Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844 Vol. IV (2009), No. 4, pp. 349-362 Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold I. Iancu Ion Iancu University of Craiova Department of Mathematics and Computer Science 13 A. I. Cuza Street, 200585, Romania E-mail: i_iancu@yahoo.com Abstract: Using Generalized Modus Ponens reasoning, we examine the values of the inferred conclusion depending on the correspondence between the premise of the rule and the observed fact. The conclusion is obtained using Fodor’s implication in order to represent a fuzzy if-then rule with a single input single output and the t- norm with threshold generated by t-norm product, as a compositional operator. A comparison study with the case when the standard t-norm product is used is made. Some comments and an example are presented in order to show how the obtained results can be used. Keywords: t-norm, t-conorm, negation, implication, fuzzy number, generalized modus ponens rule 1 Introduction The database of a rule-based system may contain imprecisions which appear in the description of the rules given by the expert. The imprecision implies the difficulty of representing the rules expressed, generally, by means of natural language. Another difficulty is the utilization of these rules in approx- imate reasoning when the observed facts do not match the condition of the rule. In order to obtain an imprecise conclusion from imprecise premises, Zadeh extends the traditional Modus Ponens rule obtain- ing Generalized Modus Ponens (GMP). An investigation of GMP inference was made by many papers: [2], [3], [4], [5], [7], [9], [14], [15], [27], [28], [29], [30], [33], [34], [35]. Also, we analyzed this type of inference in some papers: [19] , [22], [24] , [25], [26]. The proposition X is A can be understood as the quantity X satis f ies the predicate A or the variable X takes its values in the set A. The semantic content of the proposition X is A can be represented by πX = µA, where πX is the possibility distribution restricting the possible value of X and µA is the membership function of the set A. Because the majority of practical applications work with trapezoidal or triangular distributions and these representations are still a subject of various recent papers ([1], [13] and [16], for instance) we Copyright c© 2006-2009 by CCC Publications 350 I. Iancu will work with membership functions represented by trapezoidal fuzzy numbers. Such a number N = (a, b, α, β ) is defined as µN (x) =     f or x < a − α x − a + α α f or x ∈ [a − α, a]  f or x ∈ [a, b] b + β − x β f or x ∈ [b, b + β ]  f or x > b + β Let X and Y be two variables whose domains are U and V , respectively. A causal link from X to Y is represented as a conditional possibility distribution [35, 36] πY /X which restricts the possible values of Y for a given value of X . For the rule i f X is A then Y is B we have ∀ u ∈ U, ∀ v ∈ V, πY /X (v, u) = µA(u) → µB(v) where → is an implication operator and µA and µB are the possibility distributions of the propositions "X is A" and "Y is B", respectively. If µA′ is the possibility distribution of the proposition X is A′ then from the rule i f X is A then Y is B and the fact X is A′ Generalized Modus Ponens rule computes the possibility distribution µB′ of the conclusion Y is B′ as µB′ (v) = sup u∈U T ( µA′ (u) , πY /X (v, u) ) , where T is a t-norm. 2 Basic concepts The main concepts used in GMP are presented below, using the terminology of [8], [17] and [32]. Definition 1. A function T : [, ] → [, ] is a t-norm iff it is commutative, associative, non-decreasing and T (x, ) = x ∀x ∈ [, ]. Definition 2. A function S : [, ] → [, ] is a t-conorm iff it is commutative, associative, non-decreasing and S(x, ) = x ∀x ∈ [, ]. Definition 3. A function N : [, ] → [, ] is a strong negation iff it is an involutive and continuous decreasing function from [, ] to itself. Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 351 In order to represent a rule, the notion of fuzzy implication is used. We recall an axiomatic approach (formulated by Fodor in [10, 11, 12]) to the definition of fuzzy implication. Definition 4. An implication is a function I : [, ] → [, ] satisfying the following conditions: I1: If x ≤ z then I(x, y) ≥ I(z, y) for all x, y, z ∈ [, ] I2: If y ≤ z then I(x, y) ≤ I(x, z) for all x, y, z ∈ [, ] I3: I(, y) =  (falsity implies anything) for all y ∈ [, ] I4: I(x, ) =  (anything implies tautology) for all x ∈ [, ] I5: I(, ) =  (Booleanity) The following properties could be important in some applications: I6: I(, x) = x (tautology cannot justify anything) for all x ∈ [, ] I7: I(x, I(y, z)) = I(y, I(x, z)) (exchange principle) for all x, y, z ∈ [, ] I8: x ≤ y if and only if I(x, y) =  (implication defines ordering) for all x, y ∈ [, ] I9: I(x, ) = N(x) for all x ∈ [, ] is a strong negation I10: I(x, y) ≥ y for all x, y ∈ [, ] I11: I(x, x) =  (identity principle) for all x ∈ [, ] I12: I(x, y) = I(N(y), N(x)) for all x, y ∈ [, ] and a strong negation N I13: I is a continuous function. The most important families of implications are given by Definition 5. A S-implication associated with a t-conorm S and a strong negation N is defined by IS,NS (x, y) = S(N(x), y) ∀x, y ∈ [, ] A R-implication associated with a t-norm T is defined by ITR(x, y) = sup{z ∈ [, ]|T (x, z) ≤ y} ∀x, y ∈ [, ] A QL-implication is defined by IT,S,NQL (x, y) = S(N(x), T (x, y)) ∀x, y ∈ [, ] One of the most important implications is the Fodor’s implication IF (x, y) = {  i f x ≤ y max( − x, y) otherwise which is [5] a R-implication for T = min, a S -implication for S = max and a QL-implication for T = min and S = max, where min (x, y) = {  i f x + y ≤  min(x, y) i f x + y >  and max (x, y) = {  i f x + y ≥  max(x, y) i f x + y <  352 I. Iancu and N(x) =  − x. Besides, the Fodor’s implication verifies the properties I1-I12. An important class of t-norms (t-conorms) is given by the t-norms (t-conorms) with thresholds, obtained from standard t-norms (t-conorms); the number of thresholds is an integer n ≥ . First example of operators with 1-threshold were given by Pacholczyk in [31]. Various families of such t-operators can be found in [18, 20, 21, 23], where the advantage of their usage to represent the uncertain knowledge is justified. In this paper we analyze the results obtained by reasoning with imprecise knowledge using a t-norm with threshold as a composition operator. Finally we will compare these results with those obtained using the corresponding standard operators. We consider the following t-norm with a single threshold k ∈ (, )[31] Tk(x, y) = { k  − k T (  − k k x,  − k k y) i f x ≤ k and y ≤ k min(x, y) i f x > k or y > k obtained from the t-norm T (x, y). We will work with the t-norm generated by TP(x, y) = xy, which is one of the most used; it results Tk(x, y) = {  − k k xy i f x ≤ k and y ≤ k min(x, y) i f x > k or y > k 3 Main results Taking into account the following reasons, we shall work with rules having a single input single output: a) a rule with multiple consequent can be treated as a set of rules with a single conclusion; for instance, the rule i f antecedent then C and C and ....... and Cn is equivalent to the rules i f antecedent then C i f antecedent then C ................................. i f antecedent then Cn. b) a rule with multiple premise can be broken up into simple rules [6] when the rules are represented with any S-implication or any R-implication and the observations are normalized fuzzy sets. Our aim is to obtain the conclusion "Y is B′" from the rule i f X is A then Y is B and the fact X is A′ where the fuzzy sets A, A′, B and B′ are represented by trapezoidal possibility distributions. The set B′ is computed as µB′(v) = sup u∈U Tk(µA′(u), IF (µA(u), µB(v))), analyzing five cases, depending on the relation between µA and µA′. Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 353 Theorem 6. If the premise contains the observation, i. e. µA′(u) ≤ µA(u) ∀u ∈ U , then µB′(v) = µB(v) i f µB(v) ≥ . µB′(v) ∈ [µB(v),  − µB(v)) i f µB(v) < . Proof. i1) value on the set U = {u ∈ U /µA(u) ≤ µB(v)} Because IF (µA(u), µB(v)) = , we have µB′(v) = sup u∈U Tk(µA′(u), ) = sup u∈U µA′(u) ≤ µB(v). i2) value on the set U = {u ∈ U /µA(u) > µB(v) ≥ .}∪{u ∈ U /µA(u) >  − µB(v) > .} We have IF (µA(u), µB(v)) = µB(v). If k < µB(v) then µB′(v) = sup u∈U Tk(µA′(u), µB(v)) = sup u∈U min(µA′(u), µB(v)) = µB(v). For k ≥ µB(v) and U  = {u ∈ U/µA′(u) ≤ k} we have µB′(v) = sup u∈U  Tk(µA′(u), µB(v)) = sup u∈U   − k k µA′(u)µB(v) ≤ ( − k)µB(v) < µB(v). For k ≥ µB(v) and U  = {u ∈ U/µA′(u) > k} we obtain µB′(v) = sup u∈U  Tk(µA′(u), µB(v)) = sup u∈U  min(µA′(u), µB(v)) = µB(v). i3) value on the set U = {u ∈ U /µB(v) < µA(u) ≤  − µB(v)} In this case IF (µA(u), µB(v)) =  − µA(u) and therefore µB′(v) = sup u∈U Tk(µA′(u),  − µA(u)). For k < µB(v) we have  − µA(u) ≥ µB(v) > k and Tk ≡ min. It results µB′(v) = sup u∈U min(µA′(u),  − µA(u)) <  − µB(v). For µB(v) ≤ k ≤  − µB(v) we analyze the cases: i): value on the set U  = {u ∈ U /µB(v) ≤ µA(u) <  − k} Because k <  − µA(u) we obtain µB′(v) = supu∈U  Tk(µA′(u),  − µA(u)) = supu∈U  min(µA′(u),  − µA(u)) < min( − k,  − µB(v)) =  − k. i): value on the set U  = {u ∈ U /µB(v) <  − k ≤ µA(u) ≤  − µB(v)} In this case,  − µA(u) ≤ k and we study three possibilities, depending on µA′(u). i): on the set U ,  = {u ∈ U  /µA′(u) = } we obtain µB′(v) =  i): on the set U ,  = {u ∈ U  /µA′(u) ∈ (, k]} we have µB′(v) = sup u∈U ,  − k k µA′(u)( − µA(u)) < k( − k). 354 I. Iancu i): on the set U ,  = {u ∈ U  /µA′(u) > k} we get µB′(v) = sup u∈U , min(µA′(u),  − µA(u)) <  − µB(v). For k >  − µB(v) we consider the set U  = {u ∈ U/µB(v) >  − k} = {u ∈ U / − k < µB(v) < µA(u) ≤  − µB(v)} and we work with the subsets of U  for which µA′(u) = , µA′(u) ∈ (, k] and µA′(u) > k, respectively; we obtain the following corresponding results: µB′(v) = , µB′(v) < k( − k) and µB′(v) <  − µB(v). Synthesizing the previous results, one obtain the conclusion formulated in the theorem. Theorem 7. If the premise and the observation coincide, i. e. µA(u) = µA′(u) ∀u ∈ U , then µB′(v) = µB(v) i f k > . and µB(v) ≥  − k, µB′(v) ∈ [µB(v),  − k) i f k > . and µB(v) <  − k, µB′(v) = max(., µB(v)) i f k ≤ .. Proof. In this case one repeat the proof of the Theorem 6 taking account the equality µA(u) = µA′(u) ∀u ∈ U . It results: 1) if . < k ≤ µB(v) then µB′(v) = µB(v) 2) if k ≤ . ≤ µB(v) then µB′(v) = µB(v) 3) if k ≤ µB(v) < . then µB′(v) = . 4) if µB(v) ≤ k ≤ . then µB′(v) = . 5) if . ≤ µB(v) < k then µB′(v) = µB(v) 6) if µB(v) ≤ . < k then µB′(v) = µB(v) i f µB(v) ≥  − k and µB′(v) ∈ [µB(v),  − k) i f µB(v) <  − k from which we get the conclusion. Theorem 8. If the observation contains the premise, i. e. µA(u) ≤ µA′(u) ∀u ∈ U , then µB′(v) ≥ max(µB(v),  − k k µB(v)( − µB(v))) i f µB(v) ≤ min(., k) µB′(v) ≥ µB(v) otherwise. Proof. i1) value on the set U = {u ∈ U /µA(u) ≤ µB(v)} Because IF (µA(u), µB(v)) =  we have µB′(v) = sup u∈U min(µA′(u), ) = sup u∈U µA′(u) ≥ µB(v). i2) value on the set U = {u ∈ U /. ≤ µB(v) < µA(u)}∪{u ∈ U /µA(u) >  − µB(v) > .} In this case IF (µA(u), µB(v)) = µB(v) and i) for k < µB(v) we obtain µB′(v) = sup u∈U Tk(µA′(u), µB(v)) = sup u∈U min(µA′(u), µB(v)) = µB(v) i) for k ≥ µB(v) we consider two subsets of U: Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 355 i) on the subset U   = {u ∈ U/µA′(u) ≤ k} we have µB′(v) = sup u∈U  Tk(µA′(u), µB(v)) = sup u∈U   − k k µA′(u)µB(v)) ≤ ( − k)µB(v) < µB(v) i) on the subset U   = {u ∈ U/µA′(u) > k} we have µB′(v) = sup u∈U  Tk(µA′(u), µB(v)) = sup u∈U min(µA′(u), µB(v)) = µB(v). i3) value on the set U = {u ∈ U /µB(v) < µA(u) ≤  − µB(v)} . In this case IF (µA(u), µB(v)) =  − µA(u) and we analyze the following cases. i) if k < µB(v) then µB′(v) = sup u∈U Tk(µA′(u),  − µA(u)) = min(µA′(u),  − µA(u)) <  − µB(v). ii) if k ≥ µB(v) we consider the following subcases: ii) µB(v) ≤ k ≤  − µB(v) ii) on the set U   = {u ∈ U/µB(v) ≤ µA(u) <  − k} we have µB′(v) = sup u∈U  min(µA′(u),  − µA(u)) <  − µB(v) ii) on the set U   = {u ∈ U/ − k ≤ µA(u) ≤  − µB(v)} we consider two subsets: • U , = {u ∈ U  /µA′(u) ≤ k} for which we obtain µB′(v) = supu∈U ,  − k k µA′(u)( − µA(u)) ≥ supu∈U ,  − k k µA(u)( − µA(u)) ≥ max(( − k),  − kk µB(v)( − µB(v))) ≥  − k k µB(v)( − µB(v)) • U , = {u ∈ U  /µA′(u) > k} for which we have µB′(v) = sup u∈U , min(µA′(u),  − µA(u)) <  − µB(v). ii) k >  − µB(v) which defines the set U  = {u ∈ U/ − k < µB(v)} = {u ∈ U / − k < µB(v) < µA(u) ≤  − µB(v)} • for µA′(u) ≤ k we obtain µB′(v) = supu∈U   − k k µA′(u)( − µA(u)) ≥ supu∈U   − kk µA(u)( − µA(u)) ≥  − k k µB(v)( − µB(v)) • for µA′(u) > k it results µB′(v) = sup u∈U  min(µA′(u),  − µA(u)) <  − µB(v). Finally we obtain the conclusion formulated in the theorem. Theorem 9. If there is a partial overlapping between the sets A and A′ then µB′(v) =  i f core(A′)∩(U − AµB(v)) 6= /0 and µB′(v) ≥ µB(v) otherwise where Aα denotes the α -cut of A. 356 I. Iancu Proof. i1) The case core(A′)∩(U − AµB(v)) 6= /0. On the set U = {u ∈ U /µA(u) ≤ µB(v)} we have IF (µA(u), µB(v)) =  and therefore µB′(v) = sup u∈U Tk(µA′(u), ) = . i2) The case core(A′)∩(U − AµB(v)) = /0. On the set U = {u ∈ U /µA(u) > µB(v) ≥ .} we have IF (µA(u), µB(v)) = µB(v) and therefore µB′(v) = sup u∈U Tk(µA′(u), µB(v)) ≥ Tk(, µB(v)) = µB(v). If µB(v) < . we analyze three cases. Let Ũ = {u ∈ U /µA(u) = µB(v)}; card(Ũ ) =  if  < µB(v) < . i) the case Ũ ∩supp(A′) = /0 and core(A′)∩core(A) 6= /0. On the set U = {u ∈ U /µA(u) ≥  − µB(v) > .} it results µB′(v) = sup u∈U Tk(µA′(u), µB(v)) ≥ Tk(, µB(v)) = µB(v). i) the case Ũ ∩supp(A′) = /0 and core(A′)∩core(A) = /0. We consider the set U = {u ∈ U /µB(v) < µA(u) ≤  − µB(v)}; on the set U = U ∪U we have µB′(v) = sup u∈U Tk(µA′(u), IF (µA(u), µB(v))) ≥ sup u∈U Tk(µA′(u), µB(v)) ≥ Tk(, µB(v)) = µB(v). i) the case Ũ ∩supp(A′) 6= /0. On the set U we obtain µB′(v) ≥ µB(v), as in the previous case. It results that, in the case i2), µB′(v) ≥ µB(v). The same result is obtained for µB(v) ∈ {, }. We consider the negation with threshold k ∈ (, ) [31] Nk(x) = {  −  − kk x i f x ≤ k k  − k ( − x) i f x ≥ k obtained from the standard negation N(x) =  − x. Theorem 10. If the premise and the observation are contradictory, i.e. µA′(u) = Nk(µA(u)) ∀u ∈ U , then µB′(v) =  ∀v ∈ V. Proof. On the set U = {u ∈ U /µA(u) ≤ µB(v)} we have µB′(v) = sup u∈U Tk(µA′(u), ) = sup u∈U min(µA′(u), ) = sup u∈U µA′(u) = sup u∈U Nk(µA(u)) =  because there is u ∈ U with µA(u) = . 4 Interpretation and utilization of results In this section we will compare the results given by the common operators (t-norm product TP(x, y) = xy and negation N(x) =  − x) with those obtained by the corresponding operators with threshold and we will indicate some possibility of their utilization in a fuzzy reasoning system. An example of working with these results is also presented. In the case of standard operators TP and N, according to [24] we have: Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 357 Theorem 11. If the premise contains the observation, i.e µA′(u) ≤ µA(u) ∀u ∈ U , then µB′ (v) = µB(v) i f µB(v) ≥ . or (. ≤ µB(v) < .) µB′ (v) < . i f µB(v) < . Theorem 12. If the premise and the observation coincide, i.e. µA(u) = µA′(u) ∀u ∈ U , then µB′ (v) = max(µB(v), .) Theorem 13. If the observation contains the premise, i.e. µA(u) ≤ µA′(u) ∀u ∈ U , then µB′(v) ≥ µB(v) ∀v ∈ V. Theorem 14. If there is a partial overlapping between the sets A and A′, then µB′(v) =  i f core(A′)∩(U − AµB(v)) 6= /0 and µB′(v) ≥ µB(v) otherwise where Aα denotes the α−cut of A. Theorem 15. If the premise and the observation are contradictory, i.e. ∀u ∈ U µA′(u) =  − µA(u), then µB′(v) =  ∀v ∈ V . If the observation is more precise than the premise of the rule then it gives more information than the premise. However, it does not seem reasonable to think that the Generalized Modus Ponens allows to obtain a conclusion more precise than that of the rule. The result of the inference is valid if µB′(v) = µB(v), ∀v ∈ V . Sometimes, the deduction operation allows the reinforcement of the conclusion, as is specified in [28], [19] and [25]: Rule: If the tomato is red then the tomato is ripe. Observation: This tomato is very red. If we know that the maturity degree increases with respect to color, we can infer "this tomato is very ripe". On the other hand, in the example Rule: If the melon is ripe then it is sweet Observation: The melon is very ripe we do not infer that "the melon is very sweet" because it can be so ripe that it can be rotten. This examples show that if the expert has not supplementary information about the connection between the variation of the premise and the conclusion, he must be satisfied with the conclusion µB′(v) = µB(v). The Theorem 6 gives a valid result if we choose µB′(v) = µB(v) for µB(v) < .. As opposite, the corresponding Theorem 11 from the case of the standard t-norm TP does not allow to obtain a valid result if µB(v) < .. When the observation and the premise of the rule coincide the convenient behavior of the fuzzy deduction is to obtain an identical conclusion. A different conclusion indicates the appearance of an uncertainty in the conclusion. The both theorems, 7 and 12, give an uncertain conclusion, but we can choose k > . in the Theorem 7 and we obtain a better result, because the uncertainty is smaller in comparison with the result from the Theorem 12. If the observation contains the premise, because max(  − kk µB(v)( − µB(v)), µB(v)) ≥ µB(v) it results that Theorem 8 gives a better result that Theorem 13. In this case the inferred conclusion B′ is a superset of B; we can choose the first superset. 358 I. Iancu If there is a partial overlapping between the premise and the observation or the premise and the observation are contradictory then the two t-norms give the same results for the inferred conclusion. The value µB′(v) =  obtained in these cases represents an indeterminate conclusion, all elements v ∈ V having a possibility equal to 1. In the case of "partial overlapping" we propose a "mediation" between the two possible values: µB′(v) =  and µB′(v) ≥ µB(v); if B, B, ..., Bk are the supersets of B with µBk (v) ≥ µBk− (v) ≥ .... ≥ µB (v), we can choose B′ = B [ k  ] , where [x] is the greatest integer which is smaller than or equal to x. The Theorem 10 gives a waited result, that represents one of the basic properties of GMP reasoning. The results from Theorems 6-10 can be used in a fuzzy inference system as in the following example. A customer is interested to buy a computer. The quality of the computer depends on its price as is specified by the rules: Rule1: If the price is very low then the quality is below average Rule2: If the price is very very high then the quality is very good. Rule3: If the price is middle then the quality is good. The variable price has values in the following set of linguistic terms Lp = {very very low, very low, low, midle, high, very high, very very high} and the variable quality has values in the set Lq = {poor, below average, average, above average, good, very good}. We consider the universes of discourse [, ] for price and [, ] for quality. The linguistic terms are represented by the following trapezoidal fuzzy numbers: very very low = (, , , ) very low = (, , , ) low = (, , , ) middle = (, , , ) high = (, , , ) very high = (, , , ) very very high = (, , , ) poor = (, , , ) below average = (, , , ) average = (., ., ., .) above average = (, , , ) good = (., ., ., .) very good = (., ., ., .). These fuzzy numbers are depicted in the Figures 1 and 2. We consider the observations: Observation1: the price is very very low Observation2: the price is very high Observation3: the price is high The theorems 6-10, used together with the comments from this section, give the following results: 1) the conclusion obtained from Rule1 and Observation1 is "the quality is below average"; this result is obtained with Theorem 6 2) Theorem 8 is applied for Rule2 and Observation2 and gives the conclusion "the quality is good" Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 359 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 0 0.2 0.4 0.6 0.8 1 De gr ee o f m em be rs hi p v.v.Low vLow Low v.v.Highmiddle high vHigh Figure 1: Fuzzy sets for linguistic terms from the list Lp 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 De gr ee o f m em be rs hi p poor bAverage average aAverage good vGood Figure 2: Fuzzy sets for linguistic terms from the list Lq 3) using Theorem 9 for the Rule3 and Observation3 one obtain the conclusion "the quality is aver- age". As it can be observed from this example, our results allow us to obtain the inferred conclusion by a very simple calculus in comparison with the standard formula used in GMP. 5 Summary and Conclusions The results obtained in this paper explain how the Generalized Modus Ponens rule works with the Fodor’s implication and the t-norm product with threshold. Combining these results with the approxi- mations proposed in the previous section we obtain a fast answer for the value of the conclusion inferred by GMP reasoning. We worked with the t-norm product because it is one of the most used in practical applications. As it results from the previous sections, one obtain better results in the case of t-norm with threshold. In a future paper we will analyze the results given by another t-norms with threshold and 360 I. Iancu another implications. Bibliography [1] T. Allahviranloo, M. Adabitabar Firozja, Note on “Trapezoidal approximation of fuzzy numbers”, Fuzzy Sets and Systems 7, 2007, pp. 755–756. [2] B. Bouchon, Inferences with imprecisions and uncertainties in expert systems, in: A. Kandel (Ed), Fuzzy Expert Systems, Addison-Wesley, New York, 1987. [3] J. C. Buisson, H. Farreny, H. Prade, Dealing with imprecision and uncertainty in expert system DIABETO III, in: Proc. 2nd Int. Conf. on Artificial Intelligence, Marseille, 1986, pp. 705–721. [4] J. F. Baldwin, A new approach to approximate reasoning using fuzzy logic. Fuzzy Sets and Systems 2, 1979, pp. 309–325. [5] E. Czogala, J. Leski, On equivalence of approximate reasoning results using different interpolations of fuzzy if-then rules, Fuzzy Sets and Systems 11, 2001, pp. 279–296. [6] K. Demirli, I.B. Turksen, Rule break up with compositional rule if inference, IEEE International Conference on Fuzzy Systems, San Diego, 1992, pp. 949–956. [7] D. Dubois, H. Prade, Operations in a fuzzy-valued logic, Information and Control 43, 1979, pp. 224–240. [8] D. Dubois, Modeles mathematiques de l’imprecis at de l’incertain en vue d,applications aux tech- niques d’aide a la decision, These d’Etat, Univ. Scientifique et Medicale, Grenoble, 1983 [9] D. Dubois and H. Prade, Fuzzy logics and the generalized modus ponens revisited, Cybernetics and Systems 15, 1984, pp. 293–331. [10] J. Fodor, On fuzzy implication operators, Fuzzy Sets and Systems 42, 1991, pp. 293-300 [11] J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer, Dordrecht, 1994 [12] J. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69, 1995, pp. 141-156 [13] J. Fodor, B. Bede, Recent Advances in Fuzzy Arithmetics, International Journal of Computers, Communications and Control, Vol. 1, Suppl. 5, 2006, pp. 199-207 [14] S. Fukami, M. Mizumoto, K. Tanaka, Some considerations on fuzzy conditional inference, Fuzzy Sets and Systems 4, 1980, pp. 243–273. [15] B. R. Gaines, Fundations of fuzzy reasoning, Int. J. Man-Machine Stud. 8, 1976, pp. 623–668. [16] P. Grzegorzewski, E. Mrowka, Trapezoidal approximations of fuzzy numbers - revisited, Fuzzy Sets and Systems 7, 2007, pp. 757–768. [17] M. M. Gupta, J. Qi, Theory of T-norms and fuzzy inference methods, Fuzzy Sets and Systems 40, 1991, pp. 431-450 [18] I. Iancu, T-norms with threshold, Fuzzy Sets and Systems. Int. J. of Soft Computing and Intelligence 85 (1997), pp. 83-92 Generalized Modus Ponens using Fodor’s Implication and T-norm Product with Threshold 361 [19] I. Iancu, Propagation of uncertainty and imprecision in knowledge-based systems, Fuzzy Sets and Systems 94, 1998, pp. 29–43. [20] I. Iancu, On a family of t-operators, Annals of the Univ. of Craiova, Mathematics and Computer Science serie, vol XXVI(1999),pp. 84-92 [21] I. Iancu, Fuzzy connectives with applications in uncertainty management, Proc. of the 3-rd annual meeting of the Romanian Society of Mathematical Sciences, 1999, pp. 40-47 [22] I. Iancu, On a Representation of an Uncertain Body of Evidence, Annals of the Univ. of Craiova, Mathematics and Computer Science serie, vol XXX(2003),pp. 100-108 [23] I. Iancu, Operators with n-thresholds for uncertainty management. J. Appl. Math. & Computing. An Int. Journal, Vol 19(2005), No. 1-2, pp. 1-17, Publisher: Springer Berlin / Heidelberg [24] A. Giurca, I. Iancu, Approximate reasoning using Fodor’s implication, Computational Intelligence, Theory and Applications - Int. Conf. 9th Fuzzy Days in Dortmund, Springer, 2006, pp. 513–520, Berlin. [25] I. Iancu, Generalized Modus Ponens Using Fodor’s Implication and a Parametric T-norm, WSEAS Transaction on Systems 6(vol. 7), 2008, pp. 738-747 . [26] I. Iancu, Generalized Modus Ponens Reasoning for Rules with Partial Overlapping Between Premise and Observation, European Computing Conference, Malta, September 2008, pp. 37-43 [27] E. H. Mamdani, Application of fuzzy logic to approximate reasoning using linguistic systems, IEEE Trans on Computers 26, 1977, pp. 1182–1191. [28] M. Mizumoto, H.-J. Zimmerman, Comparison on fuzzy reasoning methods, Fuzzy Sets and Systems 8, 1982, pp. 253–283. [29] M. Mizumoto, Fuzzy reasoning under new compositional rule of inference, Kybernetics 12, 1985, pp. 107–117. [30] A. Nafarieh, J. M. Keller, A new approach to inference in approximate reasoning, Fuzzy Sets and Systems 41, 1991, pp. 17–37. [31] D. Pacholczyk, Introduction d’un seuil dans le calcul de l’incertitude en logique floue, BUSEFAL 32, 1987, pp. 11-18 [32] I. J. Rudas, J. Fodor, Information Aggregation in Intelligent Systems Using Generalized Operators, International Journal of Computers, Communications and Control, Vol. 1, No. 1, 2006, pp. 47-57 [33] Y. Tsukamoto, An approach to fuzzy reasoning method. In: M. M. Gupta, R. K. Ragade and R. R. Yager (Eds.) Advances in Fuzzy Sets Theory and Applications, North-Holland, Amsterdam, 1979, pp. 137–149. [34] R. R. Yager, An approach to inference in approximate reasoning, Int. J. Man-Machine Stud. 13, 1980, pp. 323–338. [35] L. A. Zadeh, Fuzzy sets as a basis for a theory of a possibility, Fuzzy Sets and Systems 1, 1978, pp. 2–28. [36] L. A. Zadeh, A theory of approximate reasoning, Machine Intelligence (Eds: J. E. Hayes, D. Mikie and L. I. Mikulich), John Wiley & Sons, New York, 1979, pp. 149–194. 362 I. Iancu Ion Iancu received his degree in Mathematics from the University of Craiova and Ph. D. in Arti- ficial Intelligence from the University of Bucharest. Currently, he is titular professor of Computer Science and Artificial Intelligence at the University of Craiova. His main topics of interest include fuzzy sets, modeling of uncertain and imprecise knowledge, approximate reasoning, soft comput- ing. A several number of research articles in the above areas have also been published in various journals and proceedings; also, he is the author of eight books.