Bipolar Logic and Bipolar Knowledge Fusion Wen-Ran Zhang Dept. of Math and CS, Georgia Southern Univ. Statesboro, Georgia 30460-8093
[email protected], Phone: (912)486-7198 Fax: (912)681-0654 Abstract: It is observed that Boolean logic is a unipolar logic defined in the unipolar space {0,1}. It is argued that a unipolar system cannot be directly used to represent and reason with the coexistence of bipolar truth. To circumvent the representational and reasoning limitations of unipolar systems, a 4-valued bipolar combinational logic BCL is introduced based on the ancient Chinese Yin-Yang philosophy. The new logic is defined in a strict bipolar space S = {-1,0}×{0,1}, which is proved a generalization of Boolean logic and a fusion of two interactive unipolar subsystems. Bipolar tautologies including modus ponens are introduced for bipolar inference. The semantics of the new logic is established, justified, and compared with unipolar systems. Bipolar relations, bipolar transitivity, and polarized reflexivity are introduced. An O(n3) algorithm is presented for bipolar transitive closure computation. In addition, the lair’s case in the ancient paradox is redressed based on bipolar logic and bipolar relations. Keywords: Bipolar Logic, Knowledge Fusion, Recovery, Tautologies, Bipolar Relation, Ancient Paradox of the Liar
1. INTRODUCTION “Every matter has two sides” is an old saying. The coexistence of both a negative and a positive side is a reality for all relations among an agent world. For instance, cooperation and competition always coexist due to the coexistence of conflict-interest and commoninterest even between married couples. It is not rare to see a good couple become enemies because of uncontrolled growth of conflicts. It is observed, however, that Boolean logic [3] is a unipolar system that cannot be directly used for bipolar representation and inference. This limitation is due to the use of truth-values in the unipolar space {0,1}. The failure of a 2-valued logic in the ancient paradox [10, p221] of the liar shows the limitation of a unipolar logic. In the ancient Chinese Taoist philosophy, yin and yang are the two sides. Yin is the feminine or negative side of a system and yang is the masculine or positive side of a system. The coexistence in equilibrium and harmony of the two sides is considered a key for the mental and physical health of a person, a society, or a system. This principle has played an essential role in the success of traditional Chinese medicine where symptoms are often diagnosed as the loss of balance of the two sides.
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It is also observed that one action on a concept A can have certain effect and certain side-effect to a concept B; or two agents can have certain cooperation and certain competition. The coexistence of both competition and cooperation or conflict-interest and common-interest between two agents A1 and A2 can be naturally represented by a bipolar relation r(A1,A2)=(r,r+)=(competition, cooperation) ∈ {-1,0} × {0,1} = {(0,0)(1,0)(0,1)(-1,1)}. Bipolar connectives can be well-defined for a competition and cooperation network among a set of agents. However, a unipolar logical system cannot be directly used for such knowledge representation. The necessity for a bipolar logic can be illustrated as in Table 1 with a multiagent world. (comp, no coop)⊗(comp, no coop) = (-1,0) ⊗(-1,0)=(0,1) =(no comp, coop) //an enemy’s enemy is a friend, ⊗ is used as a cross-pole serial conjunctive (comp, no coop)&(comp,coop) = (-1,0) &(-1,1)=(-1,0) =(comp, no coop) //& is used as a linear parallel conjunctive (comp, no coop)⊕ (no comp, coop)= (-1,0) ⊕ (0,1) =(-1,1)=(comp, coop) //coexistence of both competition and cooperation, ⊕ is used as a bipolar disjunction operator ¬(no comp, coop)=¬(0,1)=(¬(no comp) ¬coop)=(-1-0,11) = (comp, no coop) //bipolar compliment -(x,y) ≡ (-y,-x)≠¬(x,y)≡(-1-x,-1-y); //bipolar negation ≠ bipolar complement (competition,cooperation)⇒(conflict interest, common interest)≡(competition→conflict-interest, cooperation →common interest)≡¬(competition, cooperation)⊕ (conflict-interest, common interest) ≡ (¬competition ∨ conflict-interest, ¬cooperation∨common interest) Note: (comp,coop) is a bipolar constant; (competion,cooperation) is a bipolar function; (x,y) is a bipolar variable. Table 1. Necessity for a bipolar logic The above observations lead to the notions of bipolar logic and bipolar inference. Section 2 introduces a bipolar logic. Section 3 introduces bipolar axioms, rule, and tautologies for bipolar inference. Section 4 presents recovery theorems to Boolean logic. Section 5 introduces bipolar relations. Section 6 is a redress of the liar’s case in the ancient paradox. Section 7 is a brief conclusion.
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2. A 4-VALUED BIPOLAR LOGIC A closer examination of Table 1 reveals the following unipolar deficiencies as well as the necessity for bipolar representations and operations: (1) A unipolar space cannot directly accommodate the coexistence of bipolar truth. A bipolar space with a bipolar disjunction operator (like ⊕ in Table1) is necessary. (2) In the unipolar case, parallel (or logical) and serial (or relational) conjunction makes no difference. In the bipolar case, serial or relational conjunction and parallel or logical conjunction are different. For instance, (-1,0)⊗(-1,0)=(0,1); but (-1,0)&(-1,0)=(-1,0). (3) Bipolar complement (¬) and bipolar implication (⇒) operators are needed for bipolar inference. (4) It can be argued that the use of complement as negation in a unipolar system is misleading because complement is not negation for rational numbers. For instance, the rational negation of “cooperation” should be “competition” or the negation of 1 should –1. However, in a unipolar logic the negation of “cooperation” is “no cooperation” or the negation of 1 is 0 that is totally different from rational negation. A separation between negation and compliment is necessary for bipolar modeling. To circumvent the limitations of unipolar systems, we first formalize two principles for bipolar representation and computation as in Table 2. Principle 1: Bipolar Representation Principle. A bipolar logical variable should consist of both a negative pole and a positive pole, which are truth functions that capture the nature of a positive side and a negative side, respectively, in a bipolar combination. Principle 2: Bipolar Computation Principle. A bipolar logic should be able to support bipolar reasoning by allowing computational interactions between the positive and negative poles. Table 2. Polarity principles Based the above observations we have Definition 1: Given truth value sets S-={ai} and S+ = {bj}, the Cartesian product S = S-× S+ = {ai}×{bj} = {sij} = {(ai,bj)} is said a strict bipolar crisp truth space and a bipolar truth variable or function (ai,bj) is said a strict bipolar crisp truth variable or function iff (1) 0≤i,j≤n
