Generalization of Fuzzy and Classic Logic in NPL2v

Transcription

1 Generalization of uzzy and Classic Logic in NPL2v HELGA GONZAGA CLAUDIO INÁCIO DE ALMEIDA COSTA GERMANO LAMBERT-TORRES Escola ederal de Engenharia de Itajubá Av. BP303 Itajubá MG BRAZIL Abstract - This paper presents a knowledge modeling technique for the making of knowledge specialists systems basis, as well as a logic tool for the introduction of the inference motor. The use of a logic that expands the classic logic, allows the inclusion of spare knowledge, indefinites and inconsistencies in the knowledge basis. The short circuit principle, the commutative, idempotence and duality characteristics, were chosen as basic principles, together with the direction given by da Costa [1]: If the propositions we are working with be of good behavior, all the valid formulae in the classic calculus must remain unchanged. This is a warrant that the generated methodology maintain the classic characteristics already accepted and of wide practical application, whereas make possible to take advantage of fuzzy and paraconsistent logic. Key Words: Paraconsistent Logic, uzzy Logic 1 Introduction The construction of a specialist system necessarily goes through a labeled knowledge acquisition and codification phase. or the choice of facts and rules that belongs to the knowledge basis, it is necessary to consult specialists that have a wide knowledge regarding the problem to be considered. The major difficulties usually happen in this phase. The inference motor drives the knowledge base. rom the facts and rules given by the specialists, the inference motor uses logic as to arrive to conclusions and infer new facts. Actually the classic logic is the one used, which in an easy way gives to each fact the state of true or false. Then, the first difficulty happens: The specialist s knowledge, in spite of having a sound logic aspect, which is well shaped upon the classic logic, has another well inaccurate side. The rigid concepts of true and falseness are not that enough for the proper representation of the process of decision take that human beings make. The human reasoning does not deal only with both ends (true and false), but also with an infinite amount of intermediate states as to represent incomplete and partial knowledge. There are areas in which a specialist is not sure with respect to his own statements, and other ones for which he presumes his lack of knowledge regarding specific aspects of one problem. There is yet another rather common situation. After a thorough analysis of the information supplied by the specialist, contradictions are noticed. The facts and rules supplied may combine and generate propositions that are in opposition between them; that is to say, the logic states of true and false are present simultaneously in same proposition. The discovery of inconsistencies regarding the acquired knowledge carries many concerns. Because of the use of classic logic, the inference motor, because of contradictions, may drive the specialist system to loose his capacity of decision making regarding true and false, falling in the trivial trap. Aiming to improve the specialist systems, and solving these difficulties, following targets must be searched: first the extension of the classic logic so as to comprise the intermediate states between true and false, allowing a smooth transition. Second, space has to be provided for the undefined state, allowing knowledge representation. At last, the inconsistencies also should be modeled, allowing true and falseness simultaneous situations, but without the risk of becoming trivial. 2 The undamental Principles When fostering the paraconsistent logic, Newton da Costa suggested as one direction line of his work the statement that: If the propositions we work with be of good behavior, all the valid formulae from the classic calculus must remain unchanged. Such a statement aims to preserve the consistency with the existent applications and strongly accepted and used as a direction for searching new rules and logic

2 procedures, which allow to reach the proposed objectives. In working with the logic operators of conjunction (AND), disjunction (OR) and deny (NOT) the aim also is to preserve the validity of those following proprieties: Commutativeness, Idempotence, Duality and Short Circuit. The first propriety allows the fast reduction of the total number of necessary combinations for writing the true tables, without worrying about the operation order. The idempotence shows the propriety of the likely logic states do not change because of conjunction and disjunction operations. The duality makes easy the transit and the comparison between operators and logic states, keeping logic s very characteristic symmetry. Also the deny has a strong hold with this principle. Last, the short circuit principle supplies a fast way for shortening the evaluation of logic expressions as well as indicates a way to follow when bad behavior logic states occur. The short circuit is expressed by the formulae bellow: T v X = T (True OR anything = True) & X = (alse AND anything = alse) (Dual of previous) In Table 1, these proprieties are shown exactly what the true table of classic logic says. The lines (1) and (4) are shortened by the idempotence of operators (AND) and (OR). The lines (2) and (3) could be fused in but one if the commutativity be remembered. Duality is responsible for symmetry between true and false states and between the operators (AND) and (OR), with the outcome of the definition for deny. Table 1: Table True of Classic Logic T T T T (1) T T T (2) T T (3) (4) 2.1 uzzy Logic uzzy logic provides the theoretical basis for modeling the partial knowledge. The smooth transition between two classic logic states is made as follow: the actual zero value is given to the false logic state, and the actual value one is given to the true logic state. The intermediate values of the straight line show the true partial levels, as for instance 0.2 or 0.8. The notation given to each logic proposition is named aptness degree (µ). This makes possible the existence of facts that model the incomplete knowledge based in the knowledge of the specialist systems. The operators AND, OR and NOT are then redefined, as proposed by Zadeh [2]: Table 2: Table True of uzzy Logic µ p µ q min(µ p,µ q ) Max(µ p,µ q ) 1-µ q This definition enlarges the classic logic, allowing working with intermediate states of logic. It is very important to notice that in the extreme cases, with zero and one values (false and true), the table above goes straight to the true table of the classic logic. By same way, it is stressed the validity of commutative, idempotence, duality and short circuit proprieties, which warrants the validity of the practical developments already made. With these changes in the operators, it is possible go by to handle the propositions as numbers of the continuous interval from zero to one, and no longer as discreet points. 2.2 Logic of 3 States By the middle of this century, the theoretical developments have shown the necessity of accepting the existence of propositions that cannot be defined as true or false. Kleene [3], discussing such propositions, presents the tables true of the so called logic of 3 states: the true (T), the false () and the undefined (). This third logic state adds the following lines on table 1: Table 3: Table Partial True from Logic of 3 States T (?a) T (A) I T (?a) T (B) (?b) (C) I (?b) (D) I (E) So, the 3 states logic contains the whole classic logic, keeping the directive: If the propositions we work with are of good behavior, all the valid formula from the classic calculus must stay

3 unchanged. So, the lines (1) to (4) from table 1 stay valid, being only necessary to add up the lines from (A) to (E) for the situations pertaining to the new undefined logic state (). Considering that if the commutative, duality and mainly the short circuit properties should be stay valid, it is possible to notice that the half of lines (A),(B),(C) and (D) had but the option of assuming the values as described above. The only new would be the interaction between the states good behavior (true and false) and the state bad behavior (undefined). Such results are identified by (?a) (true versus undefined) and (?b) (false versus undefined). But line (E), because of the idempotence propriety, could only result in the value shown. 2.3 Unifying 3 Staes Logic and uzzy Logic Within the three proposed objectives, two were already reached. uzzy logic allowed the existence of intermediate states between the true (one) and false (zero). The minimizing and maximizing operators, as substitutes for classic operators, allow the existence and evaluation of logic expressions containing intermediate values, as 0.2 or 0.8.The 3 states logic has introduced the concept of a indefinite logic proposition, allowing the modeling of facts about which no knowledge is available. Before proceeding, it is necessary to unify these gains, by grouping them in one sole structure. Surprisingly, it is easy doing that by associating the undetermined third Logic State to the true number 0.5. So, the whole true table from the 3 states logic may be taken as a special case of the fuzzy logic. The minimizing (AND) and maximizing (OR) operators and the fuzzy deny remain valid. This explain the chosen results for entries (?a) and (?b) from table 3, and stands for Kleene s original proposition. The classic logic starts from two discreet points. The fuzzy logic generates a straight line uniting such points and the 3 states logic set up the straightline central point. 0 0,5 1 igure 1: Unification of 3 States Logic and uzzy Logic 3 Paraconsistent Logic The last objective is the inconsistency incorporation into the logic that will command the inference motor. By placing the fourth logic state (S), which represents the inconsistencies, result the following Hasse s diagram: S igure 2: Hasse s Diagram Using again the fundamental directive, so as the four chosen proprieties, it is possible to fill up the new lines from true table, made with the addition of the inconsistent state. ollowing lines will be added into true table: Table 4: Table True for Paraconsistent Logic T S S (?c) T S (I) S T S (?c) T (II) S S (?d) (III) S S (?d) (I) S S S S () S???? (I) S???? (II) The points marked with (?) need something more to be determined. or this purpose it is required to use the Noted Paraconsistent Logic of Two alues Notation (NPL2v). 3.1 Noted Paraconsistent Logic of Two alues Notation (NPL2v) In the fuzzy logic, each proposition has a notation given by a number from zero to one, named pertinence degree. In the NPL2v, each proposition is noted by two values, also in the unit interval. They represent the degree (α) and the degree (β). To represent the logic prepositions in NPL2v, it is used the unit square in the Cartesian plane (USCP):

4 Dis β 1 By the points -- pass the line Perfectly Inconsistent (LPI), that contain the points of maximum inconsistency (α=β), as per the figure bellow: LPC UD LPI S1 0 α Belief 1 IdG igure 3: Unit Square in the Cartesian Plane (USCP) In it, it is possible to represent, for instance, one proposition A(0,2;0,8), that is to say, with a degree of 0,2 and of 0,8, by a point in the USCP in these coordinates. It is possible to see also that one proposition totally true has a degree 1 and 0, as well as the whole falseness would be in the point 0 and 1. Remembering that 0.5 represents the undefined state, a totally undefined proposition would have a degree of and un equal to 0.5. inally, some propositions may be true and false at same time, originating the points of inconsistency ( 1 and 1) and ( 0 and 0). Representing these notable points in the USCP results: igure 4: Notable Points from USCP By the points T-- it is possible to trace the so called Line Perfectly Consistent (LPC) which contains points that besides representing incomplete or partial knowledge, show total coherence between and (α+β=1). A very interesting notice is that the straight line of figure 1 (unification of fuzzy logic and 3 states) now concur in some way with LPC. CD igure 5: Degrees and Lines from USCP In the above figure are indicated also tree other interesting measures: The Certainty Degree (CD), which measure the distance from one proposition up to LPI; The Uncertainty Degree (UD), which measure the distance to LPC, and finally the nondefinition Grade (IdG), represented by concentric circles, that measure the distance between one proposition and the point of total non-definition (I). 3.2 Proposal of operators AND, OR and NOT for NPL2v It is possible now to define how the operators AND, OR and NOT will work. Will be used also the operators of maximization and minimization, as in fuzzy logic, but now they have to operate over the two notations (Belief degree α and β), instead of only in the degree of pertinence µ. Mainly because of the duality, the operators are defined as follow: Table 5: Operators AND, OR and NOT proposed for NPL2v P Q -Q (α p, β p ) (α q, β q ) (1-α q,1-β q ) P & Q: ( min(α p,α q ), max(β p, β q ) ) P v Q: ( max(α p,α q ), min(β p, β q ) ) It is interesting the graphical representation of the working mechanism of these operators. Given two logic propositions, represented by two points of USPC, it is possible to draw a rectangle with sides paralleled to Cartesian axes, having the two T

5 propositions as the vertexes in diagonal opposition. The result of the conjunction of these propositions, will be the left superior vertex of the rectangle, while the disjunction will set up exactly on the right lower vertex of the rectangle. This represents actually the consecutive application of operator AND, which tends to carry on the result nearer the false state. In the same way, the operator OR tends to carry on the result for the true state. AND OR igure 6: Geometric Interpretation of Operators AND and OR The above definitions justify the results shown on table 4, signaled with (?c) and (?d). It should be noticed that, besides having two points of inconsistency and in the USPC, all results from lines (I) to () are independent of the chosen inconsistency point. q-t q qt combinations show up another 4 notable points from USPC: the nearly states shown bellow: The nearly states (qt=nearly true, q=nearly false, q-t=nearly not true and q-=nearly not false) will be used in splitting lines (I) and (II) as shown in the table bellow: Table 6: Table True of the Nearly States P Q P & Q P v Q T 0 q-t q- (I) 0 T 0 q-t q-. (II) 0 T 1 q QT (I) 1 T 1 q qt (II) 1 4 Conclusions The use of commutative, idempotence, duality and short circuit proprieties, make possible the extension of classic logic, by making a mechanism able to deal with the incomplete knowledge, with the nondefinitions and inconsistencies. The directive stated by Newton da Costa If the propositions we work with be of good behavior, all valid formula in the classic calculus must stay unchanged allow that the developed procedure, when introduced in the inference motor of a specialist system, keep the investments made in classic systems, while allow, with little changes in the logic operators, to take advantages of the new possibilities of the fuzzy and paraconsistent logics. The presented modeling uses the noted paraconsistent logic of two values notation (NPL2v), with degrees of and, to represent the facts of the knowledge base. The inference motor should use the operators AND, OR and NOT proposed on table 5, or reducibly, only for the extreme points, the true table made by the union together of all true tables presented. q- 5 Acknowledgement The authors thank INEP/SAGE, PRONEX/CNPq and CAPES for the financial support of this project. igure 7: Nearly States urthermore, in the lines (I) and (II), where both propositions are of bad behavior, a better attention is required. By applying the operators defined in table 5, it is possible to see that the result of these lines will depend of the inconsistency, whether or. The analyses of possible 6 References [1] Da Costa, Newton C. A., Sistemas ormais Paraconsistentes, Ed. UPR, Curitiba 1993 [2] Zadeh, L.A., uzzy Sets, Information and Control, ol. 8, pp , [3] Kleen, S.C., Introduction to Metamathematics, an Nostrand, 1950