Some Considerations about Fuzzy Logic Based Decision Making by Autonomous Intelligent Actor

The article presents an approach toward an implementation of a fuzzy log-ic-based decision-making process by Autonomous Intelligent Actor (AIA) (© A. Tserkovny), when an input information is defined for its “strategic target-ing” by a human operator in terms of a fuzzy incident geometry, whereas its “tactical” behavior (a navigation in space) is directed by fuzzy conditional inference rules. For implementing both elements of AIA decision-making a fuzzy logic [1] for formal geometric reasoning with extended objects is used. This fuzzy logic based fuzzification of axioms of an incidence geometry and a predicate apparatus [2] for AIA space orientation are also presented. The approach, offered in the article, extends predicates of a counter positioning of two objects and their mutual navigation into their fuzzy counterparts. The latter allows AIA to make certain “tactical” decisions.


Introduction
The article introduces the notion of an Autonomous Intelligent Actor (AIA), which, in general terms, represents an entity, which would be able to make an independent decision about strategic and tactical behavior, given an ultimate goal, defined by a human operator. For this purpose, we have proposed to use both fuzzy incident geometry paradigm for AIA strategic planning and fuzzy conditional inference rules for its local orientation (tactical behavior). The article How to cite this paper: Tserkovny, A. Note that the set of axioms of incidence geometry is just a subset of the axioms of Euclidean geometry.

Fuzzy Logic in Use
Now we present some basic operations in a fuzzy logic [1] we will use for all purposes of an article. We define the truth values of logical antecedent A and consequent B as ( ) , which we shall use as a universe of discourse in all our future exercises. Table 2 shows the operation implication Table 1. The logical operations of a fuzzy logic in use.

Name
Designation Value Tautology   A I   1 Controversy

Geometric Primitives as Fuzzy Predicates
Let us denote some incidence geometry predicates as p(a) ("a is a point"), l(a) ("a is a line"), and inc(a, b) ("a and b are incident"). Traditionally predicates are interpreted by crisp relations. The predicate expressing equality can be denotes by eq(a, b) ("a and b are equal"). to every pair of objects from N. In other words, every two objects of N are equal to some degree. The degree of equality of two objects a and b may be 1 or 0 as in the crisp case, but can as well be 0.9, expressing that a and b are almost equal.
Note that point-predicate p(.) for Cartesian point does not change when the point is rotated, i.e. rotation-invariance could be a main characteristic of "point likeness" with respect to geometric operations. In other words "point likeness" should be kept in a relevant fuzzy predicate expressing the extended subsets of R 1.  If we express the degree to which the convex hull of a Cartesian point set A is rotation-invariant as A ⊆ Dom p(.) and if p(A) = 1, then ch(A) is perfectly rotation invariant and it is a disc. And since A is assumed to be two-dimensional,

Let us define
always holds. In addition to p(.), the fuzzy line-predicate is defined as a compliment to fuzzy-point one To define the degree to which a Cartesian point set A ⊆ Dom is sensitive to rotation and since we only regard convex hulls, then a fuzzy version of the incidence-predicate inc(.,.) would be a binary fuzzy relation between Cartesian point sets A, B ⊆ Dom: Here in (2.6) we select the greater one of two convex hulls of A and B and this fuzzy relation measures the relative overlaps of them. Here A and B are considered as an incident to degree one, if |ch(A)| denotes the area occupied by ch(A). The greater inc(A, B), "the more incident" are A and B: If A ⊆ B or B ⊆ A, then inc(A, B) = 1.
Contrariwise to inc(.,.), we define a graduated equality predicate eq(.,.) between the bounded Cartesian point sets A, B ⊆ Dom as follows: where eq(A, B) measures the minimal relative overlap of A and B, whereas the ne- measures the degrees to which the two-point sets do not overlap: if eq(A, B) ≈ 0, then A and B are "almost disjoint". Then we can define the following measure of "distinctness of points" dp(.) of two extended objects from Figure 1 It is apparent, that the greater dp(A, B), the more A and B behave like distinct Cartesian points with respect to connection. Indeed, for Cartesian points a and b, we would have dp(A, B) =1. If the distance between the Cartesian point sets A and B is infinitely big, then dp(A, B) = 1 as well. If then dp(A, B) = 0.

Formalization of Fuzzy Predicates
To formalize fuzzy predicates, defined in subchapter 2.2 both implication → and conjunction operators are defined as in Table 1: In our further discussions we will also use the disjunction operator from the same If fuzzy predicate dp(…) is defined as in (2.8) and disjunction operator is defined as in (2.11), then Proof: From (2.8) we get the following: For the case, when 1 a b + < , and given (2.11) ( ) ( )

Fuzzy Axiomatization of Incidence Geometry
By using the fuzzy predicates formalized in subchapter 2.3, we propose the set of axioms as fuzzy version of incidence geometry in the language of a fuzzy logic [1] as follows: I1': In axioms I1'-I4' we also use a set of operations (2.9)-(2.11). PROPOSITION 3.
If fuzzy predicates dp(…) and inc(…) are defined like (2.20) and (2.12) respectively, then axiom I1' is fulfilled for the set of logical operators from a fuzzy logic [1]. (For every two distinct point a and b, at least one line l exists that is incident with a and b.) Given (2.9) we are getting   If fuzzy predicates dp(…), eq(…) and inc(…) are defined like (2.35), (2.16) and (2.15) respectively, then axiom I2' is fulfilled for the set of logical operators from a fuzzy logic [1]. (For every two distinct point a and b, at least one line l exists that is incident with a and b and such a line is unique) Proof: Let's take a look at the following implication: But from (2.14) we have From (2.31) and (2.32) we see, that Therefore, the following is also true Now let's look at the following implication Since from (2.12) ( ) , 1 inc a c ≤ , then with taking into account (2.34) we've gotten the following: And given, that ( ) 1 l c < we are getting If fuzzy predicate inc(…) is defined like (2.15), then axiom I4' is fulfilled for the set of logical operators from a fuzzy logic [1]. (At least three points exist that are not incident with the same line.) Proof: From (2.12) we have

Equality of Extended Lines Is Graduated
In [3] it was shown that the location of the extended points creates a constraint on the location of an incident extended line. It was also mentioned, that in traditional geometry this location constraint fixes the position of the line uniquely. And therefore, in case points and lines are allowed to have extension this is not the case. Consequently, Euclid's First postulate does not apply: Figure 2 shows that if two distinct extended points P and Q are incident (i.e., overlap) with two extended lines L and M, then L and M are not necessarily equal. Yet, in most cases, L and M are "closer together", i.e., "more equal" than arbitrary extended lines that have only one or no extended point in common. The further P and Q move apart from each other, the more similar L and M become. One way to model this fact is to allow degrees of equality for extended lines. In other words, the equality relation is graduated: It allows not only for Boolean values, but for values in the whole interval [0, 1].

Incidence of Extended Points and Lines
As it was demonstrated in [3], there is a reasonable assumption to classify an extended point and an extended line as incident, if their extended representations in the underlying metric space overlap. We do this by modelling incidence by the subset relation: Definition 1: For an extended point A, and an extended line L we define the incidence relation by where the subset relation ⊆ refers to A and L as subsets of the underlying metric space. The extended incidence relation (2.40) is a Boolean relation, assuming either the truth value 1 (true) or the truth value 0 (false). It is well known that since a Boolean relation is a special case of a graduated relation, i.e., since , we will be able to use relation (2.40) as part of fuzzified Euclid's first postulate later on.

Equality of Extended Points and Lines
As stated in previous chapters, equality of extended points, and equality of extended lines is a matter of degree. Geometric reasoning with extended points and extended lines relies heavily on the metric structure of the underlying coordinate space. Consequently, it is reasonable to model graduated equality as inverse to distance.

Metric Distance
In [3] was mentioned that a pseudo metric distance, or pseudo metric, is a map Well known examples of metric distances are the Euclidean distance, or the Manhattan distance. Another example is the elliptic metric for the projective plane defined in (2.42) [3]. The "upside-down-version" of a pseudo metric distance is a fuzzy equivalence relation w.r.t. in proposed t-norm fuzzy logic. We will use this particular fuzzy logic to formalize Euclid's first postulate for extended primitives in chapter 4. The reason for choosing a proposed fuzzy logic is its strong connection to metric distance.

Fuzzy Equivalence Relations
As mentioned above, the "upside-down-version" of a pseudo metric distance is a fuzzy equivalence relation w.r.t. the proposed t-norm ^. A fuzzy equivalence relation is a fuzzy relation   ( ) , , and therefore from (2.45) and (2.46) the following is taking place

Approximate Fuzzy Equivalence Relations
In [3] it was mentioned, that graduated equality of extended lines compels graduated equality of extended points. Figure 3(a) sketches a situation where two extended lines L and M intersect in an extended point P. If a third extended line L' is very similar to L, its intersection with M yields an extended point P' which is very similar to P. It is desirable to model this fact. To do so, it is necessary Journal of Software Engineering and Applications to allow graduated equality of extended points. Figure 3(b) illustrates that an equality relation between extended objects need not be transitive. This phenomenon is commonly referred to as the Poincare paradox. The Poincare paradox is named after the famous French mathematician and theoretical physicist Henri Poincare, who repeatedly pointed this fact out, e.g., in [3], referring to indiscernibility in sensations and measurements. Note that this phenomenon is usually insignificant, if positional uncertainty is caused by stochastic variability. In measurements, the stochastic variability caused by measurement inaccuracy is usually much greater than the indiscernibility caused by limited resolution. For extended objects, this relation is reversed: The extension of an object can be interpreted as indiscernibility of its contributing points. In the present paper we assume that the extension of an object is being compared with the indeterminacy of its boundary. Then in [3] we also shown, that for modelling the Poincare paradox we can replace a graduated context transitivity by a weaker form: is a lower-bound measure (discernibility measure) for the degree of transitivity that is permitted by q. A pair (e, dis) that is reflexive, symmetric and weakly transitive (2.50) is called an approximate fuzzy ∧ -equivalence relation. Let us rewrite (2.50) as follows , , , F a c F b c are defined in (2.46) and (2.45) correspondingly and given (2.47) we found that In [3] we also mentioned that an approximate fuzzy ∧ -equivalence relation is the upside-down version of a so-called pointless pseudo metric space ( ) , s δ :  Table 1.
is a weak form of the triangle inequality. It corresponds to the weak transitivity (2.50) of the approximate fuzzy ∧ -equivalence relation e. In case the size of the domain M is normalized to 1, e and dis can be represented by [3] ( ) ( ) If a distance between extended regions ( ) , a b δ from (2.53) and pseudo me- And from (2.55) we are getting But as it was mentioned in [3], given a pointless pseudo metric space ( ) The so defined equivalence relation on the one hand complies with the Poincare paradox, and on the other hand retains enough information to link two extended points (or lines) via a third. For used fuzzy logic an example of a pointless pseudo metric space is the set of extended points with the following measures: A pointless metric distance of extended lines can be defined in the dual space [3]:

Boundary Conditions for Granularity
As it was mentioned in [3], in exact coordinate geometry, points and lines do not have size. Therefore, distance of points does not matter in the formulation of Euclid's first postulate. If points and lines are allowed to have extension, both, size and distance matter. Figure 4 depicts the location constraint on an extended line L that is incident with the extended points A and B. The location constraint can be interpreted as tolerance in the position of L. In Figure 4(a) the distance of A and B is large with respect to the sizes of A and B, and with respect to the width of L. The resulting positional tolerance for L is small. In Figure 4(b), the distance of A and B is smaller than it is in Figure 4(a). As a consequence, the positional tolerance for L becomes larger. In Figure 4(b), A and B have the same distance than in Figure 4(a), but their sizes are increased. Again, positional tolerance of L increases. Therefore, a formalization of Euclid's first postulate for extended primitives must take all three parameters into account: the distance of the extended points, their size, and the size of the incident line.   A. Tserkovny two extended points P and Q are too close and the extended line L is too broad, then P and Q are indiscernible for L. Since this relation of indiscernibility (equality) depends not only on P and Q, but also on the extended line L, which acts as a sensor, we denote it by ( )[ ] , e P Q L , where L serves as an additional parameter for the equality of P and Q. In [3] the following three boundary conditions to specify a reasonable behavior of ( )[ ] , e P Q L were proposed: , then P and Q impose no direction constraint on L (cf. Figure 5), i.e., P and Q are indiscernible for L to degree 1: 1 e P Q L = .
2) If ( ) ( ) ( ) ( ) , s L P Q s P s Q δ < + + , then P and Q impose some direction constraint on L, but in general do not fix its location unambiguously. Accordingly, the degree of indiscernibility of P and Q lies between zero and one: In this paper we are proposing an alternative approach to one from [4]

A Euclid's First Postulate Formalization
In previous chapter we identified and formalized a number of new qualities that enter into Euclid's first postulate, if extended geometric primitives are assumed. We are now in the position of formulating a fuzzified version of Euclid's first postulate. To do this, we first split the postulate "Two distinct points determine a line uniquely." A verbatim translation of (3.4) and (3.5) into the syntax of a fuzzy logic we use yields

Fuzzy Logical Inference for Euclid's First Postulate
Similarly, to an approach from [3], we suggest to use the same fuzzy logic ( To determine the estimates of the membership function in terms of singletons from (3.14) in the form we propose the following procedure.
We also represent D as a fuzzy set forming linguistic variable, described by a triplet of the form T y is extended term set of the linguistic variable "discernibility measure "from To get an estimates of values of e(L,M) or "extended lines sameness", represented by fuzzy set S  from (3.18) given the values of ( ) , , , E a b l m or "degree of indiscernibility" and ( ) , D a b "discernibility measure", represented by fuzzy sets E  from (3.14) and D  from (3.16) respectively, we will use a Fuzzy Conditional Journal of Software Engineering and Applications Inference Rule, formulated by means of "common sense" as a following conditional clause: In other words, we use fuzzy conditional inference of the following type [7]: , , Given (2.10) and since we consider that CardX CardY CardZ = = , then expression (3.22) looks like But for practical purposes we will use another Fuzzy Conditional Rule (FCR) where P E D =  and U X Y = = , therefore from (3.25) we are getting The FCR from (3.26) gives more reliable results.

Example
To build a binary relationship matrix of type . By using (3.15) and (3.17) respectively we got (see Table 2    It is obvious that the value of fuzzy set S is laying between terms "almost average distance" and "average distance" (see Table 2), which means that ap-

Preliminary Considerations
Let consider that both Target and Object, a subject of mutual navigation, to be presented as octagons, depicted on Figure 6. We use octagons for simplification's sake only. Given the fact that we are studying a projection-based model, both targets and objects could be presented as follows: Where j is number of heights of a Target, whereas and i is number of heights of an Object. Both a target and an object could be presented in three-dimensional space as follows: On the other hand, from Figure 6 It is import to consider the following features of both target and object from

Predicates of Two Entities Mutual Relations
Considering   T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center  center  center  center   T  O  center center T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center center center center T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center  center  center  center  T  O  center  center   T  O  T  O  center center center center

A. Tserkovny
Let us also introduce the following values, which define entities coordinate derivation in space.

Entity Shape Estimation Predicates
We define the following predicates by using (4.20)-(4.22) 1) Entity has right geometric form (RGF)

Docking Positioning Predicates
We define the following predicates by using  T  O  T  O  T  O  center  center  center  center  center  center   d T O  x  x  y  y  z We have to take into account the fact that (4.33) presents idealistic case for two points in space, whereas for Target and Object from Figure 6 we have to use their real size values. For this purpose, given (4.4) and (4.5) we introduce the following.
Therefore, the real distance between Target and Object from Figure 6, given (4.33), (4.34) could be defined like that We define the degree ( , ,  from θ is about devθ . Again θ ∆ models how flexible "about devθ " is interpreted. We define . The fuzzy set of points, which are "East" of a reference point center T , using this interpretation, is displayed in Figure 9. In this figure, membership degrees  behavior. Let us formulate the "Tactical Level" of an Object decision making mechanism, the goal of which is to figure out if there is a way to achieve the ultimate outcome of its potential ACTIONs.

Feasibility of a Goal Setting
Let us define a notion of a Pre-conditions and GOALs of an Object and a Target mutual interaction. From the position of common sense, we consider a set of Size Comparison Predicates from (4.6), (4.7), Mutual Positioning Predicates from (4.8)-(4.15) and Entity Shape Estimation Predicates from (4.23)-(4.26) as a set of Pre-conditions to find out if any particular GOAL of an Object/Target setting is achievable. We also consider the set of Docking Positioning Predicates from (4.27)-(4.32) as an Object action GOALs (for the sake of current discussion).
We have to note that by the term Docking we presume unfriendly (no specific docking mechanism in place) approaching of a Target by an Object. We also presume that the size of an Object is SMALLER, than a Target one. It means that both (4.38) and (4.39) as well, as all sentences from an APPENDIX, should also include A feasibility of achieving a Target by an Object could be described by the set of predicates based logical sentences of the following structure. Let's underscore the fact, that (4.38)-(4.42) represent feasibility of a goal settings in its Boolean form and, of course, could not be always applied to a real-world situation. Therefore, we shall try to introduce its fuzzy counterpart.

Fuzzification of Feasibility of a Goal Setting
We are not pretending to generalize the way to re-interpret Size Comparison Predicates from (4.6), (4.7), Mutual Positioning Predicates from (4.8)-(4.15) and   We will use the following mapping  ( ) (4.59) On the other hand, similarly to the previous cases, to determine the estimates of the membership function in terms of singletons from (4.5) But for practical purposes we will use another Fuzzy Conditional Rule (FCR) where P Z X Y =   and As was already mentioned above, FCR from (4.70) gives more reliable results.

Example
To build a binary relationship matrix of type (4.69) we us use a conditional clause of type (4.64): P = "IF (Z is "ideal surface") AND (X is "ideal surface") AND (Y is "ideal surface"), THEN (RGF is "perfect")" (4.71) To build membership functions for fuzzy sets Z, X and Y we use (4.47), (4.53) and (4.59) respectively.
In (4.47) the membership functions for fuzzy set Z (for instance from Table 5) would look like: Same membership functions we use for fuzzy sets X and Y. Note, that the membership function for fuzzy set RGF from Table 5  , , , R A z x y A rgf shown in Table 6.
Suppose that the current value of "horizontal derivation", represented by a fuzzy set X' from (4.53), is defined as " , Table 6. Binary relationship matrix of a current example. which means (4.61) is not fully satisfactory and an Entity has to re-orient itself to successfully "dock" to another one.
The summary of this presentation: 1) The AIA strategic targeting could be based on an axiomatic geometry and extended objects representation.
3) The AIA tactical behavior could be defined by fuzzification of an element of a predicate based logical sentences, very limited subset of which are presented in an APPENDIX.

Conclusion
In this work it was shown that the AIA strategic targeting could be based on approximate geometric behavior of extended objects, described by fuzzy predicates. The axiom system of Boolean Euclidean geometry was fuzzified and formalized in the language of fuzzy logic, presented in [1]. Based on the same fuzzy logic, we formulated a special form of positional uncertainty, namely positional tolerance that arises from geometric constructions with extended primitives. We also addressed Euclid's first postulate, which lays the foundation for consistent geometric reasoning in all classical geometries by considered extended primitives and gave a fuzzification of Euclid's first postulate by using the same fuzzy logic. Fuzzy equivalence relation "Extended lines sameness" is introduced. We also use the fuzzy logic from [1] for fuzzy conditional inference determination, elements of which were "Degree of indiscernibility" and "Discernibility measure "of extended points. We also presented some logical principles of AIA orientation, which will allow an implementation of its fuzzy "tactical" decision making.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.