On the Independence of the Formal System L *

Transcription

1 6 International Journal of Fuzzy Systems, Vol. 4, No., June On the Independence of the Formal System L * Daowu Pei Astract The formal system L * of fuzzy propositional logic has een successfully applied to the joint research of fuzzy logic and fuzzy reasoning. The current paper further studies the syntactic structure of the system L *, presents several new results with respect to this system. Especially, a simplified system which contains only seven axioms is proposed, and the system is found to e equivalent to the original system. Also, the independence of the simplified system is proved in the present paper. Keywords: Fuzzy logic; Formal system L * ; Independence; Nilpotent minimum logic. Introduction In order to provide suitale logic foundation for fuzzy reasoning and for more general approximate reasoning, the formalization of fuzzy logic has een studied intensively during the past two decades [, 4, 5, 7, ]. Recently, several interesting formal deductive systems of fuzzy propositional logic ased on t-norms (triangular norms) [4] were proposed. In each of t-norm ased fuzzy logic, a suitale t-norm is taken as the interpretation of the conjunction connective; The residuated implication and negation induced y this t-norm are employed as the interpretations of the implication and negation connectives, respectively [4, 5, 7]. Wang [] proposed the formal system L * in 997 which is a formalization of the so-called R interval which is a fuzzy logic ased on the so-called standard nilpotent minimum t-norm [, ]. And Hájek [5] proposed the asic logic system BL in 998 which is a common formalization of all fuzzy logics ased on continuous t-norms. Besides Esteva and Godo [] proposed the monoidal t-norm ased logic system MTL with three extensions WNM, IMTL and NM in. The system MTL is a common formalization of all fuzzy logics ased on left continuous t-norms [5]. It has een proved that Wang's system L * can e also considered as a sche- Corresponding Author: Daowu Pei is with the Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, P R China. peidw@6.com Manuscript received 4 Jan. ; revised 5 Oct. ; accepted June.. matic extension of the system MTL [4]. The aforementioned logics have een applied to fuzzy control, expert systems and some other domains [5, 6, 5,, 7]. In literature, much attention has een paid to the syntax and semantics of these logics [4, 5, 7, ]. These studies have greatly enriched the contents of logic and algera, and also provided suitale logic asis for uncertainty reasoning [5, 5, 6, -4]. On the other hand, according to the widely accepted viewpoints toward logics, the independence of a formal system is a very important logic property [4, 6, 6]. In view of the importance of the formal system L *, the present paper further focuses on its syntax. The main purpose of this study is to simplify the structure of the system. With the latest results of the related areas, a truth constant is introduced into the system as a new -ary propositional connective, and then a simplified version NM * of the system L * is proposed. Also, Pei [6] proposed an open prolem: whether the simplified version NM * is independent or not. In this paper, we will try to solve the open prolem.. The System L * and Its Main Properties In 997, in order to set a sound logic foundation for fuzzy reasoning, Wang [] proposed a new formal system L * for fuzzy propositional calculus ased on the detail analysis of shortages of fuzzy logic and fuzzy reasoning [-9,,, 7]. Suppose that S = {p, p,...} is the set of all atomic formulas, the formula set F(S) is the (,, )-type free algera generated y S. Elements of F(S) are called formulas. Definition : The system L * consists of the inference rule MP (modus ponens) and the following twelve axioms: (L * ) A (B A) (L * ) ( A B) (B A) (L * ) (A (B C)) (B (A C)) (L * 4) (B C) ((A B) (A C)) (L * 5) A A (L * 6) A A B (L * 7) A B B A (L * 8) (A C) (B C) (A B C) (L * 9) (A B C) (A C) (B C) (L * ) (A B) ((A B) A B) (L * ) A (B A B) TFSA

2 Daowu Pei: On the Independence of the Formal System L * 7 (L * ) (A B) (A C) (A B C) where P Q is the areviation of the formula ( P Q). The original system L * proposed y Wang [] contains fourteen axioms and two inference rules (the rule MP and the so-called intersection rule), and it does not contain the axiom L *. He and Wang [6] proved that five axioms are redundant, i.e., they can e derived from the other nine axioms L * -L * 9. In order to otain the completeness of the system L *, Wang [] added the axiom L * into the system L *. Pei [] pointed out: if two new axioms L * and L * are added into the system L *, then the intersection rule is redundant. It has een oserved that the system L * is a formalization of the fuzzy logic system W = ([,],,,, ), where a = a, a = max(a, ), a = min(a, ),, a a = R ( a, ) =. a, a > This logic system is called R interval, or the revised Kleene logic system [-]. The operator R is called R implication, or the revised Kleene implication which has many excellent properties [,, 4, ]. R implication indeed is the residuated implication induced y the nilpotent minimum t-norm [, ]. Therefore, the standard nilpotent minimum algera proposed y Esteva and Godo [] is the same as R interval. Wang [] introduced some familiar syntactic concepts such as proof, theorem, provale equivalence in the system L *. Then a series of important theorems, derived rules (such as HS rule, sustitution rule, etc) and properties of the system are proved. Pei [] introduced a new connective and the corresponding power operations A n into the system L *, and some new theorems including the generalized deduction theorem of the system L * are given then. In order to prove the completeness of the system L *, one needs to provide suitale semantics for the system. Naturally, one should consider valuations of the formula set F(S), i.e., (,, )-type algeraic homomorphisms from F(S) to the R interval W. The set Ω of all such valuations is called the semantics of the system L * [-]. Moreover, in order to otain the completeness of the system L *, one needs to introduce suitale algeraic structures as generalizations of the revised Kleene logic system W, which should contain the L * -Lindenaum algera [6, ]. Wang [] proposed the following concept of the so-called R algeras. Definition : Suppose that M is a (,, )-type algera, where is a unary operation, and and are inary operations. If there exists a partial ordering on M such that (M, ) ecomes a ounded distriutive lattice, is the supremum with respect to, is an order-reversing involution with respect to, and satisfies the following conditions for all a,, c M, (R) a = a (R) a = a, a a = (R) c (a ) (a c) (R4) a ( c) = (a c) (R5) a c = (a ) (a c), a c = (a ) (a c) (R6) (a ) ((a ) a ) = where is the greatest element of M, then M is called an R algera. Pei and Wang [7] called the linear ordered R algeras R chains, and called the corresponding algera structures otained from R algeras y deleting the property R6 weak R algeras. Naturally, oth L * -Lindenaum algera and R interval are R algeras. Every Boole algera (B,, ',, ) with respect to the negation operation a = a ' and Kleene implication R K forms an R algera, where a = R K (a, ) = a '. The completeness theorem of the system L * has een proved y Pei and Wang [8]. In their paper, authors uilt the semantics of the system L * on general R algeras, i.e., for any R algera M, the set Ω(M) consisting of all (,, )-type algeraic homomorphisms (or M-valuations) from the formula set F(S) to M is called the M-semantics of the system L *. By using some algeraic tools such as filters and congruence relations, authors proved the following theorems. Theorem [8]: The system L * is complete, i.e., for each formula A, the following conditions are equivalent: (i) A is provale in the system L * ; (ii) A is a tautology in each R chain; (iii) A is a tautology in each R algera. Theorem [8]: The system L * is standard complete, i.e., for each formula A, A is provale in the system L *, i.e., A, if and only if A is a tautology in R interval, i.e., for each v Ω, we have v(a) =. Furthermore, Pei and Wang [9] proved the semantical completeness of the system L *. These results showed that the syntax and the semantics of the system L * are harmonious. Thus we can expect that it will e more widely applied to fuzzy reasoning and other areas.. Simplification of the System L * In the system L *, the following two equivalent propo-

3 8 International Journal of Fuzzy Systems, Vol. 4, No., June sitions are meaningful. Theorem : In the system L *, the axiom L * 9 can e replaced y one of the following three formulas: (L * 9) (A B C) (A B) (A C) (L * 9) (A B) (B A) (L * 9) ((A B) C) (((B A) C) C). Proof: (i) L * 9 L * 9 holds oviously. (ii) L * 9 L * 9. [A B A B] [(A B A) (A B B)] [(B A) (A B)] L * 9. (iii) L * 9 L * 9. We have (A B) L * 9, (B A) L * 9. Thus L * 9 L * 9. (iv) L * 9 L * 9. [(B C) (B C C)] [(B C) ((A B C) (A B) (A C))]. Similarly, [(C B) ((A B C) (A B) (A C))]. Thus ((A B C) (A B) (A C)). Following Hájek [4] and Esteva and Godo [] we take the so-called strong conjunction &, implication, conjunction and the truth value constant (for revity, we write for this constant) as four asic connectives. Suppose that S = {p, p,...} is the set of all propositional variales (or, atom formulas), F (S) is the formula set generated y S and connectives, &,,. In F (S), connectives and are defined as follows: A = A, A B = ((A B) B) ((B A) A). Also, one new axiom is added for characterizing : (L * ) A. Thus we can otain the following conclusion. Theorem 4: In the system L *, the axiom L * can e replaced y the following formula: (L * ) (A& B ) (A B A& B). Proof: In fact, we have (A& B ) (A B A&B) ~ (A B) (A B (A B)) ~ (A B) ((A B) A B). Esteva and Godo [] proposed a new formal deductive system MTL for the so-called the monoidal t-norm ased fuzzy logic. Definition : The system MTL consists of the inference rule MP and the following ten axioms: (Ax) (A B) ((B C) (A C)) (Ax) A&B A (Ax) A&B B&A (Ax4) A B A (Ax5) A B B A (Ax6) A&(A B) A B (Ax7) (A (B C)) (A&B C)) (Ax8) (A&B C) (A (B C)) (Ax9) ((A B) C) (((B A) C) C) (Ax) A In addition, the so-called involutive monoidal t-norm ased logic IMTL is a schematic extension of MTL y adding the following axiom: (Ax) A A. And the so-called nilpotent minimum logic NM is a schematic extension of IMTL y adding the following axiom: (Ax) (A&B ) (A B A&B). The following conclusion uilds a ridge etween oth researches of Chinese and foreign scholars in this area. Theorem 5 [4]: The system L * and NM are two equivalent axiom systems. Based on the aove consideration and the results given y Pei [6], now we propose a simplified system for the system L *. Definition 4: The system L * () consists of the inference rule MP and the following seven axioms: (Ax * ) (A B) ((B C) (A C)) (Ax * ) A B A (Ax * ) A B B A (Ax * 4) A ((A B) A B) (Ax * 5) ((A B) C) (((B A) C) C) (Ax * 6) ( A B) (B A) (Ax * 7) (A B) ((A B) A B). In Pei [4], the notation L * is used to denote the axiom system otained from the system L * y deleting the axiom L *. Also Pei [6] used the notation IMTL * to denote the axiom system otained from the system L * () y deleting the axiom Ax * 7, and the notation NM * to denote the axiom system otained from L * () y sustituting the axiom Ax * 7 y Ax. It is worth pointing that the aove selection of seven axioms mainly ased on the consideration of formal deduction, and the merits of the systems NM and L *. In addition, the axiom Ax * 6 is one of three axioms of the classical propositional calculus, and also common one of the system L * and the famous axioms of the Łukasiewicz infinite-valued axiom system, and it can e replaced y the axiom Ax. Also, the axiom Ax * 6 is a formalization of one of ten properties aout fuzzy implications []. The system L * () oviously simplifies the system L *, or equivalently, the system NM.

4 Daowu Pei: On the Independence of the Formal System L * 9 Oviously, the axioms Ax * -Ax * 7 are all theorems of the system NM. In order to prove the equivalence of the systems L * () and NM, it is only need to prove all theorems of the system NM are theorems of the system L * (). From the corresponding results of Pei [6], the following theorem is ovious. Theorem 6: The following formulas are theorems of the system L * (): (T) (A B) ( B A) (T) A ((A B) A) (T) A ((A B) B) (T4) (A (B C)) (B (A C)) (T5) (B C) ((A B) (A C)) (T6) A A (T7) A A (T8) A A (T9) (A B) (B A) (T) ( A B) ( B A) (T) (T) ( A) A (T) A ( A) (T4) A (T5) A (T6) A (A B) (T7) A (B A) (T8) A ( A B) where is the formula. Now we introduce the strong conjunction connective & into the system L * () as follows: A&B = (A B). Aout formulas containing the connective &, we can easily otain the following conclusions. Theorem 7: The following formulas are theorems of the system L * (): (T9) A&B A (T) A&B B&A (T) (A (B C)) (A&B C)) (T) (A&B C) (A (B C)). Theorem 8: Four axiom systems L *, NM, NM * and L * () are equivalent. In fact, the axioms Ax-Ax and Ax *, T8, T9, Ax * - Ax * 4, T, T, Ax * 6, Ax * 9 and T are the same, respectively. Moreover, y the previous results the axiom Ax and Ax * 8 are equivalent. Thus two axiom system L * and L * () are equivalent. The following conclusion reported y Pei [6] can e seen as a direct consequence of Theorem 8. Theorem 9 [6]: Two systems IMTL and IMTL * are equivalent. 4. The Independence of the Simplified System Pei [6] proved the independence of the simplified system IMTL *, and proposed the prolem of the independence of the system NM *. In this section the independence of the simplified system L * (), or equivalently, NM * will e proved. Usually, one uses the so-called arithmetic interpretation method to prove the independence of the classical logic system [6]. Here, we will also use this method and the method used y Gottwald [4] which is presented as follows. Suppose that K is a sound logic calculus, where there exist n axioms A, A,..., A n. In order to prove that the axiom A cannot e inferred from the other axioms A,..., A n, we only need to uild such a multiple-valued logic system S (or model), i.e., a truth value set and interpretations (or valuations) of propositional connectives of K such that all inference rules of K are sound in the model S, or S-sound, axioms A,..., A n are all S-valid, and A is not S-valid. The present paper only contains the inference rule MP. The S-soundness of this rule means that B is S-valid whenever A and A B are S-valid. We note that this method to prove independence indeed is a kind of method to prove derivality. Strictly speaking, a formula A is independent to a formula set Γ if and only if A and A are not Γ- consequences. However, in many cases proofs for derivality indeed are the same as proofs for independence. The following proofs elong to this kind of proofs. Denote Δ = {Ax *,..., Ax * 7}, Δi = Δ\{Ax * i}, i =,..., 7. Now we give some results aout the independence of the simplified systems L * (). By making use of some models and conclusions given y Pei [6], we can prove the independence of the axioms Ax *, Ax * 4, Ax * 5, Ax * 6 and Ax * 7 in the simplified system L * (). Lemma : The axiom Ax * is underivale from Δ. In fact, we can use the model S = ([,],,, ) (see Pei [6], P. ) to prove this lemma, where a = min(a, ), a = max(a, ),, a a = ( a), a= or =., otherwise The connectives, and are interpreted as, and respectively. Pei [6] has proved: the MP rule is S-sound, the axiom Ax * is not valid in S, and Ax * -Ax * 6 are valid in S. Furthermore, we can verify that the axiom Ax * 7 is

5 International Journal of Fuzzy Systems, Vol. 4, No., June also valid in S y considering the following four cases: (i) Case : a ; (ii) Case : a = ; (iii) Case : = ; (iv) Case 4: otherwise where a = v(a) and = v(b) for aritrary valuation v. Lemma : The axiom Ax * 4 is underivale from Δ4. Similarly, we can use the model S = ([,],,, ) (see Pei [6], P. S) to prove this lemma, where a = min(a, ), a = max(a, ),, a a =. ( a), a > The formulas, A B and A B are interpreted as, and v(a) v(b) respectively. Lemma : The axiom Ax * 5 is underivale from Δ5. For the axiom Ax * 5, let S e a lattice showed y the Figure (see S of Pei [6], P. ). a c d Figure. The model S. In the model S, the operation is defined y the following Tale. Tale. Negation in S. x a B c d x A d c The operation is defined as follows, x y x y =. x y, otherwise The connectives, and are interpreted as, and, respectively. Lemma 4: The axiom Ax * 6 is underivale from Δ6. Proof: Set S4 = ([,],,, ) (see S4 of Pei [6], P. ), where a = min(a, ), a = max(a, ),, a a =., a> The connectives, and are interpreted as, and respectively. Lemma 5 [4]: The axiom Ax * 7 is underivale from Δ7. We point out that the model given y Pei [6] cannot e applied to prove the independence of Ax * and Ax * (see Lemmas and, S of Pei [6], P. ). Therefore, we use the arithmetic interpretation method to complete our proofs. Lemma 6: The axiom Ax * is underivale from Δ. Proof: We use the arithmetic interpretation method to prove this lemma. The model S5 is constructed y the following truth value tale: x y Tale. Truth value tale of S5. x y x y x y In this model, is interpreted as. Now we take a suset E = {, } of S5. Then we can verify that the suset E is closed with respect to the rule MP in the sense that for any valuation v we have v(b) E whenever v(a) E and v(a B) E. Moreover, we can verify that for any valuation v, v(ax * i) E for i 7 and i through listing the truth value tale for every axiom, and there exists a valuation v such that v (Ax * ) E y the following example: v (A) =, v (B) =, v (A B A) = = E. This shows that the axiom Ax * cannot e derived from Δ. Lemma 7: The axiom Ax * is underivale from Δ. Similarly, we can construct a model S6 y the arithmetic interpretation method to prove this lemma. The model S6 is constructed y the following truth value Tale. In this model, is interpreted as. Now we take a suset E = {} of S6. Then we can verify that the suset E is closed with respect to the rule MP in the sense that for any valuation v we have v(b) E whenever v(a) E and v(a B) E. Moreover, we can verify that for any valuation v, v(ax * i) E for i 7 and i through listing the truth value tale for every axiom, and there exists a valuation v such that v (Ax * ) E y the following example: v (A) =, v (B) =, v (A B B A) = = E. This shows that the axiom Ax * cannot e derived from Δ.

6 Daowu Pei: On the Independence of the Formal System L * x y Tale. Truth value tale of S6. x y x y x y Summarizing the previous lemmas we otain the following conclusions. Theorem : The simplified axiom system L * () is independent. Since the equivalence of the system NM * and L * (), we also proved the independence of NM *, and solved an open prolem proposed in Pei [6]. Furthermore, ased on the previous lemmas, as a direct consequence, we also otained the independence of the system INTL *. Theorem [4]: The simplified axiom system IMTL * is independent. 4. Conclusions The simplification and independence of a fuzzy logical system are two interesting and important prolems. This paper has solved the open prolem proposed y Pei [6]: how to find simple and independent axioms of the formal deductive system L *, which is proposed y Wang [], or equivalently NM * proposed y Pei [6]. As introduced in this study, the new simplified system of the system L * only include seven axioms. Similar to the open question proposed y [6], an interesting prolem is worthy of further consideration: to find more simple, or even, the most simple axioms of the monoidal t-norm ased logic system MTL, the asic logic system BL or Höhle's monoidal logic system ML [, 4]. Also, in the future, the connections etween fuzzy logic and fuzzy control [5], or fuzzy decision making [8, 9, ] will e two essential and promising application fields. Acknowledgment This work is supported y National Science Foundation of P. R. China (Grant Nos. 879, 78). The author would like to thank Professor Wen-June Wang, the Editor-in-Chief of International Journal of Fuzzy Systems, the anonymous area editor and referees for their valuale comments and recommendations. Finally, special thanks to my son, Zhi Pei, PhD, from Zhejiang University of Technology, for his careful proofreading of the paper and valuale advices. References [] D. Duois and H. Prade, Fuzzy sets in approximate reasoning, Part, Fuzzy Sets and Systems, vol. 4, pp. 4-, 99. [] F. Esteva and L. Godo, Monoidal t-norm ased logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, vol. 4, pp. 7-88,. [] J. C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems, vol. 69, pp. 4-56, 995. [4] S. Gottwald, A Treatise on Many-Valued Logics, Research Studies Press LTD, Baldock,. [5] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 998. [6] Y.-Y. He and G.-J. Wang, The canonical valuation mediums and simplification of the system L *, Chinese Science Bulletin, vol. 4, no. 6, pp , 998 (in Chinese). [7] E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer, Dordrecht,. [8] S.-H. Lee, W. Pedrycz, and G. Sohn, Design of similarity and dissimilarity measures for fuzzy sets on the asis of distance measure, International Journal of Fuzzy Systems, vol., no., pp. 67-7, 9. [9] J. M. Merigó, Fuzzy multi-person decision making with fuzzy proailistic aggregation operators, International Journal of Fuzzy Systems, vol., no., pp. 6-74,. [] D.-W. Pei, The operation and deductive theorem in the formal deductive system L *, Fuzzy Systems and Mathematics, vol. 5, no., pp. 4-9, (in Chinese). [] D.-W. Pei, The characterization of residuated lattices and regular residuated lattices, Acta Math. Sin., vol. 45, 7-78, (in Chinese). [] D.-W. Pei, A logic system ased on strong regular residuated lattices and its completeness, Acta Math. Sin., vol. 45, pp , (in Chinese). [] D.-W. Pei, R implication: characteristics and applications, Fuzzy Sets and Systems, vol., pp. 97-,. [4] D.-W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems, vol. 8, pp ,. [5] D.-W. Pei, On the strict logic foundation of fuzzy reasoning, Soft Computing, vol. 8, pp , 4. [6] D.-W. Pei, Simplification and independence of axioms of fuzzy logic systems IMTL and NM, Fuzzy Sets and Systems, vol. 5, pp. -, 5.

7 International Journal of Fuzzy Systems, Vol. 4, No., June [7] D.-W. Pei and G.-J. Wang, A new kind of algeraic systems for fuzzy logic, J. Southwest Jiaotong Univ., vol. 5, pp , (in Chinese). [8] D.-W. Pei and G.-J. Wang, The completeness and applications of the formal system L *, Sci. China (Ser. F), vol. 45, pp. 4-5, (in Chinese). [9] D.-W. Pei and G.-J. Wang, The extensions L * n of the system L * and their completeness, Information Sciences, vol. 5, pp ,. [] Z. Pei and L. Zheng, A novel approach to multi-attriute decision making ased on intuitionistic fuzzy sets, Expert Systems with Applications, vol. 9, pp ,. [] G.-J. Wang, A formal deductive system for fuzzy propositional calculus, Chinese Sci. Bull., vol. 4, pp. 4-45, 997. [] G.-J. Wang, On the logic foundation of fuzzy reasoning, Information Sciences, vol. 7, pp , 999. [] G.-J. Wang, Non-classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, (in Chinese). [4] G.-J. Wang, Full implication triple I method for fuzzy reasoning, Sci. China (Ser. E), vol. 9, pp. 4-5, 999 (in Chinese). [5] W. Wang, Y. Wang, and Y. Zhang, Operations and properties of fuzzy logic systems, International Journal of Fuzzy Systems, vol., no. 4, pp. 7-79,. [6] X.-J. Wang, An Introduction to Mathematical Logic, Beijing Normal University Press, Beijing, 98 (in Chinese). [7] M. S. Ying, Declarative semantics of programming in residuated lattice-valued logic, Sci. China (Ser. E), vol. 4, pp , (in Chinese). Daowu Pei received the PhD degree in mathematics in from Sichuan University, Chengdu, Sichuan province, P.R. China. From to, he was a postdoctoral researcher in the Department of Applied Mathematics, Xi an Jiaotong University, Xi an, Shaanxi province, P.R. China. He worked in the Department of Mathematics, Yancheng Teachers College, Yancheng, Jiangsu province, P.R.China, from 98 to, where he was an associate professor in 997 and full professor in. Then he worked in the Department of Computer Science and Engineering, Tongji University, Shanghai, P.R.China, and served as a PhD supervisor in mathematics and computer science. He is currently a professor with the Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang province, P.R.China. His research interests include non-classical logic, approximation reasoning, decision making under uncertainty and rough set theory. He is an author or co-author of over sixty technical papers.