptual Information System

Let X, Y ⊆ O denotes sets of perceptual objects. Sets X, Y ⊆ O are considered near each other if the sets contain perceptual objects with at least partial matching descriptions. A perceptual object x∈O is something presented to the senses or knowable by the mind [6]. In keeping with the approach to pattern recognition suggested by Pavel [11], the features of an object such as contour, colour, shape, texture, bilateral symmetry are represented by probe functions. A probe function can be thought of as a model for a sensor. A probe makes it possible to determine if two objects are associated with the same pattern without necessarily specifying which pattern (classification). A detailed explanation about probe functions vs. attributes in the classification of objects is given in [16] table 2. Table 2 Sample perceptual information systems Sys1

Sys2

X

φ3

d

φ1

φ2

Y

φ1

φ2

φ3

d

x1

0

x2

0

1

0.1

1

y1

0

2

0.2

0

1

0.1

0

y2

1

1

0.25

x3

0

1

2

0.05

0

y3

1

1

0.25

0

x4

1

3

0.054

1

y4

1

3

0.5

0

x5

0

1

0.03

1

y5

1

4

0.6

1

x6

0

2

0.02

0

y6

1

4

0.6

1

x7

1

2

0.01

1

y7

0

2

0.4

0

y8

0

3

0.5

1

y9

0

3

0.5

1

For representing results of a perception we use the notion of a perceptual information system. In general, an information system is a triple S = < Ob, At, {Valf}f ∈ At> where Ob is a set of objects, At is a set of functions representing either object features or object attributes, and each Valf is a value domain of a function f ∈ At , where f : Ob →P(Valf ) (P(V alf ) is a power set of Valf ). If f(x)≠ φ, for all x ∈Ob and f ∈ At, then S is total. If card(f(x)) = 1 for every x∈Ob and f∈At, then S is deterministic. Otherwise S is nondeterministic. In the case, when f(x) = {v}, {v} is identified with v. An information system S is real valued if Valf = R for every f∈ At.Very often a more concise notation is used: , especially when value domains are understood, as in the case of real valued information systems. Since we discuss results of perception, as objects we consider perceptual objects while f ∈ At are interpreted as probe functions. Two examples of perceptual systems are given in Table 2.

Definition 2: Nearness Relation [23] Let be a perceptual system and let X, Y ⊆ O. The set X is perceptually near to the set Y (X~F Y ), if and only if there are x ∈ X and y ∈ Y such that x ~F y. Definition 3: Weak Nearness Relation [23] Let be a perceptual system and let X, Y ⊆ O. The set X is weakly near to set Y within the perceptual system (X؄F Y), if there are x ∈ X and y ∈ Y and there is B ⊆ F such that x؄B y .If a perceptual system is understood, then we say shortly that a set X is weakly near to set Y. Definition 4: Weak Tolerance Nearness Relation [20] Let be a perceptual system and let X, Y ⊆ O, ε∈[0, 1]. The set X is perceptual near to the set Y within the perceptual system (X، Y), if there are x ∈ X and y ∈ Y and there is φ ∈ F, ε R such that x ؄B, ε y .If a perceptual system is understood, then we say shortly that a set X is perceptual weakly near to set Y in a weak tolerance sense of nearness. Definition 5: Tolerance Perceptual Near Sets Let be a perceptual system and let X ⊆ O,A set x is a tolerance perceptual near set iff there is Y ⊆ O ,such that X،F Y .The family of near sets of perceptual system is denote by NearF(O). III TOLERANCE MEASURE OF NEARNESS The section briefly considers tolerance measure of nearness, since there is interest not only in identifying behaviours that belong to near sets but also to determine the degree that sets are near to each other. Obviously, the greater the number of perceptual objects in near sets that have matching descriptions, the greater the degree of nearness of the sets. This observation leads to the various measures of nearness of sets using the nearness relations. In this article, we restrict the presentation to a nearness measure based on the weak tolerance nearness relation. Definition 6: Tolerance Nearness Measure Let be a perceptual system and let X, Y ⊆ O. X ≠ Y. Let B ⊆ F, φ ∈ B, ε ∈ [0,1]. A measure of nearness of sets X and Y is computed using

µ φ ,ε

(X

,Y

{( x , y ) ∈

)=

X × Y | x  φ ,ε y }

⎛ ⎞ ⎜ X ∪ 2 ⎟ 2 ⎝ ⎠

i. e for each (x, y) ∈ X × Y, it is the case that |φ(x)- φ(y)|≤ ε IV NEARNESS OF SWARM BEHAVIOURS Swarm-bot is self-assembling and self-organizing robot colony composed of number (30-35) of smaller devices, called s-bots [7].Each s-bot is a fully autonomous mobile robot capable of performing basic tasks such as autonomous navigation, perception of environment and grasping of objects. In addition to these features, one s-bot is able to communicate with other s-bots and physically connect to them in flexible ways, thus forming a so-called swarm-bot. Such a robotic entity is able to perform tasks in which a single s-bot has major problems, such as exploration, navigation, and transportation of heavy objects on very rough terrain (this hardware structure is combined with a distributed adaptive control architecture loosely inspired upon ant colony behaviors). A swarm behavior description is defined by means of probe functions (sensor readings) that represent behaviour features. V RESULTS AND ANALYSIS Actor critic learning algorithms [24][25] in context of approximation spaces is used as a case study to analyse nearness of swarm behaviours i.e actor critic algorithm[24] ,rough actor critic algorithm and near actor critic algorithm is used for simulation. For calculating the nearness the attribute reward has been considered. One such sample ethogram having the attributes(O-objects,S-states,a-actions,p-prefrence,vvaluestate,d-decision)obtained during simulation is shown in table 3. Table 3 Sample Ethograms Table 1

O

s

a

p

v

d

0 1 2 3 4 5

0 0 1 0 1 0

0 1 0 2 1 0

31 21 14 11 10 7

3 6 5 4 5 4

1 0 1 0 1 0

Table 2

0 1 2 3 4 5

0 0 0 0 0 0

0 1 0 1 1 0

47 38 23 24 21 12

3 6 5 6 6 6

1 0 1 0 1 0

At a time two ethogram tables were taken as input to nearness algorithm 1 and µφ,ε (X,Y) values are accumulated for different pairs of ethograms and the results were produced as show in the Fig.1 and 2 . The figures (i.e Fig.1 – Fig.2) are the plots of number of episodes vs avg. nearness at a given value of tolerance, discount rate and learning rate of actor critic ,rough actor critic and near actor critic algorithms .In the plots the green line shows the value of nearness of near actor critic algorithm , the blue line shows the value of nearness of rough actor critic algorithm and the red line shows the value of nearness of actor critic algorithm .When tolerance=0 the observed behaviour pairs are less and the variation in nearness is large from actor critic to that of rough and near actor critic, but when tolerance >0 i.e{0.1,0.3,0.5..} the number of behaviour pairs is large and the nearness first increases during learning and later it becomes steady as shown in Fig.2.The nearness of behaviour patterns at various tolerance values,learning rate and discount factor are also analyzed(not shown)and it is found that nearness of behaviour patterns are increasing at every episode initially and than decreases and later it becomes steady. Algoritm1: Degree of Nearness Input : Table1,Table2, φ,ε . Output: µφ,ε (X,Y). Initialize X ∈ Table1, Y ∈ Table2, ε, count← 0; Compute denom ←(|X‫ڂ‬Y| ) 2

Choose x ∈ X; y ∈ Y ; while ($ x ∈ X) do while ($ y ∈ Y ) do if x ؄φ, ε y then count ← count + 1; end Choose y ∈ Y ; end Choose x ∈ X; Compute µφ,ε (X,Y) = count/denom; end

Fig. 1 No.of episodes Vs Average Nearness

Fig. 2 No.of episodes Vs Average Nearness

VI CONCLUSION The problem considered in this paper is how to measure the degree of nearness of behaviours of swarms that learn. The proposed approach to comparing bahaviours and extracting pattern information in different ethograms takes advantage of recent studies of the nearness of objects and near sets. Behavior patterns are near each other if they have similar descriptions. The contribution of this paper is a framework for determining the nearness of behaviours represented in two or more ethograms.The plots that we obtained from our simulation shows that in cooperating learning the behaviour patterns in various time scales are quite close to each other and reflects the fact of closeness of behaviour patterns in each and every episode. This helps us to analyze the extent to which the organisms (machines) are learning from each other. REFERENCES .[1]. V. Efremovi_c, Geometry of proximities 1, Mat. Sb. 31 (73) (1952) 189{200, in Russian.

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