Chinese Journal of Electronics Vo1.19, No.3, July 2010 Topological Characterization of Consistency of Logic Theories in n-valued Lukasiewicz Logic Luk(n)* SHE Yanhonglr', WANG Cuojuu ' r' and HE Xiaoli 2 (1.Coll ege of Mathemat ics and Information Sci ence, Shaanxi Normal University, Xi 'an 710062, China) (2.College of Science, Xi 'an Shiyou University, Xi'an 710065, China) (3.Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China) Abstract - Let ( F(S), p) be the n-valued logic metric space, the present paper characterizes consistency of logic theories in the propositional logic system Luk(n) by means of topological concepts in the n -valued logic metric space. It is proved that a closed theory r is consistent iff it con- tains no interior points, iff it possesses the truth-forgotten property, iff it contains no non-empty regular sphere. Key words - n-valued propositional logic Luk(n), n- valued logic metric space, Theory, Consistency, Interior point, Regular sphere . I. Introduction Let r be a logic theory in any logic system, a fund amen tal qu estion is: whether r is consistent or not ? An int erest ing way, among oth ers, for answering thi s question is the way of consid- ering r as a subset of a logic metric space and then checking the consi st en cy of r by analyzing its topological properties. The authors did this in Ref.[l] for logic theories in 2-valued propositional logic and some int eresting results were ob tained, e.q. , we proved that a logically closed theory r is consistent if and only if r contains no int erior points in the logic metri c space (F( S) , p), where F (S) is the free algeb ra of type (-' , ---» generated by the set S consisting of all atomic formulas, i.e., F(S) is the set consisting of all well-formed formulas. p is a ps eudo-metric on F(S) which can be semantically defined in quantitative logic. The aim of the present paper is to generalize the results obt ained in Ref.[l] to the n-valued Lukasiewicz prop ositional logic syst em Luk(n) where the McNaught on functions will play an important role as the Bool ean functions did in Ref.[l]. All the concepts of truth degrees of formulas, of similarity de- grees between formulas, and of pseudo-metric are similar to the corresponding concepts in 2-valued propositional logic, while proofs of corresponding theorems are much mor e complicated as we will see below. In Section II, basic conce pts and results concern ing the logic system Luk(n) ar e list ed. In Section III, the concepts of truth degree , similarity degree and pseudo-metric in Luk(n) are introduced . In Section IV, three theorems for checking con- sistency of logic theories by means of their topological proper- ties are proved. Lastly, Section V is the conclusion. II. The Lukasiewicz n-valued Propositional Logic System Luk(n) Definition 1[ 41 . The axioms of Luk(n) are as follows: (1) A ---> (B ---> A); (2) (A ---> B) ---> ((B ---> C) ---> (A ---> C))j (3) ((A ---> B) ---> B) ---> ((B ---> A) ---> A)j (4) (-,A ---> -,B) ---> (B ---> A) j (5) (n - l)A == nA, (6) (kA k-l) n == nA k , where k = 2, . . . , n - 2, and kin - 1 does not hold. In addition , kA == A EB ... EB A (kA' s), A k == A Q9 .•. Q9 A (k A's ). A == B = (A ---> B) Q9 (B ---> A), A Q9 B = -,(A ---> -,B ), A EBB = -, AV B,A ,B E F (S ). The inference rule of Luk(n) is Modus ponens (MP for short) . Throught the present paper, we assume that i; ={0, n 1'..., = ,I} . Let A(pI,'" ,Pm) be a formula con taining m at omic formu- lae PI,'" ,pm, then A induces a function A : L': ---> Ln, the value of A at the point ( Xl , ''', X m) E L': can be obtained by substituting Xl, ... ,X m for PI, .. . , Pm in A(PI , . .. ,Pm) and explaining the operators -', ---> , Q9 , EB, V and 1\ as follows respec- tively: -'X = 1- x. ' Manuscript Received Nov . 2008; Accepted Oct. 2009. Thi s work is supported by theNational Natur al Science Foundation of China (No.10771129) and the 211 Const ruc tion Found ati on and Innovation Funds of Graduate Progr ams of Sh aanxi Normal University (No.2008CXB017) .