Research Article | Open Access
Alex Tserkovny, "Fuzzy Logic for Incidence Geometry", The Scientific World Journal, vol. 2016, Article ID 9057263, 18 pages, 2016. https://doi.org/10.1155/2016/9057263
Fuzzy Logic for Incidence Geometry
The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects “as if they were points.” Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid’s first postulate. Fuzzy equivalence relation “extended lines sameness” is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy “degree of indiscernibility” and “discernibility measure” of extended points.
In [1–4] it was mentioned that there are numerous approaches by mathematicians to restore Euclidean Geometry from a different set of axioms, based on primitives that have extension in space. An approach, aimed at augmenting existent axiomatization of Euclidean geometry with grades of validity for axioms (fuzzification), is also presented in [1–4]. But in contrast with [1–4], where the Lukasiewicz logic was only proposed as the basis for “fuzzification” of axioms and no proofs were presented for both fuzzy predicates and fuzzy axiomatization of incidence geometry, we use fuzzy logic from  for all necessary mathematical purposes to fill up above-mentioned “gap.”
2. Axiomatic Geometry and Extended Objects
2.1. Geometric Primitives and Incidence
Similarly to [1–3, 6–8] we will use the following axioms from . These axioms formalize the behaviour of points and lines in incident geometry, as it was defined in :(I1)For every two distinct points p and q, at least one line l exists that is incident with p and q.(I2)Such a line is unique.(I3)Every line is incident with at least two points.(I4)At least three points exist that are not incident with the same line.
The uniqueness axiom (I2) ensures that geometrical constructions are possible. Geometric constructions are sequential applications of construction operators. An example of a construction operator is the following: Connect: point × point → line.
Take two points as an input and return the line through them. For connect to be a well-defined mathematical function, the resulting line needs always to exist and needs to be unique. Other examples of geometric construction operators of 2D incidence geometry are the following: Intersect: line × line → point. Parallel through point: line × point → line.
The axioms of incidence geometry form a proper subset of the axioms of Euclidean geometry. Incidence geometry allows for defining the notion of parallelism of two lines as a derived concept but does not permit expressing betweenness or congruency relations, which are assumed primitives in Hilbert’s system . The complete axiom set of Euclidean geometry provides a greater number of construction operators than incidence geometry. Incidence geometry has very limited expressive power when compared with the full axiom system.
The combined incidence axioms (I1) and (I2) state that it is always possible to connect two distinct points by a unique line. In case of coordinate points a and b, Cartesian geometry provides a formula for constructing this unique line:
As it was shown in [1–4], when we want to connect two extended geographic objects in a similar way, there is no canonical way of doing so. We cannot refer to an existing model like the Cartesian algebra. Instead, a new way of interpreting geometric primitives must be found, such that the interpretation of the incidence relation respects the uniqueness property (I2).
Similarly to [1–4] we will refer to extended objects that play the geometric role of points and lines by extended points and extended lines, respectively. Section 3 gives a brief introduction on proposed fuzzy logic and discusses possible interpretations of fuzzy predicates for extended geometric primitives. The fuzzy logic from  is introduced as possible formalism for approximate geometric reasoning with extended objects and based on extended geometric primitives fuzzification of the incidence axioms (I1)–(I4) is investigated.
3. Fuzzification of Incidence Geometry
3.1. Proposed Fuzzy Logic
Let, and continuous function , which defines a distance between and . Notice that , where and . When normalized, the value of is defined as follows:
It is clear that. This function represents the value of “closeness” between two values (potentially antecedent and consequent), defined within single interval, which therefore could play significant role in formulation of an implication operator in a fuzzy logic. Before proving that is defined asand is from (2), let us show some basic operations in proposed fuzzy logic. Let us designate the truth values of logical antecedent and consequent as and , respectively. Then relevant set of proposed fuzzy logic operators is shown in Table 1. To get the truth values of these definitions we use well-known logical properties such as and the like.
In other words in  we proposed a new many-valued system, characterized by the set of base union () and intersection () operations with relevant complement, defined as In addition, the operators and are expressed as negations of the and correspondingly. For this matter let us pose the problem very explicitly.
We are working in many-valued system, which for present purposes is all or some of the real interval . As was mentioned in , the rationales there are more than ample for all practical purposes; the following set of 11 values is quite sufficient, and we will use this set in our illustration. Table 2 shows the operation implication in proposed fuzzy logic.
3.2. Geometric Primitives as Fuzzy Predicates
It is well known that in Boolean predicate logic atomic statements are formalized by predicates. Predicates that are used in the theory of incidence geometry may be denoted by (“a is a point”), (“a is a line”), and (“a and b are incident”). The predicate expressing equality can be denoted by (“a and b are equal”). Traditionally predicates are interpreted by crisp relations. For example, eq: is a function that assigns 1 to every pair of equal objects and 0 to every pair of distinct objects from the set N. Of course, predicates like or which accept only one symbol as an input are unary, whereas binary predicates, like and , accept pairs of symbols as an input. In a fuzzy predicate logic, predicates are interpreted by fuzzy relations, instead of crisp relations. For example, a binary fuzzy relation is a function eq: , assigning a real number to every pair of objects from N. In other words, every two objects of 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 may as well be 0.9, expressing that a and b are almost equal. In [1–4] the fuzzification of , , , and predicates was proposed.
Similarly to [1–4] we define a bounded subset as the domain for our geometric exercises. Predicates are defined for two-dimensional subsets , of Dom and assume values in . We may assume two-dimensional subsets and ignore subsets of lower dimension, because every measurement and every digitization introduces a minimum amount of location uncertainty in the data . For the point-predicate the result of Cartesian geometry involves a Cartesian point that does not change when the point is rotated: rotation-invariance seems to be a main characteristic of “point likeness” with respect to geometric operations; it should be kept when defining a fuzzy predicate expressing the “point likeness” of extended subsets of . As a preliminary definition letbe the minimal and maximal diameter of the convex hull of , respectively. The convex hull regularizes the sets A and B and eliminates irregularities. denotes the centroid of , and denotes the rotation matrix by angle (Figure 1(a)) [1–4].
Since A is bounded, and exist. We can now define the fuzzy point-predicate by
expresses the degree to which the convex hull of a Cartesian point set A is rotation-invariant: if , then is perfectly rotation-invariant; it is a disc. Here, always holds, because A is assumed to be two-dimensional. Converse to , the fuzzy line-predicate
Let us express the degree to which a Cartesian point set is sensitive to rotation. Since we only regard convex hulls, disregards the detailed shape and structure of A but only measures the degree to which A is directed.
A fuzzy version of the incidence-predicate is a binary fuzzy relation between Cartesian point sets:measures the relative overlaps of the convex hulls of A and B and selects the greater one. Here denotes the area occupied by . The greater , “the more incident” A and B: if or , then , and A and B are considered incident to degree one.
Conversely to , a graduated equality predicate between the bounded Cartesian point sets can be defined as follows: where measures the minimal relative overlap of A and B, whereas measures the degrees to which the two point sets do not overlap: if , then A and B are “almost disjoint.”
The following measure of “distinctness of points”, , of two extended objects tries to capture this fact (Figure 1(b)). We definewhere expresses the degree to which and are distinct: the greater , the more A and B behave like distinct Cartesian points with respect to connection. Indeed, for Cartesian points a and b, we would have . If the distance between the Cartesian point sets A and B is infinitely big, then as well. If , then .
3.3. Formalization of Fuzzy Predicates
In our further discussions we will also use the disjunction operator from the same table:
Proof. Let us rewrite (8) in the following way:Let us define and , and given (10) we have got the following:Therefore, given (15), we have the following.
Let us use (19) in the expression of in (15) and first find the following:In the meantime we can show that the following is also taking place:From (21) we are gettingBut from (18) we have the following:
Corollary 3. If fuzzy predicate is defined as (23), then the following type of transitivity is taking place:where and is partially ordered space that is either or vice versa (note: both conjunction and implication operations are defined in Table 1).
Proof. From (17) we havethenMeanwhile, from (17) we have the following:Case 1 (). From (26) we haveFrom (27) and (28) we have to prove thatBut (29) is the same as , from which we get
From definition of implication in fuzzy logic (11) and since for condition is taking place, therefore .
Case 2 (). From (26) we haveFrom (27) and (30) we have to prove thatBut (31) is the same as , from which we get
From definition of implication in fuzzy logic (11) and since for condition is taking place, therefore .
3.4. Fuzzy Axiomatization of Incidence Geometry
Proposition 5. If fuzzy predicates and are defined like (37) and (13), respectively, then axiom is fulfilled for the set of logical operators from a fuzzy logic . (For every two distinct points a and b, at least one line l exists, i.e., incident with a and b.)
Proposition 6. If fuzzy predicates , , and are defined like (37), (17), and (16), respectively, then axiom is fulfilled for the set of logical operators from a fuzzy logic  (for every two distinct points a and b, at least one line l exists, i.e., incident with a and b, and such a line is unique).
Proof. Let us take a look at the following implication:But from (27) we haveAnd from (16)From (40) and (41) we see that , which means thattherefore the following is also true:Now let us take a look at the following implication: . Since , we are getting . Taking into account (43) we have the following:Since, from (16), , then with taking into account (44) we have gotten the following: Since , from (45) we are getting Finally, because we have
Proposition 7. If fuzzy predicates and are defined like (17) and (16), respectively, then axiom is fulfilled for the set of logical operators from a fuzzy logic . (Every line is incident with at least two points.)
Proposition 8. If fuzzy predicate is defined like (16), then axiom is fulfilled for the set of logical operators from a fuzzy logic . (At least three points exist that are not incident with the same line.)
3.5. Equality of Extended Lines Is Graduated
In  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. It is not true in case of extended points and lines. 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,” that is, “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 for not only Boolean values, but also values in the whole interval .
3.6. Incidence of Extended Points and Lines
As it was demonstrated in , 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 9. For an extended point P and an extended line L we define the incidence relation bywhere the subset relation refers to P and L as subsets of the underlying metric space.
The extended incidence relation (54) 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, that is, since , we will be able to use relation (54) as part of fuzzified Euclid’s first postulate later on.
3.7. Equality of Extended Points and Lines
As stated in previous sections, equality of extended points and equality of extended lines are 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.
3.7.1. Metric Distance
In  it was mentioned that a pseudo metric distance, or pseudo metric, is a map from domain M into the positive real numbers (including zero), which is minimal and symmetric and satisfies the triangle inequality:d is called a metric, if the following takes place:Well-known examples of metric distances are the Euclidean distance or the Manhattan distance. The “upside-down-version” of a pseudo metric distance is a fuzzy equivalence relation with respect to a proposed t-norm. The next section introduces the logical connectives in a proposed t-norm fuzzy logic. We will use this particular fuzzy logic to formalize Euclid’s first postulate for extended primitives in Section 5. The reason for choosing a proposed fuzzy logic is its strong connection to metric distance.
3.7.2. The t-Norm
Proposition 10. In proposed fuzzy logic the operation of conjunction (10) is a t-norm.
Proof. The function is a t-norm if the following hold:(1)Commutativity: .(2)Associativity: .(3)Monotonicity: .(4)Neutrality: .(5)Absorption: .Commutativity. Considertherefore
Case 1 (). Considerfrom where we have thatIn other words and .
For the case we are getting results similar to (59): that is, and . Consider
Monotonicity. If , then, givenwe are getting and , whereas, for the case , .
Neutrality. Considerfrom which the following is apparent:Absorption. Consider since .
3.7.3. Fuzzy Equivalence Relations
As mentioned above, the “upside-down-version” of a pseudo metric distance is a fuzzy equivalence relation with respect to the proposed t-norm . A fuzzy equivalence relation is a fuzzy relation on a domain M, which is reflexive, symmetric, and -transitive:
(1) Reflexivity. comes from (66) because .
(2) Symmetricity. . Consider the following:buttherefore .
(3) Transitivity. -lattice.
From (66) letthenButNow compare (72) and (69). It is apparent that . The same is true for . And lastly , when . Given that , we are getting the proof of the fact that .
Note that relation is called a fuzzy equality relation, if additionally separability holds: . Let us define a pseudo metric distance for domain M, normalized to 1, asFrom (66) we are getting
3.7.4. Approximate Fuzzy Equivalence Relations
In  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 is very similar to L, its intersection with M yields an extended point which is very similar to P. It is desirable to model this fact. To do so, it is necessary 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 Poincare, who repeatedly pointed this fact out, for example, in , 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. Gerla shows that modelling the Poincare paradox in graduated context transitivity may be replaced by a weaker form : Here is a lower-bound measure (discernibility measure) for the degree of transitivity that is permitted by q. A pair that is reflexive, symmetric, and weakly transitive (75) is called an approximate fuzzy -equivalence relation. Let us rewrite (75) as follows:where are defined in (72) and (69) correspondingly. ButFrom (77) in order to satisfy condition (76) we havethat is, we haveBy using (79) in (77) we are getting that . From (69) we are getting and subsequently inequality (76) holds.
In  it was also mentioned that an approximate fuzzy -equivalence relation is the upside-down-version of a so-called pointless pseudo metric space :Here, is a (not necessarily metric) distance between extended regions and is a size measure and we are using an operation disjunction (12) also shown in Table 1. Inequality is a weak form of the triangle inequality. It corresponds to the weak transitivity (75) 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