_{1}

^{*}

In this work, we present an account of our recent results on applications of rough mereology to problems of 1) knowledge granulation; 2) granular preprocessing in knowledge discovery by means of decision rules; 3) spatial reasoning in multi-agent systems in exemplary case of intelligent mobile robotics.

Rough Mereology (RM, for short) is a paradigm in the domain of Approximate Reasoning, which is an extension of Mereology, see [1-3]. RM is occupied with a class of similarity relations among objects which are expressed as rough inclusions, see [4-8]. A rough inclusion is a ternary predicate, read as: The object x is a part in object y to a degree of. For a detailed discussion of rough inclusions along with an extensive historic bibliography, see [

Rough inclusions can be induced from continuous tnorms, see [8,9] whose classical examples are

Rough inclusions can be obtained from residual implications cf. [

Then,

is a rough inclusion. A particular case of continuous tnorms are Archimedean t-norms which satisfy the inequality for each. It is wellknown, cf. [8,9], that each Archimedean t-norm T admits a representation

where the function is continuous decreasing with and is the pseudo-inverse to f_{T}, i.e.,. There are two basic Archimedean t-norms: L and P. Their representations are 1.1.2. (4)

and

For an Archimedean t-norm T, we define the rough inclusion µ^{T} on the interval [0,1] by means of 1.1.3. (6)

Specific recipes are 1.1.4. (7)

and

Both residual and Archimedean rough inclusions satisfy the transitivity condition, see [

(Trans) if then

An important domain where rough inclusions play a dominant role in our mining of knowledge is the realm of information/decision systems in the sense of Pawlak, cf., [8,10]. For a recent view on information systems in a broader perspective of applications, see [

For a rough inclusion of the form µ_{T}, we define a rough inclusion by means of

Then, is a rough inclusion with the associated ingredient relation of identity and the part relation empty. For the Łukasiewicz t-norm, the rough inclusion is given by means of the formula, cf. [

We introduce the set With its help a new form of (11) is 1.1.5. (12)

Formula (12) witnesses that reasoning based on the rough inclusion is the probabilistic one. Rough inclusions of type satisfy the transitivity condition (Trans) and are symmetric. Formula (12) can be abstracted for set and geometric domains. For finite sets A, B, we let 1.1.6. (13)

where denotes the cardinality of the set X. For bounded measurable sets A, B in an Euclidean space_{}

defines a rough inclusion µ^{G}, where is the area (Lebesgue measure) of X. Both µ^{S} and µ^{G} are symmetric but not transitive. Default value is 1 in case A is empty or of measure 0.

Granular computing goes back to Lotfi A. Zadeh and its idea lies in aggregating objects with respect to a chosen similarity measure into “granules” of satisfactorily similar objects, see [5,6,8,12,13].

For a given rough inclusion μ, the granule of the radius r about the center u is defined as the class of property ^{}

where

Hence, an object v belongs in the granule if and only if.

For the exemplary information system in

Clearly, granules of all radii between 0 and 1 about all objects in a given universe U, provide a quasi-metric topology on U.

Quasi-metric topology mentioned above may be applied in the build-up of mereogeometry, cf. [8,14,15]. Elementary geometry was defined by Tarski as a part of Euclidean geometry which can be described by means of 1st order logic. There are two main aspects in formalization of geometry: one is metric aspect dealing with the distance underlying the space of points which carries geometry and the other is affine aspect taking into account linear relations. In the Tarski axiomatization for 2- and 3-dimensional elementary geometry, recalled in [

of the betweenness relation. The only logical predicate required is the identity “=”. Equidistance relation denoted (or, as a congruence:) means that the distance from x to y is equal to the distance from u to z (pairs x, y and u, z are equidistant). Betweenness relation is denoted (x is between y and z). Van Benthem [

Observe that the mereological distance differs essentially from the standard distance: the closer are objects, the greater is the value of means X = Y whereas means that X,Y are either externally connected or disjoint, no matter what is the Euclidean distance between them. We use to define in our context the relation N of nearness proposed in [

Here, means that X is closer to U than V is to U. We introduce a betweenness relation in the sense of Van Benthem T_{B} modeled on betweenness proposed in [

The principal example bearing, e.g., on our approach to robot control deals with rectangles in 2D space regularly positioned, i.e., having edges parallel to coordinate axes. We model robots (which are represented in the plane as discs of the same radii in 2D space) by means of their safety regions about robots; those regions are modeled as squares circumscribed on robots. One of advantages of this representation is that safety regions can be always implemented as regularly positioned rectangles. Given two robots a, b as discs of the same radii, and their safety regions as circumscribed regularly positioned rectangles A, B, we search for a proper choice of a region X containing A and B with the property that a robot C contained in X can be said to be between A and B. In this search we avail ourselves with the notion of betweenness relation T_{B}.

Taking the rough inclusion defined in (14), for two disjoint rectangles A,B, we define the extent, , of A and B, as the smallest rectangle (the convex hull) containing the union. Then we have the claim: the only object between C and D in the sense of the predicate T_{B} is the extent of C and D.

For a proof, as linear stretching or contracting along an axis does not change the area relations, it is sufficient to consider two unit squares A, B of which A has as one of vertices whereas B has with as the lower left vertex (both squares are regularly positioned).

Then the distance κ between the extent and either of A, B is. For a rectangle with x in, y in, we have that

For, we find that

and, similarly, , i.e., Ф is increasing in x, y reaching the maximum when R becomes the extent of A, B.

We return to this result when we discuss planning and navigation problems in behavioral robotics for teams of autonomous mobile robots.

Planning is concerned with setting a trajectory for a robot endowed with some sensing devices which allow it to perceive the environment in order to reach by the robot a goal in the environment at the same time bypassing obstacles.

A method of potential field, see [

The basic property of the potential is that its density (=force) increases in the direction toward the target. We observe this property in our construction. We apply the geometric rough inclusion

where is the area of the region x. In our construction of the potential field, region will be squares: robots are represented by squares circumscribed on them (simulations were performed with disk-shaped Roomba robots, the intellectual property of iRobot Inc.).

Geometry induced by means of a rough inclusion can be used to define a generalized potential field: the force field in this construction can be interpreted as the density of squares that fill the workspace and the potential is the integral of the density. We present now the details of this construction.

We construct the potential field by a discrete construction. The idea is to fill the free workspace of a robot with squares of fixed size in such a way that the density of the square field (measured, e.g., as the number of squares intersecting the disc of a given radius r centered at the target) increases toward the target. To ensure this property, we fix a real number, the field growth step in the interval (0, square edge length); in our exemplary case the parameter field growth step is set to 0.01. The collection of squares grows recursively with the distance from the target by adding to a given square in the -th step all squares obtained from it by translating it by k• field growth step (with respect to Euclidean distance) in basic eight directions: N, S, W, E, NE, NW, SE, SW (in the implementation of this idea, the floodfill algorithm with a queue has been used, see [

where d is the field growth step, k is the number of generations. Hence,

and thus

The potential can be taken as. The force field is the negative gradient of,

Hence, the force decreases with the distance r from the target slower than traditional Coulomb force. It has advantages of slowing the robot down when it is closing on the target. Parameters of this procedure are: the field growth step set to 0.01, and the size of squares which in our case is 1.5 times the diameter of the Roomba robot.

A view of mereological potential field is given in

A robot should follow the path proposed by planner shown in

We recall that on the basis of the rough inclusion μ, and mereological distance κ, geometric predicates of nearness and betweenness, are redefined in mereological terms. Given two robots a, b as discs of same radii, and their safety regions as circumscribed regularly positioned rectangles A, B, we search for a proper choice of a region X containing A and B with the property that a robot C contained in X can be said to be between A and B. We recall that the only object X with these property is the extent, i.e., the minimal rectangle containing the union, i.e., the convex hull of the union. For details of the exposition which we give now, please consult [19,20]. For robots a, b, c, we say that the robot b is between robots a and c, in symbols

in case the rectangle is contained in the extent of, i.e.1.1.7. (28)

This allows as well for a generalization of the notion of betweenness to the notion of partial betweenness which models in a more realistic manner spatial relations among a, b, c; we say in this case that the robot b is between robots a and c to a degree of at least r, in symbols,

i.e.1.1.8. (30)

For a team of robots, , an ideal formation IF on T is a betweenness relation (between^{..}^{.}) on the set T of robots. In implementations, ideal formations are represented as lists of expressions of the form

indicating that the object r0 is between r1, r2, along with a list of expressions of the form

indicating triples which are not in the given betweenness relation. To account for dynamic nature of the real world, in which due to sensory perception inadequacies, dynamic nature of the environment, etc., we allow for some deviations from ideal formations by allowing that the robot which is between two neighbors can be between them to a degree in the sense of (29). This leads to the notion of a real formation. For a team of robots, , a real formation RF on T is a betweenness to degree relation (betweendeg^{...}) on the set T of robots. In practice, real formations will be given as a list of expressions of the form,

indicating that the object r0 is to degree of δ in the extent of r1, r2, for all triples in the relation (between—deg^{...}), along with a list of expressions of the form,

indicating triples which are not in the given betweenness relation.

Description of formations, as proposed above, can be a list of relation instances of large cardinality, cf., examples below. The problem can be posed of finding a minimal set of instances sufficient for describing a given formation, i.e., implying the full list of instances of the relation. This problem turns out to be NPhard, see [_{—}predicate param1... paramN))

For each object on a list (and for a formation as a whole) an extent can be derived and in facts, in most cases only extents of those objects are considered. We have defined two possible types of objects: 1) Identifier: robot or formation name (where formation name can only occur at top-level list as the first element); 2) Predicate: a list in LISP meaning where first element is the name of given predicate and other elements are parameters; number and types of parameters depend on given predicate. Minimal formation should contain at least one robot. For example(formation2 roomba0)

To help understand how predicates are evaluated, we need to explain how extents are used for computing relations between objects. Suppose we have three robots (roomba0, roomba1, roomba2) with roomba0 between roomba1 and roomba2 (so the predicate is satisfied). We can draw an extent of this situation as the smallest rectangle containing the union oriented as a regular rectangle, i.e., with edges parallel to coordinate axes. This extent can be embedded into bigger structure: it can be treated as an object that can be given as a parameter to predicate of higher level in the list hierarchy. For example, cf.

We can easily find more than one situation of robots that fulfill this example description. That is one of the features of our approach: one s-expression can describe many situations. This however makes very hard to find minimal s-expression that would describe already given arrangement of robots formation (as stated earlier in this chapter, the problem is NP-hard). Typical formation description may look like below, see [

(cross

(set

(max—dist 0.25 roomba0 (between roomba0 roomba1 roomba2))

(max—dist 0.25 roomba0 (between roomba0 roomba3 roomba4))

(not—between roomba1 roomba3 roomba4)

(not—between roomba2 roomba3 roomba4)

(not—between roomba3 roomba1 roomba2)

(not—between roomba4 roomba1 roomba2)

)

)

This is a description of a formation of five Roomba robots arranged in a cross shape. The max-dist relation is used to bound formation in space by keeping all robots close one to another. We show a screen-shot of robots (submittted by Dr Paul Osmialowski) in the initial formation of cross-shape in a crowded environment, see Figures 4 and 5. These behaviors witness the flexibility of our definition of a robot formation: first, robots can change formation, next, as the definition of a formation is

relational, without metric constraints on robots, the formation can manage an obstacle without losing the prescribed formation (though, this feature is not illustrated in figures in this chapter).

We recall that for a decision system with the decision d, a decision rule is an implication 1.1.9. (35)

where is a set of attributes, v_{a} is a value of the attribute a; the formula is the descriptor whose meaning is the set; these elementary meanings define the meaning of the formula r by interpreting the disjunction of descriptors as the union of meanings, the conjunction of descriptors as the intersection of meanings. Coefficients of accuracy and coverage are defined for a decision algorithm D, trained on a training set T_{r} and tested on a test set T_{st}, as, respectively, quotients 1.1.10.

1.1.11. where ncorr is the number of test objects correctly classified by D and ncaught is the number of test objects classified, and,

where ntest is the number of test objects. Thus, the product

gives the measure of the fraction of test objects correctly classified by D. We have already mentioned that accuracy and coverage are often advised to be combined in order to better express the trade-off between the two: one may have a high accuracy on a relatively small set of caught objects, or a lesser accuracy on a larger set of caught by the classifier objects. Michalski [

where A stands for accuracy and C for coverage. With the symbol MI, we denote the Michalski index as defined in (36).

Our idea of augmenting existing strategies for rule induction consists in using granules of knowledge. The principal assumption we can make is that the nature acts in a continuous way: if objects are similar with respect to judiciously and correctly chosen attributes, then decisions on them should also be similar. A granule collecting similar objects should then expose the most typical decision value for objects in it while suppressing outlying values of decision, reducing noise in data, hence, leading to a better classifier. The idea of a rough mereological granular classifier was proposed and refined in [22,23]. We give a resume of the most important ideas and results in this area. We assume that we are given a decision system from which a classifier is to be constructed; for a discussion on the subject of data classification, cf., e.g. [

We intend that becomes a new universe of the decision system whose name will be the granular reflection of the original decision system. It remains to define new attributes for this decision system. Each granule g in is a collection of objects; attributes in the set can be factored through the granule g by means of a chosen strategy, i.e., for each attribute q in, the new factored attribute 1.1.13. (38)

is defined. In effect, a new decision system

is defined. The object v^{g} with values is called the granular reflection of the granule g. Granular reflections of granules need not be objects found in data set; yet, the results show that they mediate very well between the training and test sets. We include in

can be taken as the covering. As the strategy we adopt the majority voting. On the basis, again, of

In order to demonstrate results of this approach on real data, we consider a standard data set the Australian Credit Data Set from University of California at Irvine Machine Learning Repository and we collect the best results for this data set by various rough set based methods in

We begin with a classifier in which granules computed by means of the rough inclusion μ_{L} form a granular reflection of the data set and then to this new data set the exhaustive classifier available in RSES is applied.

1) The data set is input;

2) The training set is chosen at random. On the training set, decision rules are induced by means of exhaustive, covering and LEM algorithms implemented in the RSES system;

3) Classification is performed on the test set by means of classifiers of pt. 2;

4) For consecutive granulation radii r, granule sets are found;

5) Coverings are found by a random irreducible choice;

6) For granules in, for each r, we determine the granular reflection by factoring attributes on granules by means of majority voting with random resolution of ties;

7) For found granular reflections, classifiers are induced by means of algorithms in pt. 2;

8) Classifiers found in pt. 7, are applied to the test set;

9) Quality measures: accuracy and coverage for classifiers are applied in order to compare results obtained, respectively, in pts. 3 and 8.

In

We can compare results expressed in terms of the Michalski index MI as a measure of the trade-off between accuracy and coverage; for template based methods, the best MI is 0.891, for covering or LEM algorithms the best value of MI is 0.804, for exhaustive classifier MI is equal to 0.907 and for granular reflections, the best MI value is 0.915 with few other values exceeding 0.900. What seems worthy of a moment’s reflection is the number of rules in the classifier. Whereas for the exhaustive classifier in nongranular case, the number of rules is equal to 5597, in granular case the number of rules can be surprisingly small with a good MI value, e.g., at r = 0.5, the number of rules is 293, i.e., 5 percent of the exhaustive classifier size, with the best MI of all of 0.915. This compression of classifier seems to be the most impressive feature of granular classifiers. This initial, simple granulation, suggests further ramifications. For instance, one can consider, for a chosen value of ε in [0,1], granules of the form.

and repeat with these granules the procedure of creating a granular reflection and building from it a classifier. Another yet variation consists in mimicking the performance of the Łukasiewicz based rough inclusion and introducing a counterpart of the granulation radius in the form of the catch radius,. The granule is then dependent on two parameters: ε and and its form is 1.1.14. (40)

Results of classification by granular classifier induced from the granular reflection obtained by means of granules (39) are shown in

A particular but important problem in data analysis is the treatment of missing values. In many data, some values of some attributes are not recorded due to many factors, like omissions, inability to take them, loss due to some events etc. Our idea is to embed the missing value into a granule: by averaging the attribute value over the granule in the way already explained, it is hoped the the average value would fit in a satisfactory way into the position of the missing value. We will use the symbol *, commonly used for denoting the missing value; we will use two methods for treating *, i.e, (1) either * is a don’t care symbol meaning that any value of the respective attribute can be substituted for *, hence, * = v for each value v of the attribute, or (2) * is a new value on its own, i.e., if * = v

then v can only be *; see [_{L} in accordance with the chosen method for treating *. In plain words, objects with missing values are in a sense absorbed by close to them granules and missing values are replaced with most frequent values in objects collected in the granule; in this way the method (1) or (2) is combined with the idea of a frequent value, in a novel way. We have thus four possible strategies: 1) Strategy A: in building granules * = don’t care, in repairing values of *, * = don’t care; 2) Strategy B: in building granules * = don’t care, in repairing values of *, * = *; 3) Strategy C: in building granules * = *, in repairing values of *, *= don’t care; 4) Strategy D: in building granules * = *, in repairing values of *, * = *. We show how effective are these strategies, see [

Strategy A reaches the accuracy value for data with missing values within 94 percent of the value of accuracy without missing values (0.9407 to 1.0) at the radius of 0.875. With Strategy B, accuracy is within 94 percent from the radius of .875 on. Strategy C is much better: accuracy with missing values reaches 99 percent of accuracy in no missing values case from the radius of 0.625

on. Strategy D gives results slightly better than C with the same radii. We conclude that the essential for results of classification is the strategy of treating the missing value of * as * = * in both strategies C and D; the repairing strategy has almost no effect: C and D differ very slightly with respect to this strategy.

Rough inclusions used in Sections 5.1 - 5.3 in order to build classifiers do take, to a certain degree, into account the distribution of values of attributes among objects, by means of the parameters ε and. The idea that metrics used in classifier construction should depend locally on the training set is, e.g., present in classifiers based on the idea of nearest neighbor, see, e.g., a survey [

and

The weak variant of the rough inclusion is defined, see [

These similarity measures will be applied in building granules. Tests are done with the Australian credit data set; the results are validated by means of the 5-fold cross validation (CV-5). For each of t-norms: M, P, L, three cases of granulation are considered:

1) Granules of training objects (GT);

2) Granules of rules induced from the training set (GRT);

3) Granules of granular objects induced from the training set (GGT).

In this approach, training objects are made into granules for a given ε. Objects in each granule g about a test object u, vote for decision value at u as follows: for each decision class c, the value 1.1.16. (44)

is computed where the weight is computed for a given t-norm T as 1.1.17. (45)

The class c* assigned to u is the one with the largest value of p. Weighted voting of rules in a given granule g for decision at test object u goes according to the formula, where 1.1.18. (46)

and weight is computed as

The optimal (best) results in terms of accuracy of classification are collected in

These results show that rough mereological granulation provides better or at least on par with best results by

other methods accuracy of classification at the radically smaller classifier size measured as the number of decision rules in it.

In [

The idea of a granular rough mereological logic consists in measuring the meaning of a unary predicate in the model which is a universe of an information system against a granule defined by means of a rough inclusion. The result can be regarded as the degree of truth (the logical value) of the predicate with respect to the given granule. The obtained logics are intensional as they can be regarded as mappings from the set of granules (possible worlds) to the set of logical values in the interval [0,1], the value at a given granule regarded as the extension at that granule of the intension. For an information/decision system, with a rough inclusion ν, e.g., , on subsets of U and for a collection of unary predicates Pr, interpreted in the universe U (meaning that for each predicate Ф in Pr the meaning [Ф] is a subset of U), we define the intensional logic by assigning to each predicate Ф in Pr its intension defined by its extension at each particular granule g, as 1.1.19. (48)

With respect to the rough inclusion the formula (47) becomes

A formula Ф interpreted in the universe U of a system is true at a granule g with respect to a rough inclusion ν if and only if 1.1.20. (50)

and Ф is true if and only if it is true at each granule g. A rough inclusion ν is regular when holds if and only if. Hence, for a regular ν, a formula Ф is true if and only if for each granule g. A particularly important case of a formula is that of decision rules; clearly, for a decision rule in the descriptor logic, the rule r is true at a granule g with respect to a regular rough inclusion ν if and only if

Analysis of decision rules in the system can be given in a more general setting of dependencies. For two sets of attributes, one says that D depends functionally on C when, symbolically denoted C⥟D, where is the Xindiscernibility relation defined as, see [

Functional dependence can be represented locally by means of functional dependency rules of the form 1.1.22. (53)

whereis the formula and. We introduce a regular rough inclusion on sets defined as, where r = 1 if

, if, otherwise r = 0.

Then one proves that is a functional dependency rule if and only if α is true in the logic induced by, see [

We have presented a survey of recent results on rough mereological approach to knowledge discovery. Presented have been applications to problems of data mining (in the form of granular classifiers), intelligent robotics, in the form of mereological potential field based planners, and, a theory of robot formations based on rough mereogeometry. Finally, we have outlined the intensional rough mereological logic for reasoning about knowledge in information/decision systems. These results show that rough mereology can be a valuable tool in the arsenal of artificial intelligence. Results on classification show that granular mereological preprocessing reduces the size of classifiers at as good quality of classification as best obtained in the past by other methods. Formations of autonomous mobile robots defined by means of mereogeometry allow for flexible managing of environments, as shown in respective figures, due to relational character of that definition. Mereological potential fields also deserve an attention due to simplicity of their definition and close geometric relation to the location of the goal and obstacles. We do hope that this survey will give researchers in the fields of Knowledge Discovery and related ones an insight into this new paradigm and will prompt them to investigate it and apply it on their own. There are many problems to investigate, among them, it seems, one of most interesting is whether it is possible to, at least approximately, predict optimal values of parameters of granular classifiers, i.e., of the radius of granulation, the parameter ε, and the catch radius. It this happens, the goal of this survey is achieved.

The author wishes to thank his former students, Dr Piotr Artiemjew and Paul Osmialowski who cooperated with the author in the tasks of implementing author’s ideas and test them on real data and autonomous mobile robots, respectively.