Experimental Results for Information Systems Based on Accesses Locality via Intuitionistic Fuzzy Metrics

The notion of an intuitionistic fuzzy metric space is a natural generalization of the fuzzy metric space concept which provides mechanisms to measure the degree of nearness and remoteness between two elements of a fuzzy set according to a parameter t. In this work we show how, interpreting t as a value representing the evolution time of an information system, we can use effective prediction tools in systems that show a strong locality component and where operations require the coordination of actions over the set of elements. Furthermore we show how we can tune the fuzzy metric results in order to predict access histories working on variations of the fuzzy constructions. To this end we study the suitability of a set of continuous t-norms and t-conorms to build fuzzy constructions. We have evaluated the metrics suitability according to their computation time and to the sensitiveness for different representative cases. INTRODUCTION Since the theory of fuzzy sets introduced by L.A. Zadeh [1] appeared in 1965 it has been used in a range of areas of mathematics. One of these areas, fuzzy logic, has allowed to apply fuzzy behaviour to implement industrial control devices and to use multivalued logic concepts to real scenarios. A fuzzy set is a set whose elements may be divided into the ones that belong to the set, the ones that do not belong to the set and the ones for which it is not possible to decide without a certain degree of uncertainty whether they belong to the set or not. Following Zadeh’s idea, K. Atanassov [2] introduced the concept of intuitionistic fuzzy set to allow grouping elements according to degrees of nearness and remoteness. Fuzzy topology is another example of use of Zadeh’s theory. Authors of this field have pursued the definition of a fuzzy metric space from different points of view (see [3-6], etc.). This work deals with the use of fuzzy metric concepts to achieve accesses optimization in information systems in general. Among the variety of information systems, we choose those based on access locality in the sense of nearness among the elements of the set. This characteristic appears quite often in basic information systems (compilation, physical memory accesses, transaction isolation, etc.) and also suits finely the way human organizations are structured (headquarters and geographically scattered delegations for instance). *Address correspondence to this author at the Instituto Universitario de Matemática Pura y Aplicada IUMPA-UPV, Universidad Politécnica de Valencia. 46071 Valencia, Spain; E-mail: [email protected] After a brief review to the results presented in the starting point section using a quasi-metric lattice, we will show that the Kramosil-Michalek definition of fuzzy metric [3] is the one that is better suited for our purposes. The metric formed by the chosen set and a distance (t-norm) on the elements of the set allows to use fuzzy metric and intuitionistic fuzzy metric techniques to improve our previous results. As a core introduction for our work we will display a series of measurements with experiments using several continuous t-norms and t-conorms that allow us to construct the fuzzy metric (and the intuitionistic fuzzy metric) in a way that its results approximate to different real systems behaviours. STARTING POINT In [7] we tackled the problem of detecting data access patterns with several degrees of locality using a quasi-metric lattice. The quasi-metric lattice is based upon a quasi-metric space (X,d)where X is the non-empty set of objects of the system and d is a quasi-metric on X. Using the quasi-metric space and the induced order x d y d(x, y) = 0 on the set X, we obtained an ordered set (X, d ) that allowed us to build a quasi-metric lattice. Definition 1 (Quasi-Metric Lattice) A quasi-metric lattice is a triple (X,d, ) such that (X,d) is a quasi-metric space and (X, ) is a lattice such that for x, y, z X : d(x z, y z) d(x, y) and d(x z, y z) d(x, y) For each x X , denote by k(x) the number of uses of x in [0,T], where T is the instant of time when we want to preAccesses Locality via Intuitionistic Fuzzy Metrics The Open Cybernetics and Systemics Journal, 2008, Volume 2 159 dict x’s value reliability and x is an object in an information system such as a replicated database object for example (see [7]). Now, for each x X with k(x) > 0, we construct a function also denoted by x from [0,T] into N U {0} as follows: x(t) = 0, if t = 0; x(t) = 1, if 0 < t t1(x); x(t) = 2, if t1(x) < t t2(x); ....................................... x(t) = k(x) 1, if t(k(x) 2)(x) < t t(k(x) 1)(x); x(t) = k(x), if t(k(x) 1)(x) < t T. If k(x) = 0, we define x : [0, T] [0, 1] by x(t) = 0, if 0 t T. x(t) represents object x history of accesses during time. Next we are interested in obtaining a function v from X into [0,1] such that v(x) provides a sufficiently satisfactory value of the probability of “use” of x and satisfying the following reasonable and obviously desirable fact to model locality: Proximity and Frequency Condition If x, y X satisfy that 0 < k(y) k(x) and for each j {1, ..., k(y)} there is i {1, ..., k(x)}with t j (y) ti(x ) then v(y) v(x); in addition, if for some j {1, ..., k(y)} there is i {1, ..., k(x)} with t j (y) < ti(x ) then v(y) < v(x). i.e., v allows us to compare two elements histories in a way that if the second element history adds closer to T accesses in between the first element history then v value for the second element is greater than for the first object. A relatively easy function which is a suitable candidate to provide an efficient model in our study, and whose construction is suggested by the function x(t) given above, is the function v : X [0,1] defined as follows: If k(x) = 0, then v(x) = 0, and if k(x) > 0, then v(x) = 2 j t(k (x ) ( j 1))(x ) T . j=1 k (x ) In [7] it is shown how v satisfies the nearness and frequency condition and how the quasi-metric lattice offers an adequate framework to explain the pattern accesses properties by grouping objects in classes [x] = {y X :v(x) = v(y)} in a way that if we compare two classes [x] [y] v(x) v(y) , then X := {[x] : x X} admits a lattice structure and ( X,d, ) is a quasi-metric lattice. MODEL EXTENSION IN TIME: FUZZY METRICS As a natural continuation of the initial study we pretend to take advantage of the intermediate accesses values. We will start the extension approach by recalling some basic definitions. Definition 2 (Metric) A metric on a set X is a real valued function d : X X R such that for all x, y, z X : (i) d(x, y) 0 ; (ii) d(x, y) = 0 x = y ; (iii) d(x, y) = d(y, x) ; (iv) d(x, y) d(x, z) + d(y, z) . Definition 3 (T-Norm [8]) A continuous t-norm is a binary operation :[0,1] [0,1] [0,1] such that: is commutative and associative, is continuous, a 1 = a for all a [0,1] and a b c d when a c and b d (a, b, c, d [0,1] ) Definition 4 (T-Conorm [8]) A continuous t-conorm is a binary operation :[0,1] [0,1] [0,1] such that: is commutative and associative, is continuous, a 0 = a for all a [0,1] and a b c d when a c and b d (a, b, c, d [0,1] ). Definition 5 (Fuzzy Metric Space: Kramosil-Michalek [3]) A triple (X,M, ) is a fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a fuzzy set on X 2 [0, ) such that (with x, y, z X and t, s > 0): 1) M(x, y, 0) = 0. 2) M(x, y, t) = 1 for all t > 0 x = y. 3) M(x, y, t) = M(y, x, t). 4) M(x, z, t) M(z, y, s) M(x, y, t + s). 5) M(x, y, _) : [0, ) [0,1] is left continuous. BASIS OF THE FUZZY METRIC EXTENSION While in our initial approach we choose k(x) as the number of uses of an object x between 0 and T and the instant of time when the metric is calculated. Now we choose t, where 0 t < T , and define k(x,t)= i(x) if t (ti(x)-1,ti(x)] with i(x) k(x), and k(x,t)=k(x) if t [tk(x),T]. (We take t0=0). Thus, k(x,t) is a function of the computed accesses until the instant t. Then, we define v : X [0, ) [0,1] by: v(x,0) = 0, 160 The Open Cybernetics and Systemics Journal, 2008, Volume 2 Castro-Company and Romaguera v(x, t) = 2 j t(k (x,t ) ( j 1))(x ) T . j=1 k (x,t ) if 0 < t T , and v(x,t) = 1 if t > T. Let’s suppose that v(x,T) offers a “reasonable” value of probability of x being accessed at instant T. Then we can compare v(x,t) and v(y,t) and if they show similar values then we can try to advance the prediction of y’s class, which will most possibly be [y] = [x]. This comparison is represented by the fuzzy metric space defined by the triple (X,M, ) where X is a non-empty set, a continuous t-norm and M a fuzzy set in X X (0, ) defined by: M(x, y, t) = v(x, t) v(y, t) In [9] we show the following result, which establishes that (M, ) is a fuzzy metric as defined by KramosilMichalek for an adequate v. Proposition 1: Let v : X [0, ) [0,1] be any function such that for each x X v(x,_) is a left continuous nondecreasing function (i.e., v(x, t) v(x, s) if t s ). Then (X,M, ) is a fuzzy metric space where: (a) M(x, y, 0) = 0. (b) M(x, x, t) = 1 for each t > 0 and for each x X. (c) M(x, y, t) = v(x, t) v(y, t) if x y for each t > 0. Notice that our function v defined at the beginning of the subsection satisfies Proposition 1, i.e., for each x, v is left continuous and non-decreasing. EVALUATED T-NORMS We have considered different continuous t-norms (check Dubois-Prade [10]) and compared them according to their results with the fuzzy metric in v(x,t): • Minimum: min(x, y) := x if x y y if y < x • Product: (x, y) := xy • Lukasiewicz: W(x, y) := max{x + y 1, 0}. • T-norm families: different parameter values will be compared. – Frank family: logs (1+ (s 1)(s 1) s 1 ) where s>0, s 1. – Hamacher family: xy + (1 )(x + y xy) where 0. – Sugeno-Weber family: max{ x + y 1+ xy 1+ , 0} where 1. – Schweizer-Sklar family: (max{x p + y p 1, 0}) . – Yager family: max{1 ((1 x) + (1 y) ) , 0} where p (0, ) . – Dombi family: 1 1+ (( 1 x x ) + ( 1 y y ) ) where (0, ) . – Dubois-Prade family: xy max(x, y, ) where [0,1] . Comparisons using continuous t-norm families allow us to tune the predictions precision (check related figures to see how the parameter affects the evaluation). COMPARISON OF RESULTS We evaluate v in [0,T] and we arbitrarily set T = 1000 to allow the prediction to range from no uses to plenty of them. t instants have been chosen uniformly scattered through the interval. Our tests are based on comparisons of v values during [0,T] for two different objects x, y (fuzzy set elements) using continuous t-norms. These differences are achieved applying localized variations in the first object x to obtain y. That is how we model element accesses with degrees of nearness as it happens in systems with strong locality components. Computing the variation of y is a simple implementation of the Proximity and Frequency condition. We have tried 3 kinds of variations as we display in the following subsections. Random Variations – Random Figs. (1-3) show n additional accesses to element y are performed randomly through the interval [0,T]. Fig. (2) shows v(x,t) and v(y,t) values: The final value for both experiments is high because accesses are performed throughout the end of the study for both elements. Fig. (3) shows the fuzzy metric results for the minimum, product and Lukasiewicz t-norms. Figs. (4-6) are examples of the results obtained for the t-norm families. For the case of the Hamacher family, there is no abrupt change in the family behaviour when we introduce changes Accesses Locality via Intuitionistic Fuzzy Metrics The Open Cybernetics and Systemics Journal, 2008, Volume 2 161 in the parameter value. For close to 1 values, Hamacher family gets close to the product results because as the parameter is placed in at the denominator, increments mean that M values will decrease. As there is no upper bound for this family allows us to get lower values than the ones obtained using the Lukasiewicz t-norm for the metric construction. 0 200 400 600 80