YinYang Bipolar Cognition and Bipolar Cognitive Mapping - CiteSeerX

INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.YANGSKY.COM/YANGIJCC.HTM), VOL. 3, NO. 3, SEPTEMBER 2005

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YinYang Bipolar Cognition and Bipolar Cognitive Mapping (Invited Paper) Wen-Ran Zhang

Abstract— Modern scientific advances draw computational and reasoning power primarily from unipolar cognition. Unipolar cognition is based on unipolar logical systems defined in the unipolar space {0, 1} or [0,1]. Aristotle’s bivalent logic (300BC), Boolean logic, and fuzzy logic belong to unipolar systems. Despite of the great success of unipolar cognition, it can be argued that unipolar systems use a bottom-up approach and bipolar cognition is necessary in the understanding of micro- and/or macrocosms as equilibrium worlds with a top-down holistic approach. A computational framework is outlined in this paper for YinYang bipolar cognition and bipolar cognitive mapping. In bipolar cognition, a universe is either an equilibrium or quasior non-equilibrium. Since any n-polar equilibrium (including quasi- or non-equilibrium) can be converted to one or more bipolar equilibriums, bipolarity as a generic and inherent part of equilibrium is inseparable from bipolar truth. This bipolar worldview is originated from the ancient Chinese Daoist YinYang theory (600BC). This work introduces a family of YinYang bipolar logical systems for modeling bipolar equilibriums. A number of basic laws related to bipolar equilibrium relations are formally proved. A bipolar modal logic is proposed. The notions of life, existence, energy, stability, and a mathematical characterization of bipolar disorder are presented. Basic ideas are illustrated with bipolar cognitive mapping in international rec 2004-2005 Yang’s Scientific Research Institute, lations. Copyright ° LLC. All rights reserved. Index Terms— YinYang bipolar cognition, bipolar logic, bipolar fuzzy logic, bipolar modal logic, laws of equilibrium, bipolar cognitive mapping, equilibrium energy, life and existence, bipolar disorder.

I. I NTRODUCTION

I

T CAN be observed that YinYang equilibrium (including quasi- or non-equilibrium) exists in a microcosm (e.g. a quantum world) as well as in a macrocosm (e.g. the universe), in a social or spiritual world (with competition and cooperation or self-negation and self-assertion) as well as in a physical world (with centripetal and centrifugal forces) [17]. A healthy agent can be described as a YinYang equilibrium of selfnegation (for adapting to the world) and self-assertion (for self-assurance). An agent world or society can be described as the equilibrium of competition and cooperation. A political system can be characterized as the equilibrium of the left and right wings. Market economy is the equilibrium of the “bears” Manuscript received July 23, 2004; revised August 19, 2004; October 3, 2004. Wen-Ran Zhang, Department of Computer Science, Georgia Southern University, Statesboro, Georgia 30460-7997, USA. Email: [email protected] Publisher Item Identifier S 1542-5908(05)10306-6/$20.00 c Copyright °2004-2005 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on August 31, 2004 at http://www.YangSky.com/ijcc33.htm

and “bulls”. The financial status of a business can be captured with the in (profit) and out (cost). Every relation between two agents or agencies is the equilibrium of conflict and common interests even for a married couple or for two allied countries. While international relations form a global equilibrium of cooperation and competition among human societies on the earth, objects and systems in the universe and/or the universe itself form a global equilibrium of centripetal and centrifugal forces. If we consider the universe as an egg and equilibrium as a chicken, we have the question: “which one created the other?” Since equilibrium or quasi- equilibrium is natural reality, unavoidability, and/or necessity, it is a form of truth that is essential to both physical and social sciences especially to information sciences [16,17]. Evidently, bipolarity, like fuzziness as a generic and inherent part of equilibrium, is inseparable from the truth. This bipolar view leads to YinYang bipolar cognition with strict bipolarity that meet the following five conditions [17,18]: 1) Coexistence: The existence of one side conceptually depends on another within one object or relation. For instance, without competition there would be no cooperation or vice versa. 2) Equilibrium: The two sides form a dynamic equilibrium, which are interactive but not counteractive. 3) Symmetry: A negation operation converts or adjusts one side to another in symmetry. 4) Linearity: Without bipolar interaction the two sides can recover to Boolean logic linearly. 5) Integrity: With bipolar interaction the two sides can recover to Boolean logic as a non-linear combination. In bipolar cognition, natural reality or truth in the universe is either an equilibrium or quasi- or non-equilibrium. Unipolar cognition is based on unipolar logical systems defined in the unipolar space {0, 1} or [0, 1]. Unipolar logical systems include Aristotle’s bivalent logic (300BC), Boolean logic, fuzzy logic, and their unipolar extensions. Despite of the great success of unipolar cognition, it can be argued that unipolar systems is a bottom-up approach and bipolar cognition is necessary for an holistic understanding of micro- and/or macrocosms as equilibriums with a top-down approach. It can be shown that a unipolar system cannot be used for representing and visualizing bipolar truth due to the lack of polarity. To illustrate, let φ = (φ− , φ+ ) =(negative, positive) be a bipolar fuzzy function defined in the space [−1, 0]×[0, 1],

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we have the bipolar mapping φ(x) = (φ− (x), φ+ (x)) = (negative(x), positive(x)) : x ⇒ [−1, 0] × [0, 1] and ∀x(thing(x) ⇒ φ(x)). Using a unipolar space [0, 1], the bipolar coexistence cannot be adequately expressed even with a group of sentences such as ∃x(thing(x) ⇒ (negative(x) ∧ ¬(positive(x)); ∃x(thing(x) ⇒ (¬negative(x) ∧ (positive(x)); and ∃x(thing(x) ⇒ (negative(x) ∧ (positive(x)). Further, without strict bipolarity, the bipolar interaction of two equilibrium sets can not be characterized. Section 2 presents a comparison and distinction between bipolar spaces and unipolar spaces. Section 3 introduces a zero-order bipolar logic and a bipolar fuzzy logic. Section 4 introduces a bipolar 1st -order lift. Section 5 presents a bipolar modal transformation of the 1sr-order lift. Section 6 introduces bipolar crisp and fuzzy equilibrium relations and their link to equivalence relations. Section 7 presents basic definitions of life, existence, energy, and stability. Section 8 includes a few conclusions. II. C OMPARISONS AND D ISTINCTIONS The 4- 9- and infinite-valued strict bipolar spaces B1 , B2 , and BF are sketched as partial orderings and compared with six other spaces in Fig. 1(a)-(i) [14-16]. A strict bipolar space is a Cartesian product B = B − × B + , where B + is a unipolar truth space, and B − is the negative truth space symmetrical to B + . The strict bipolar space B1 captures bipolar truth of one matter in 2-levels on each side, where (0, 0) = (negative-pole false, positive-pole false) or (−F, F ); (−1, 0) = (negativepole true, positive-pole false) or (−T, F ); (0, 1) = (negativepole false, positive-pole true) or (−F, T ); (−1, 1) = (negativepole true, positive-pole true) or (−T, T ). The bipolar space B2 captures bipolar truth of one matter in three levels (true, partially true, and false) for each pole; and the bipolar space BF captures bipolar truth of one matter with infinite number of gray levels for each pole. Evidently, B1 is a polarization of Boolean space [4]; B2 is a polarization of Kleene’s 3-valued logical space [7], and BF is a polarization of Zadeh’s fuzzy space [8]. It is clear from Fig. 1 that the bipolar family of Bn and BF are essentially different structures from unipolar systems. The 4-valued algebra in [5], the 6-valued algebra in [9,10], and the 8-valued power-set are not qualified to be bipolar in the strict sense. Although they are defined in a bipolar space, they violate the polarity principle of coexistence. For instance, the values -1, 0, 1, are not bipolar in the strict sense but they are admissible in one or more of these approaches. We refer them as loosely defined bipolar approaches. Strict bipolarity also distinguishes bipolar models from Belnap’ 4-valued logic [3], Ginsberg’s bilattice [6], and Atanassov’s intuitionistic fuzzy logic [1]. Belnap and Ginsberg use four truth-values from a bilattice. Bilattice provides an important knowledge structure that assumes different semantics from a bipolar model. As Ginsberg stated in [6, p43] a bilattice represents truth in one side and information about the truth

in another with the space {true, false, no information, contradiction} where bipolar coexistence and bipolar interaction are not observed. In the same light, Atanassov’s intuitionistic fuzzy logic [1] uses one side as a fuzzy logic and another as a guard for consistency, where bipolarity is not observed either. The difference can be further clarified by comparing a bilattice with the 9-valued bipolar space B2 and the fuzzy bipolar space BF . It is clear from Fig. 2 that the bipolar family of Bn and BF are generally different structures from a bilattice [6]. A bilattice is defined in the first quadrant without explicit bipolarity; a strict bipolar model is defined as a bipolar Cartesian product in the second quadrant with explicit bipolar representation for bipolar coexistence, interaction, and equilibrium. Without explicit bipolarity, non-linear bipolar equilibrium relations and bipolar sets (crisp or fuzzy) [1519] cannot be formulated for a multiagent world. On the other hand, B1 can be easily coerced to a bilattice by taking the absolute values while a bilattice cannot be coerced to B1 without additional definitions for negation and non-linear bipolar conjunction. The unipolar and bipolar crisp and fuzzy spaces can be characterized with the following two equations bipolar crisp space = bivalent space + bipolarity; (1) bipolar fuzzy space = unipolar fuzzy space + bipolarity ≡ bipolar crisp space + fuzziness ≡ bivalent space + bipolarity + fuzziness. (2) III. B IPOLAR L OGIC AND BIPOLAR FUZZY L OGIC A 4-valued bipolar combinational logic BCL1 and its fuzzy counterpart BCLF are defined in the bipolar space B1 = {−1, 0} × {0, 1} and BF = [−1, 0] × [0, 1], respectively, with the following definitions [15-18]: • • • • • •

Disjunction: (a, b) ⊕ (c, d) ≡ (a ∨ c, b ∨ d) ≡ (−(|a| ∨ |c|), b ∨ d) ≡ (− max(|a|, |c|), max(b, d)); Serial Conjunction: (a,b)⊗(c,d) ≡ (-max(|a| ∧ |d|,|b| ∧ |c|),max(|a| ∧ |c|,b∧d)); Parellel Conjunction: (a,b)m(c,d)≡(-(|a| ∧ |c|),b∧d); Negation: -(a,b) ≡ (-b,-a); Complement: ¬(a,b) ≡ (¬a,¬b) ≡ (-1-a,1-b); Implication: (a,b)⇒(c,d)≡(a→c,b→d)≡ (¬a∨c, ¬b∨d).

Table 1 shows a set of zero-order bipolar axioms. Zero-order soundness and completeness is proved in [16]. Serial(φ,ψ) is defined as ⊗ is applicable between φ and ψ. IV. A L IFT TO B IPOLAR P REDICATE L OGIC Based on BA1-BA5, BAR, and BR1, the bipolar combinational logic BCL1 can be “lifted” to a bipolar predicate logic [17] with the introduction of two predicate axioms (BA6,BA7) and one predicate generalization (BR2) as in Table 2 with bipolar quantification assuming the tradition that a bipolar variable x occurring in a formula φ is free if it is not within the scope ∀x or ∃x, otherwise, it is said to be bound.

ZHANG, YINYANG BIPOLAR COGNITION AND BIPOLAR COGNITIVE MAPPING

1

(-1,1)

?

1 x

0

0

(a)

(b)

(-1,0) -

0

(-2,1)

(-1,1)

(0,1) 0

1

-True

+True

(0,1)

(-1,0) (0,0) (e)

(d)

+True

+True

(-1,1) (-1,2)

(-1,y) (0,2)

(-1,1)

(-1,0)

-True

-1

+

(c) (-2,2)

(-2,0)

55

(-1,0)

(0,1)

(x,1) (x,y)

(x,0)

(0,0)

-True

(0,1)

(0,y) (0,0)

(f)

(g)

k

k

T

true

false

⊥ (h)

t

t

(i)

Fig. 1. Different truth spaces and their order of magnitude [adapted from 16,17]: (a) Boolean space (Boole [4]); (b) Fuzzy space (Zadeh [8]); (c) 4-valued quantitative universe (De Kleer [5]); (d) 6-valued bipolar space (Zhang [10]); (e) 4-valued strict bipolar space B1 (Zhang [12-17]]); (f) 9-valued strict bipolar space B2 [12-17]; (g) Infinite-valued strict bipolar fuzzy space BF [17]; (h) 4-valued logical space (Belnap [3]); (i) 4-valued bilattice (Ginsberg [6]).

V. A “T RANSFORMATION ” TO M ODAL L OGIC The “lifted” BCL1 can be “transformed” from a quantificational language to a bipolar modal logic. According to the Leibnizian idea, necessity is what is true at every possible world and possibility is what is true at some. Linguistically, a sentence of the form ¤A (necessarily A) is true if and only if A itself is true at every possible world; and a sentence of the form ♦A (possible A) is true just in case A is true at some possible world. In the bipolar case, true should be either negative pole true (−1, 0), positive pole true (0, 1), or bipolar true (−1, 1), and false should be negative pole false (0, 1), positive pole false (−1, 0), or bipolar false (0, 0), which are the complement of the polarized true values. Let the necessity modality ¤(a, b) = (a, b) and possibility

modality ♦(a, b) = ¬¤¬(a, b). Let A be a modal sentence, Pn be one of denumerably many one-place predicates and α = (α− , α+ ) ∈ B1 be a bipolar variable, so that Pn (α) is an atomic formula. We define the mapping =, from the language of bipolar modal logic to the quantificational language of bipolar predicate logic, as follows. 1) =(Pn ) = Pn (α), for n = 0, 1, 2,... 2) =(a, b) = (a, b), ∀(a, b) ∈ B1 = { − 1, 0} × {0, 1}. 3) =(−(a, b)) = −(a, b), ∀(a, b) ∈ B1 = {−1, 0}×{0, 1}. 4) =(¬A) = ¬=(A). 5) =(A&B) = =(A)&=(B). 6) =(A ⊗ B) = =(A) ⊗ =(B). 7) =(A ⊕ B) = =(A) ⊕ =(B). 8) =(A ⇒ B) = =(A) ⇒ =(B). 9) =(A ⇔ B) = =(A) ⇔ =(B).

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TABLE I B IPOLAR A XIOMS ( ADAPTED FROM [15])

Linear Axioms: BA1: (φ-,φ+)⇒((ϕ-,ϕ+)⇒(φ-,φ+)); BA2: ((φ-,φ+)⇒((ϕ-,ϕ+)⇒(χ-,χ+)))⇒(((φ-,φ+)⇒(ϕ-,ϕ+))⇒((φ-,φ+)⇒(χ-,χ+))); BA3: (¬(φ-,φ+)⇒(ϕ-,ϕ+)) ⇒ ((¬(φ-,φ+)⇒¬(ϕ-,ϕ+)) ⇒ (φ-,φ+)); BA4: (a) (φ-,φ+)&(ϕ-,ϕ+)⇒(φ-,φ+); (b) (φ-,φ+)&(ϕ-,ϕ+)⇒(ϕ-,ϕ+); BA5: (φ-,φ+)⇒((ϕ-,ϕ+)⇒((φ-,φ+)&(ϕ-,ϕ+))); Rule of Inference – Bipolar Modus Ponens (BMP): BMP: ((φ-,φ+) & ((φ-,φ+)⇒(ϕ-,ϕ+)))⇒(ϕ-,ϕ+); Non-Linear Bipolar Augmentation Rule (BAR): (φ − , φ + ) ⇒ (ϕ − , ϕ + ), (ψ − ,ψ + ) ⇒ ( χ − , χ + ) (φ − , φ + ) ⊗ (ψ − ,ψ + ) ⇒ (ϕ − , ϕ + ) ⊗ ( χ − , χ + ) or [((φ-,φ+)⇒(ϕ-,ϕ+))&((ψ-,ψ+)⇒(χ-,χ+))]⇒ [((φ-,φ+)⊗(ψ,ψ+)) ⇒((ϕ-,ϕ+)⊗(χ-,χ+))], ∀(φ-,φ+),(ψ-,ψ+),(ϕ-,ϕ+),(χ-,χ+), Serial((φ-,φ+),(ψ-,ψ+)), Serial((ϕ-,ϕ+),(χ-,χ+)) TABLE II A LIFT OF BCL1

TO A

1 ST- ORDER BIPOLAR LOGIC [17].

BA6: ∀x(φ− (x),φ+ (x))⇒ (φ− (t),φ+ (t)) ; where t is term free for x in φ. BA7: ∀x((φ− ,φ+ ) ⇒ (ϕ− ,ϕ+ ))⇒((φ− ,φ+ ) ⇒ ∀x(ϕ− ,ϕ+ )); where φ contains no free occurrences of x. BR2-Generalization: (φ− ,φ+ ) ⇒ ∀x(φ− ,φ+ ).

10) =(¤A) = ∀α=(A). 11) =(♦A) = ∃α=(A). Thus = associates with each sentence A in the bipolar modal language a unique formula =(A) in the quantification language by replacing each atomic sentence Pn by Pn (α) and putting ∀α and ∃α respectively for occurrences of ¤ and ♦. Bipolar modality supports bipolar reasoning with the forms of =(A ⊗ B) = =(A)⊗=(B) and =(A⊕B) = =(A)⊕=(B) in addition to the bipolar forms isomorphic to their unipolar counterparts. Thus the bipolar modal logic enables reasoning with bipolar equilibria as a form of unavoidability, necessity, or truth. VI. E QUILIBRIUM VS . E QUIVALENCE AND S IMILARITY A. Bipolar Relations Bipolar relations( crisp or fuzzy) are defined in [17] as in the following: A bipolar (binary) relation R from X to Y , where X = {xi }, 0 < i ≤ m, and Y = {yj }, 0 < j ≤ n, is a collection of ordered pairs or subsets of X × Y characterized by a membership function µR (xi , yj ) which associates with each ordered pair (xi , yj ) a relationship strength using a strict bipolar variable. Formally, we have R

= {µR (xi , yj )|∀i, j, 0 < i ≤ m, 0 < j ≤ n,

Given two bipolar relations (crisp or fuzzy) R1 and R2 in X, where X = {xi } with size n, if ∀i, j, 0 < i, j ≤ n, µR1 (xi , xj ) ⊕ µR2 (xi , xj ) = µR2 (xi , xj ), we say that R1 is contained in (smaller than or equal to) R2 denoted by R1 ⊂ R2 or R1 ≤≤ R2 . The ⊕ − ⊗ or max −⊗ Composition of an m × n bipolar relation (crisp or fuzzy) R ∈ X × Y and an n × k bipolar relation Q ∈ Y × Z, denoted by R ⊗ Q, is defined as R ⊗ Q(x, z)

max(µR (x, y) ⊗ µQ (y, z)), y

∀x, y, z, x ∈ X, y ∈ Y, z ∈ Z, where max is equivalent to ⊕ and the n-fold composition of R with itself is denoted by Rn through this paper. The bipolar disjunction operator ⊕ on m × n bipolar relations R and Q denoted by R ⊕ Q, is defined by R ⊕ Q = [(rij ⊕ qij )], ∀i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ n, where rij , qij are the (i, j)-th element of R and Q, respectively. If Ω is used as a binding operator for an m × n bipolar relation R, ∀i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ n, we have R

≡

(R− , R+ ) = R− ΩR+ = [rij ]

− + + − = [rij ]Ω[rij ] = R+ ΩR− = [rij ]Ω[rij ]

µR : (xi , yj ) ⇒ B}, where ⇒ is a bipolar mapping. When B = Bn we say R is a strict bipolar crisp relation; when B = BF we say R is a strict bipolar fuzzy relation. Evidently, a bipolar crisp relation [15] is a special case of a bipolar fuzzy relation. In the follows we reference both as bipolar relations.

=

− + = [(rij , rij )];

(¬R− , R+ ) =

− + [(¬rij , rij )],

(R− , ¬R+ ) =

− + [(rij , ¬rij )],

¬R

− + = (¬R− , ¬R+ ) = [(¬rij , ¬rij )].

Then, the identities in TableIII hold.


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TABLE III S OME IDENTITIES FOR BIPOLAR FUZZY RELATIONS [17]. Laws Identity Null Idempotency Commutativity Associativity Distributivity Negation Augmentation

Inverse

⊗ ⊕ [I]⊗R≡R; //[I] and R are both n×n [(0,0)]⊕R≡R; //R is m×n [(0,0)]⊗R≡ [(0,0)]; //[(0,0)] and R are both n×n [(-1,1)]⊕R≡ [(-1,1)]; //R is m×n R⊕R≡R; //R is m×n; R⊗Q≡Q⊗R; //R and Q are both n×n R⊕Q≡Q⊕R; // R,Q are both m×n; (R⊗Q)⊗W≡R⊗(Q⊗W); (R⊕Q)⊕W≡R⊕(Q⊕W); //R is m×n, Q is n×w, W is w×x; //R,Q,W are all m×n R⊗(Q⊕W) ≡R⊗Q⊕R⊗W; //R is m×n; N and W are both n×x -[I]⊗R ≡ -R; //[I] and R are both n×n (-1,1)⊗R ≡ [((-1,1)⊗rij )] [(-1,0)]⊕R≡ [(-1, r+ ij )], ∀i,j, 1≤i≤m, 1≤j≤n; − + − + ≡[((-max(|rij |,rij ),max(|rij |, rij ))], [(0,1)]⊕R≡ [(r− ,1)], ∀i,j, 1≤i≤m, 1≤j≤n; ij ∀i,j, 1≤i≤m, 1≤j≤n; // R is m×n //[(-1,0)], [(0,1)],R are all m×n (R− ,R+ ) ⊕ (¬R− ,¬R+ ) ≡ [(-1,1)]; (R− ,R+ ) ≡ ¬(¬R) // all matrices are m×n

B. Equilibrium Relations Bipolar reflexivity, symmetry, and transitivity are defined in [15] as in the following: A bipolar relation R (crisp or fuzzy) in X, where X = {xi }, 0 < i ≤ n, is bipolar symmetric if, ∀i, k, 1 ≤ i, k ≤ n, we have µR (xi , xk ) = µR (xk , xi ). It is positive pole reflexive if, ∀i, 1 ≤ i ≤ n, we have µR (xi , xi ) = (n, 1). It is negative pole reflexive if, ∀i, 1 ≤ i ≤ n, we have µR (xi , xi ) = (−1, p). It is bipolar reflexive if, ∀i, 1 ≤ i ≤ n, we have µR (xi , xi ) = (−1, 1). It is ⊕-⊗ or max-⊗ bipolar transitive if, ∀i, j, k, 1 ≤ i, j, k ≤ n, we have µR (xi , xk ) ≥≥ max(µR (xi , xj ) ⊗ µR (xj , xk )). xj

where ≥≥ is a transitive bipolar comparison operator; (a, b) ≥≥ (c, d) iff |a| ≥ |c| and b ≥ d or iff (a, b) ⊕ (c, d) = (a, b). Note that, a bipolar transitive closure can be computed with an O(n3 ) algorithms [17]. Bipolar symmetry and transitivity are easy to understand. Negative bipolar reflexivity provides a measure for self-adjustability to equilibrium and harmony. The notions of bipolar sets and equilibrium relations are defined in [15] as in the follows: Let C = {ci } and C ⊆ X, 0 < i ≤ n, C is a coalition subset of X and a relation µ(cj , ck ) is a coalition relation if, ∀j, k, 0 < j, k ≤ n, µ(cj , ck ) = (0, 1). Let H = {hi } and H ⊆ X, 0 < i ≤ n, H is a harmony subset of X and µ(hj , hk ) is a harmony relation if, ∀j, k, 0 < j, k ≤ n, µ(hj , hk ) = (−1, 1). Let C1 and C2 be coalition sets, C1 = {c1i } and C2 = {c2j }, 0 < i ≤ n and 0 < j ≤ p; let F = C1 ∪ C2 and F ⊆ X; F is a conflict subset of X and

µ(c1k , c2z ) is a conflict relation if, ∀k, z, 0 < k ≤ n and 0 < z ≤ p, µ(c1k , c2z ) = { − 1, 0}. A bipolar relation R in X, where X = {xi }, 1 ≤ i ≤ n, is an equilibrium relation [15] if it is (1) bipolar symmetric; (2) positive pole or bipolar reflexive, and (3) bipolar transitive. More specifically, a positive pole-reflexive equilibrium relation is a P -type equilibrium relation; a negative pole reflexive equilibrium relations is a N -type equilibrium relation; and a bipolar reflexive equilibrium relation is an N P type equilibrium relation. Theorem 1: The transitive closure of any negative pole reflexive and symmetric bipolar relation R is an N -type equilibrium relation. The transitive closure of any positive pole reflexive and symmetrical bipolar relation R is a P type equilibrium relation. The transitive closure of any bipolar reflexive and symmetric bipolar relation R is an N P -type equilibrium relation. Proof : The theorem follows from that (1) the transitive closure of any symmetrical bipolar relation is still symmetrical because any symmetrical relation is a bigraph; (2) the closure does not change the original reflexivity since (−1, x), (y, 1), and (−1, 1) are already closed, respectively, on the negative pole, positive pole, or both poles. ¥ Now we are ready to prove the harmony laws (HLaws) presented in [15] without proofs: Theorem 2: The following laws hold. HLaw1: The bipolar transitive closure < of any negative pole reflexive bipolar relation R must be bipolar reflexive. HLaw2: Any N -type equilibrium relation must also be an N P -type equilibrium relation and any equilibrium relation is positive-pole reflexive or a P -type equilibrium relation. HLaw3: The negative pole R− and the positive pole R+ of any N P -type equilibrium relation R in X, where X = {xi }, 1 ≤ i ≤ n, must meet the condition R− = −R+ or R+ = −R− , and therefore, all clusters from R must be in a harmonic state. HLaw4: A harmony subset must be disjoint by neutral (0, 0) relationships with any other types of clusters if any other clusters coexist with a harmony subset in an equilibrium relation. HLaw5: The transitive closure of any bipolar relation in a harmony set must be an NP-type equilibrium relation.

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Proof : HLaw1 follows from that µR (xi , xi ) ⊕ µR (xi , xi ) ⊗ µR (xi , xi ) = (−1, v) ⊕ (−1, v) ⊗ (−1, v) = (−1, v) ⊕ (−1, 1) = (−1, 1), which is derived from cycling the negative pole reflexive edge (−1, v) twice (2n) using Eq. (8). For HLaw2, since an N -type equilibrium relation must also be N P -type following HLaw1, any equilibrium relation is either: (1) P -type but not N -type, or (2) N P -type. Since an N P -type is both N -type and P -type by definition, any equilibrium relation must be P -type. HLaw3 follows from that, given any µR (xi , xj ) = (u, v), if |u| > v, (u, v) must be (−1, 0), that would violate the bipolar transitivity requirement

=

(u, v) ≥≥ µR (xi , xj ) ⊗ µR (xj , xj ) (−1, 0) ⊗ (−1, 1) = (−1, 1)

because the reflexive edge µR (xj , xj ) must be harmonic for any N P -type equilibrium relation. Similarly, if |u| < v, (u, v) must be (0, 1), that would violate the bipolar transitivity requirement

=

(u, v) ≥≥ µR (xi , xj ) ⊗ µR (xj , xj ) (0, 1) ⊗ (−1, 1) = (−1, 1).

Since each relationship in R is either (0, 0) or (−1, 1), all clusters induced from R+ must be in a harmonic state. For HLaw4, according to the augmentation law, (−1, 1) ⊗ (−1, 0) = (−1, 1) and (−1, 1) ⊗ (0, 1) = (−1, 1). Therefore, if agent A and B are in a harmonic relation, B and C have any non-neutral relationship will necessarily imply A, B, and C are all in a harmonic relation based on the transitivity of an equilibrium relation. Since only when (a, b) = (0, 0) we have (−1, 1) ⊗ (a, b) = (0, 0) (null law), the only way for other type of clusters to coexist with a harmony cluster is to have a (0, 0) relation with it. Otherwise, other clusters have to join the harmony subset. For HLaw5: First, given any two nodes x and y with a harmonic relationship (−1, 1), the transitive cycle x → y → x always results in bipolar reflexivity. Secondly, harmony implies symmetry. ¥ On the equilibrium laws (ELaws) [15] we have the following formal proofs: Theorem 3: The following laws hold. ELaw1: R+ of any crisp bipolar transitive relation R in X, where X = {xi }, 1 ≤ i ≤ n, is unipolar transitive. ELaw2: R+ of any crisp equilibrium relation R in X, where X = {xi }, 1 ≤ i ≤ n, is an equivalence relation. ELaw3: For any n × n crisp bipolar transitive relation R, |R− | ∪ R+ results in a transitive unipolar relation. ELaw4: For any crisp equilibrium relation R, |R− | ∪ R+ results in an equivalence relation. Proof : For ELaw1: Based on the bipolar transitivity definition (Eq. 7), if R+ were not unipolar transitive, R would not be bipolar transitive.

For ELaw2: First, any crisp equilibrium relation is positive pole reflexive. Secondly, equilibrium implies bipolar symmetry. Thirdly, positive pole transitivity R+ follows from bipolar transitivity. For ELaw3: Since R is bipolar transitive, let µR be a mapping function of R in set X with size n, we must have, ∀i, j, k, 1 ≤ i, j, k ≤ n, µR (xi , xk ≥≥ max(µR (xi , xj ) ⊗ µR (xj , xk )). xj

Then, let R1 = |R− | ∪ R+ be a unipolar relation and, let ρR1 be the mapping function of R1, we must have ρR1 (xi , xk ) ≥ maxxj (ρR1 (xi , xj ) ∧ ρR1 (xj , xk )), therefore |R− | ∪ R+ must be unipolar transitive. For ELaw4: First, positive (unipolar) reflexivity follows from the definition of an equilibrium relation. Secondly, unipolar symmetry follows from the fact that the ∨-operation does not change the property of symmetry. Thirdly, its transitivity follows from bipolar transitivity. ¥ On the bipolar partitioning laws (BPLaws) [15] we have: Theorem 4:The following laws hold. BPLaw1: |R− | ∪ R+ of any equilibrium relation R in X, where X = {xi }, 1 ≤ i ≤ n, induces disjoint subsets including coalition subset(s) not in conflict, harmony subset(s), and conflict subset(s) (if any) in X. BPLaw2: Given any equilibrium relation R in X, where X = {xi }, 1 ≤ i ≤ n, R+ induces disjoint subsets including coalition subset(s) that are in or not in conflicts and harmony subset(s) (if any) in X. BPLaw3: Given any equilibrium relation R in X, where X = {xi }, 1 ≤ i ≤ n, let the set of disjoint subsets induced from (|R− |∪R+ ) be S1; let the set of disjoint subsets induced from R+ be S2. i S1∩S2 results in a set S3 of disjoint harmony subset(s) and coalition subset(s) that are not in any conflict in X. ii S1 ∪ S2 results in a set S4 of disjoint conflict subset(s), coalition subsets in the conflict subset(s), coalition subset(s) not in conflict, and harmony subset(s). iii S1 − S2 results in a set S5 of disjoint conflict subset(s), each of which contains two coalition subsets. iv S2−S1 results in a set S6 of disjoint coalition subset(s) that are involved in a conflict. Proof : For BPLaw1: First, |R− | ∪ R+ is an equivalence relation(ELaw4). Non-neutral relationships (−1, 0), (0, 1), and (−1, 1) in R become related in |R− | ∪ R+ and neutral (0, 0) relationships in R become. Second, coalition subsets, conflict subsets, and harmony subsets neutral to each other in R become unrelated to each other. Third, based on HLaw4, a harmony subset must be disjoint with other clusters. Fourth, two and at most two coalition subsets can join a conflict set because (1) if coalition A is in conflict with B and B is in conflict with C implies A and C is a coalition due to the fact that (−1, 0) ⊗ (−1, 0) = (0, 1) (“an enemy’s enemy is a friend”); and (2) if any element in a coalition is in conflict with an element in another coalition, all elements in the first coalition must be in conflict with every one in the second coalition due to the fact that (0, 1) ⊗ (−1, 0) = (−1, 0) (“a friend’s enemy is an enemy”).


For BPLaw2: First R+ is an equivalence relation (ELaw2) that induces disjoint partitions. Second, (0, 1) and (−1, 1) relationships in R become 1 in R+ , the (−1, 0) relationships are excluded. Therefore, the partitions induced from R+ can only be coalition or harmony subsets. BPLaw3 follows from BPLaw1 and 2 directly. ¥ On the local equivalence laws (LELaws) [15], we have Theorem 5: The following laws hold. LELaw1: Given any equilibrium relation R in X, where X + = {xi }, 1≤i≤n, if R+ ∩ |R− | = [|r− ij |∧rij ], ii,j, 1≤i,j≤n, is not null it must be a local equivalence, which induces disjoint harmony subsets. LELaw2: Given any equilibrium relation R in X, where X = {xi }, 1≤i≤n, if (|R− |∪R+ )-(R+ ∩|R− |) is not null, it must be a local equivalence in X, which induces either disjoint conflictfree coalition subsets, or disjoint conflict subsets, or disjoint subsets of both types LELaw3: Given any equilibrium relation R in X, where X = {xi }, 1≤i≤n, if (|R− |∪R+ ) − |R− | is not null it must be a local equivalence in X that induces one or more coalition subsets that are either involved or not involved in a conflict. LELaw4: Given any equilibrium relation R in X, where X = {xi }, 1≤i≤n, we have R+ -(R+ ∩ |R− |) ≡ (|R− |∪R+ ) − |R− |. Proof : LELaw1: First, following the 5th harmony law a transitive binary relation on a harmony set must be an N P type equilibrium relation. Secondly, only a harmonic relationship (−1, 1) in R results in a nonzero value in R+ ∩ |R− |. Thirdly, following the 4th harmony law, harmonic clusters must be disjoint with each other and any other clusters. Thus, if R+ ∩ |R− | is not null, it must be a local equivalence that identifies one or more disjoint harmony subsets. LELaw2: It follows from 1) (|R− | ∪ R+ ) is an equivalence relation (Theorem 6); 2) Following last theorem if (R+ ∩|R− |) is not null it is an local equivalence that identifies harmonic relationships and induces one or more disjoint harmony subsets; and 3) the subtraction (|R− | ∪ R+ ) − (R+ ∩ |R− |) simply removes the harmonic type relationships without affecting the unipolar reflexivity, symmetry, and transitivity among other types of true relationships that are positively related or negatively related, because Truth(1,0)=1 and Truth(0,1)=1. LELaw3: It follows from 1) |R− | ∪ R+ is an equivalence relation; 2) (−1, 0) and (−1, 1) values in R lead to 1 in |R− |; and 3) The subtraction removes disjoint harmonic relationships and conflict relationships without affecting the reflexivity, symmetry, and transitivity among coalition subsets. LELaw4: It follows from 1) (0, 1) and (−1, 1) in R lead to 1 in R+ ; (−1, 1) values in R lead to 1 in (R+ ∩ |R− |); 2) (−1, 0),(0, 1), and (−1, 1) in R all lead to 1 in (|R− | ∪ R+ ); (−1, 0) and (−1, 1) in R lead to 1 in |R− |; and 3) R+ −(R+ ∩|R− |) and (|R− |∪R+ )−|R− | both identify the (0, 1)-values in R. ¥

59

On the necessary and sufficient conditions (besides reflexivity, symmetry, and bipolar transitivity) for a bipolar relation R to be an equilibrium relation, we have Theorem 6: Given a bipolar relation R in X, where X = {xi }, 1 ≤ i ≤ n, the following conditions are necessary and sufficient for R to be an equilibrium relation: 1) R+ is an equivalence relation; 2) |R− | ∪ R+ is an equivalence relation; 3) if (R+ ∩|R− |) is not null it must be a local equivalence; 4) if (|R− | ∪ R+ ) − (R+ ∩ |R− |) is not null it must be a local equivalence; 5) if (|R− | ∪ R+ ) − |R− | is not null it must be a local equivalence; and 6) R+ − (R+ ∩ |R− |) ≡ (|R− | ∪ R+ ) − |R− |, null or not null. Proof : It follows from 1) Positive pole reflexivity of R follows from that R+ is an equivalence relation; 2) Bipolar transitivity of R follows from its reflexivity and the fact (R+ ∩ |R− |), (|R− | ∪ R+ ) − (R+ ∩ |R− |), and R+ − (R+ ∩ |R− |) ≡ (|R− | ∪ R+ − |R− |) are all local equivalence if not null. That is, the disjoint harmony sub-relations in X are bipolar transitive; the disjoint coalition sub-relations are bipolar transitive, and the disjoint conflict subrelations are bipolar transitive. Since there are no other types of subrelations besides the three types, R must be bipolar transitive. 3) Symmetry of R follows from the fact that R is transitive and both R+ and |R− | ∪ R+ are equivalence relations. ¥ Similarly, given a bipolar fuzzy relation R in X, where X = {xi }, 1 ≤ i ≤ n, the following conditions are necessary and sufficient for R to be a fuzzy equilibrium relation [19]: 1) R+ is a fuzzy similarity relation; 2) |R− | ∪ R+ is a fuzzy similarity relation; 3) If (R+ ∩ |R− |) is not null it must be a local similarity relation; 4) If (|R− | ∪ R+ ) − (R+ ∩ |R− |) is not null it must be a local similarity relation; 5) If (|R− | ∪ R+ ) − |R− | is not null it must be a local similarity relations; and 6) R+ − (R+ ∩ |R− |) ≡ (|R− | ∪ R+ ) − |R− |. Based on the above conditions, it is clear that the concept of equilibrium find no equivalence in bivalent or fuzzy representation including S4 and S5 modality. The use of |R− | must necessarily lose bipolarity and therefore, bipolar truth cannot be visualized using a unipolar representation. A bipolar counterpart of S4 and S5 modality remains a challenge because a bipolar equilibrium is a non-linear fusion of many equivalence or similarity relations. C. Examples Example 1: A bipolar equilibrium relation can be used for modeling a multiagent competition and cooperation network. Its application in developing a bipolar conceptual graph or cognitive map

60


(CM) of such a network is illustrated in Fig. 2(a). The CM is first represented as a positive pole reflexive and symmetrical bipolar relation R, and then the bipolar transitive closure < of R is computed. (0, 0) or the energy of φ(A) = φ− (A) + φ+ (A) 6= 0}, where φ is a bipolar truth function or a bipolar equilibrium. Based on this condition, existence, life, equilibrium energy, and stability of an object or agent can be defined in Table IV. This view conforms with the principle in Chinese medicine, where a the health of a patient (agent) has YinYang bipolar equilibrium

of self-negation (for adapting to the world) and self-assertion (for self-assurance) in a harmonic state. Bipolar disorder can be described as the lack of bipolar equilibrium and bipolar harmony. Table V shows four fuzzy equilibrium classes of Fig. 5b and Table VI shows the what-if equilibrium energy and stability analysis, where equilibrium energy [15,18] of an equilibrium relation is defined as Given an equilibrium or quasi-equilibrium relation R = (R− , R+ ), the summation of all elements in R− , denoted ε− (R) is the negative energy of R; the summation of all elements in R+ , denoted ε+ (R), is the positive energy of R; the polarized total, denoted ε(R) = (ε− , ε+ ), is the bipolar energy of R; the absolute total, denoted |ε|(R) = − + |ε total energy of R. The summation εimb (R) = Pm|+|ε Pn|, is the + − (r − |rij |) is the energy imbalance of R. ij i=1 j=1 While energy values provide unnormalized measures for cooperation and competition among clusters, we now define normalized measures for different clusters or quasi-equilibrium relations as in the following definitions: Given ε(R) = (ε− , ε+ ) where R is a quasi-equilibrium relation, (1) if R is a relation in an equilibrium class E(c),

62


(-1,0) S. Vietnam

Japan

N. Vietnam (0,1)

(0,1)

(0,1)

Cuba

 ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )   ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )     ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )     ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )   ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )     ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 )   ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 )     ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 )   ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 )     ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 )     ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 )   ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( -1 ,0 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) ( 0 ,1 ) 

(0,1)

(-1,0) (0,1) Nato Allies

(0,1)

(0,1) S. Korea

US

USSR

(0,1)

(-1,0) (0,1) China

Taiwan

E. European Allies

(0,1) N. Korea

(-1,0)

(a)

(b)

Partition(R) results in {H,F,C,A,B} where H = {}; // no harmony sets F = {{1,2,3,4,5,6,7,8,9,10,11,12}}; // one conflict set C = {{1,2,3,4,5,6}{7,8,9,10,11,12}}; // two coalition sets A={{1,2,3,4,5,6}{7,8,9,10,11,12}}; // two coalition sets in conflict B = {} //no conflict-free coalition sets

(c) Fig. 4.

(a) A bipolar crisp CM in the 1950s (adapted from [15]). (b) Equilibrium relation of (a). (c) Partition(E1). TABLE IV Dplr∨ (φ− (A), φ+ (A)) = φ− (A) ∨ φ+ (A) DEFINES AN EXISTENCE OR LIFE FUNCTION .

(φ− (A), φ+ (A)) (0, 0) (0, y)

Dplr∨ (φ− (A), φ+ (A)) = φ− (A) ∨ φ+ (A) 0 y>0

(x, 0)

|x| > 0

(x, y)

|x| > 0, y > 0, |x| 6= y

(−1, 1)

1

Existence/life of A No life or no existence Life with yin-deficiency (non-equilibrium) Life with yang-deficiency (non-equilibrium) Life with unbalanced yin-yang (quasi-equilibrium) Full life with balanced yin-yang in harmony (equilibrium)

the common interest of c with the others is measured as c+ = ε+ /|ε|(R), and the conflict interest is measured as c− = 1 − c+ = |ε− |/|ε|(R); (2) if R is a relation in a fuzzy coalition, the coalition strength is measured as c+ = ε+ /|ε|(R), and the coalition weakness is measured as c− = 1−c+ = |ε− |/|ε|(R); (3) if R is a relation in a fuzzy conflict set, the conflict strength is measured as c− = |ε− |/|ε|(R); (4) if R is a relation in a fuzzy harmony set, the harmony level is measured as h± = (ε− /n2 , ε+ /n2 ), where n is the size of the harmony set. Given an equilibrium relation or quasi-equilibrium R, the stability of R [15,18] is measured by a real value in [0, 1] as defined in the following: Stability(R) = (|ε|(R) − |εimb (R)|)/|ε|(R).

Energy Imbalance (I) 0 y

|x|

|x|

|x| + y

|x + y|

Stability S = (T − I)/T Undefined 0 (bipolar disorder) 0 (bipolar disorder) 1 − |x + y|/(|x| + y)

2

0

1

conflict set has two fuzzy coalition sets involved; (3) a fuzzy conflict relationship has more conflict strength than common interest. We assume the weakness measure of a coalition and the common interest measure of a conflict can cancel each other, and therefore, they are not taken into consideration. The harmony level measure is based on the rational that (1) the strongest harmony relationship strength is (−1, 1) and the weakest is (0, 0); (2) there are n2 bilateral relationships for a harmony set with size n; (3) h± = (ε− /n2 , ε+ /(n2 ) is the bipolar average closed in [−1, 0] × [0, 1]. It can be observed that fuzzy coalition, conflict, and harmony correlate with bipolar transitivity in the following natural ways: •

The conflict strength of a fuzzy conflict set is defined based on the rational that (1) a fuzzy coalition relation has less coalition weakness and more coalition strength; (2) a fuzzy

Energy Total(T ) 0 y

•

The higher the transitivity among a fuzzy coalition set the stronger the coalition. The higher the transitivity among a fuzzy conflict set the stronger the two involved coalitions and the stronger the


63

(0,0.3)

(-0.2,0.2)

3. Japan

(-0.6,0.8)

(-0.2,0.9)

2. Nato Allies

(-0.9,0.1)

(-0.5,0.6)

(-0.4,0.8)

(-0.2,0.8)

1.US

5. S. Korea

(0,0.6)

(-0.9,0.9)

1) 0.9) 0.8)

(-0.7 0.9) (-0.6 1) (-0.6 0.7)

(-0.7 0.8) (-0.6 0.7) (-0.5 1)

(-0.4 0.6) (-0.3 0.5) (-0.3 0.4)

(0,0.3)

9. China

4. Taiwan

(-0.7 0.9) (-0.6 0.8) (-0.6 0.7) (-0.3 0.5) (-0.6 1) (-0.1 0.1)

8. E. European countries (0,0.6)

(-0.1,0.9)

(-0.5,0.4) (-0.2,0.6)

 (-0.8  (-0.7   (-0.7   (-0.4  (-0.7   (-0.2  (-0.8   (-0.5  (-0.9   (-0.7  (-0.9 

(-0.3,0.6)

7. Russia

(-0.1,0.6)

(-0.1,0.9)

11. Cuba

6. Vietnam

(0,0.3) 10. N. Korea

(-0.8,0.2)

(-0.2 0.2) (-0.1 0.1) (-0 0)

0.6) (-0.3 0.5) 0.9) (-0.6 0.8) 0.2) (-0.1 0.1)

(-0.3 0.4) (-0.6 0.7) (-0 0)

(0 1) (-0.3 0.5) (0 0)

(0 0) (-0.1 0.1) (0 1)

0.8) (-0.7 0.7) 0.6) (-0.4 0.5)

(-0.6 0.7) (-0.3 0.4)

(-0.4 0.4) (-0.1 0.2)

(-0.7 0.7) (-0.4 0.5)

(0 0) (0 0)

0.9) (-0.8 0.8) 0.5) (-0.6 0.4) 0.7) (-0.8 0.6)

(-0.7 0.8) (-0.5 0.4) (-0.7 0.6)

(-0.5 0.5) (-0.3 0.1) (-0.5 0.3)

(-0.8 0.8) (-0.8 0.4) (-0.8 0.6)

(-0.1 0.1) (0 0) (-0.1 0.3)

(-0.8 0.8) (-0.7 0.7) (-0.6 0.7)

(-0.5 0.6) (-0.4 0.5) (-0.3 0.4)

(a)

(-0.9 0.9) (-0.8 0.8) (-0.7 0.8)

(-0.7 0.5) (-0.6 0.4) (-0.5 0.4)

(-0.9 0.7) (-0.8 0.6) (-0.7 0.6)

(-0.4 0.4) (-0.1 0.2) (-0.5 0.5) (-0.3 0.1) (-0.5 0.3) (-0.7 0.7) (-0.4 0.5) (-0.8 0.8) (-0.8 0.4) (-0.8 0.6) (0 0) (0 0) (-0.1 0.1) (0 0) (-0.1 0.3) (-0.6 1) (-0.4 0.6) (-0.7 0.9) (-0.5 0.5) (-0.7 0.7)

(-0.4 0.6) (-0.1 1) (-0.5 0.6) (-0.3 0.2) (-0.5 0.4)

(-0.7 0.9) (-0.5 0.6) (-0.8 1) (-0.6 0.6) (-0.8 0.8)

(-0.5 0.5) (-0.3 0.2) (-0.6 0.6) (-0.2 1) (-0.4 0.6)

(-0.7 0.7) (-0.5 0.4) (-0.8 0.8) (-0.4 0.6) (-0.6 1)

                

(b) Respect to the interest of the US (No. 1), the 11 countries can be clustered with a bipolar α-cut as F = {10,11}; // those with more conflicts than common interests C = {1,2,3,4,5,8}; //those with more common interests than conflicts H1 = {7,9}; //those with significant equal common interests and conflicts H2 = {6}; //those with insignificant equal common interests and conflicts

No 1 2 3 4 5 6 7 8 9 10 11

(c)

1 2 3 4 5 6 … 12 ((1 9 1)(1 1))((1 9 1 2)(1 2))((1 9 3)(1 3))((1 9 1 4)(1 4))((1 9 1 5)(1 5))((1 11 6)(1 9 1 11 6))((1 9 7)(1 9 7))((1 9 8)(1 8))((1 9)(1 9)) ((1 5 10)(1 9 1 5 10))((1 11)(1 9 1 11)) ((2 1 9 1)(2 1))((2 1 9 1 2)(2 2))((2 1 9 3)(2 1 3))((2 1 9 1 4)(2 1 4))((2 1 9 1 5)(2 1 5))((2 1 11 6)(2 1 9 1 11 6))((2 1 9 7)(2 1 9 7))((2 1 9 8)(2 1 8)) ((2 1 9)(2 1 9))((2 1 5 10)(2 1 9 1 5 10))((2 1 11)(2 1 9 1 11)) ((3 9 1)(3 1))((3 9 1 2)(3 1 2))((3 1 9 3)(3 3))((3 9 1 4)(3 1 4))((3 9 1 5)(3 1 5))((3 1 11 6)(3 9 1 11 6))((3 1 9 7)(3 9 7))((3 1 9 8)(3 1 8)) ((3 1 9)(3 9))((3 1 5 10)(3 9 1 5 10))((3 1 11)(3 9 1 11)) ((4 1 9 1)(4 1))((4 1 9 1 2)(4 1 2))((4 1 9 3)(4 1 3))((4 1 9 1 4)(4 4))((4 1 9 1 5)(4 1 5))((4 1 11 6)(4 1 9 1 11 6))((4 1 9 7)(4 1 9 7)) ((4 1 9 8)(4 1 8))((4 1 9)(4 1 9))((4 1 5 10)(4 1 9 1 5 10))((4 1 11)(4 1 9 1 11)) ((5 1 9 1)(5 1))((5 1 9 1 2)(5 1 2))((5 1 9 3)(5 1 3))((5 1 9 1 4)(5 1 4))((5 1 9 1 5)(5 5))((5 1 11 6)(5 1 9 1 11 6))((5 1 9 7)(5 1 9 7))((5 1 9 8)(5 1 8)) ((5 1 9)(5 1 9))((5 10)(5 1 9 1 5 10))((5 1 11)(5 1 9 1 11)) ((6 11 1)(6 11 1 9 1))((6 11 1 2)(6 11 1 9 1 2))((6 11 1 3)(6 11 1 9 3))((6 11 1 4)(6 11 1 9 1 4))((6 11 1 5)(6 11 1 9 1 5))((6 11 1 9 1 11 6)(6 6)) ((6 11 1 9 7)(6 11 1 9 7))((6 11 1 8)(6 11 1 9 8))((6 11 1 9)(6 11 1 9))((6 11 1 9 1 5 10)(6 11 1 5 10))((6 11 1 9 1 11)(6 11)) ((7 9 1)(7 9 1))((7 9 1 2)(7 9 1 2))((7 9 1 3)(7 9 3))((7 9 1 4)(7 9 1 4))((7 9 1 5)(7 9 1 5))((7 9 1 11 6)(7 9 1 11 6))((7 9 1 9 7)(7 7))((7 9 1 8)(7 8)) ((7 9 1 9)(7 9))((7 9 1 5 10)(7 9 1 5 10))((7 9 1 11)(7 9 1 11)) ((8 9 1)(8 1))((8 9 1 2)(8 1 2))((8 1 9 3)(8 1 3))((8 9 1 4)(8 1 4))((8 9 1 5)(8 1 5))((8 1 11 6)(8 9 1 11 6))((8 1 9 7)(8 7))((8 1 9 8)(8 8))((8 1 9)(8 9)) ((8 1 5 10)(8 9 1 5 10))((8 1 11)(8 9 1 11)) ((9 1)(9 1))((9 1 2)(9 1 2))((9 1 3)(9 3))((9 1 4)(9 1 4))((9 1 5)(9 1 5))((9 1 11 6)(9 1 11 6))((9 1 9 7)(9 7))((9 1 8)(9 8))((9 1 9)(9 9)) ((9 1 5 10)(9 1 5 10))((9 1 11)(9 1 11)) ((10 5 1)(10 5 1 9 1))((10 5 1 2)(10 5 1 9 1 2))((10 5 1 3)(10 5 1 9 3))((10 5 1 4)(10 5 1 9 1 4))((10 5)(10 5 1 9 1 5))((10 5 1 9 1 11 6)(10 5 1 11 6)) ((10 5 1 9 7)(10 5 1 9 7))((10 5 1 8)(10 5 1 9 8))((10 5 1 9)(10 5 1 9))((10 5 1 9 1 5 10)(10 10))((10 5 1 9 1 11)(10 5 1 11)) ((11 1)(11 1 9 1))((11 1 2)(11 1 9 1 2))((11 1 3)(11 1 9 3))((11 1 4)(11 1 9 1 4))((11 1 5)(11 1 9 1 5))((11 1 9 1 11 6)(11 6))((11 1 9 7)(11 1 9 7)) ((11 1 8)(11 1 9 8))((11 1 9)(11 1 9))((11 1 9 1 5 10)(11 1 5 10))((11 1 9 1 11)(11 11))

(d) Fig. 5. (a) A bipolar fuzzy CM during the post cold war era (adapted from [18]). (b) ⊕-∆ bipolar transitive closure of Fig. 3(a) (An 11 × 11 P -type equilibrium relation) [18]. (c)Bipolar fuzzy clusters [18]. (d) Bipolar transitive paths [18].

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conflicts between the two coalitions. The higher the transitivity among a fuzzy harmony set the higher the harmony level.

Since we have R1 ∧ R2 >> R1 × R2 >> R1∆R2, among all ⊕ − ⊗ transitive equilibrium relations, ⊕ − ∧ transitivity leads to stronger coalition, stronger conflict, and higher level of harmony, respectively, among a fuzzy coalition set, a fuzzy conflict set, and a fuzzy harmony sets. Based on equilibrium energy and stability, the following commonsense coordination laws can be derived mathematically [18] for multiagent coordination. 1) Law1 Always try to reduce negative energy and increase positive energy among a cooperative agent group for stronger coalition. 2) Law2 Reduce negative energy with competitive agents for higher stability of a conflict group. 3) Law3 Increase negative energy with competitive agents for more competition in a conflict group. 4) Law4 Increase positive energy with non-cooperative and non-competitive agents for more cooperation. 5) Law5 Decrease negative energy and increase positive energy with partially cooperative and partially competitive agents for more cooperation. 6) Law6 Decrease negative energy and increase positive energy with cooperative and competitive agent groups for more cooperation and less competition. 7) Law7 Cooperative agents sometimes need to sacrifice for collective goals. 8) Law8 Competitive agent groups sometimes need to compromise for higher stability. 9) Law9 Cooperation is needed for competition; coordination is necessary for both cooperation and competition. VIII. C ONCLUSIONS A theory of bipolar cognition has been presented based on bipolar logic, bipolar fuzzy logic, bipolar predicate logic, bipolar modal logic, and equilibrium relations. Different models are compared. Some basic laws related to equilibrium relations have been formally proved. It has been argued that equilibrium is unavoidability, necessity, natural reality and a form of bipolar truth that is different from unipolar truth. Bipolar truth leads to a holistic world model of equilibrium for life, existence, energy, and stability. Bipolar truth finds no equivalence from unipolar truth. A bipolar counterpart of S5 modality remains a challenge. This challenge is mainly due to the fact that a crisp equilibrium relation is a non-linear fusion of many equivalence relations while S5 modality relies on a single equivalence relation. The theory of bipolar cognition is a first step toward an expandable world model of bipolar equilibrium. While preliminary applications of the theory have been illustrated with cognitive mapping, bipolar cognition has potential impacts on many fields including artificial intelligence, computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social science.

R EFERENCES [1] Atanassov, K., “Intuitionistic Fuzzy Sets.” Fuzzy Sets and systems. 20, 1986, 87-96. [2] Axelrod, R., Structure of Decision. Princeton University Press, Princeton, New Jersey, 1976. [3] Belnap, N. D., “A useful 4-valued logic.” in Modern Uses of MultipleValued Logic, G. Epstein and J. M. Dunn (eds). Reidel, 8-37, 1977. [4] Boole, G., An Investigation of the Laws of Thoughts. MacMillan, London, 1854. Reprinted by Dover Books, New York. [5] De Kleer, J., and Brown, J., “Qualitative physics based on confluence.” Artificial Intelligence, 24, 1984, 7-83. [6] Ginsberg, M. L., “Bilattices and Modal Operators.” J. of Logic and Computation, Oxford University Press, 1(1):41-69, 1990. [7] Kleene, S. C., Introduction to Metamathematics. New York: Van Nostrand, 1952. [8] Zadeh, L. A., “Fuzzy sets.” Information and Control, 8, 338-353, 1965. [9] Zhang, W., Chen, S., Chen, K., Zhang, M., and Bezdek, J. C., “On NPN Logic.” Proc. 18th IEEE Int’l. Sym. on MVL, Palma de Mallorca, Spain, May 24-26, 1988, 381-388. [10] Zhang, W., Chen, S., and Bezdek, J. C., “POOL2: A Generic System for Cognitive Map Development and Decision Analysis.” IEEE Trans. on SMC., Vol. 19, No. 1, Jan./Feb. 1989, 31-39. [11] Zhang, W., Chen, S., Wang, W., and King, R., “A Cognitive Map Based Approach to the Coordination of Distributed Cooperative Agents”. IEEE Trans. on SMC, Vol. 22, No. 1, 1992, p103-114. [12] Zhang, W., “NPN Fuzzy Sets and NPN Qualitative-Algebra: A Computational Framework for Bipolar Cognitive Modeling and Multiagent Decision Analysis.” IEEE Trans.on SMC Vol. Vol. 26, No. 8, 1996, p561-575. [13] Zhang, W., “Yin-Yang Equilibrium Relations: A Mathematical Model for Bipolar Partitioning and Coordination.” Proc. of Int’l Conf. on Artificial Intelligence. June, Las Vegas, Nevada, 2000. P1427-1434. [14] Zhang, W., “Bipolar Logic and Bipolar Fuzzy Logic.” Proc. of 2002 Int’l Conf. of NAFIPS, June, New Orleans, Louisiana. [15] Zhang, W., “Equilibrium Relations and Bipolar Cognitive Mapping for Online Analytical Processing with Applications in International Relations and Strategic Decision Support.” IEEE Trans. on SMC – Par B, Vol. 33. No. 2, April, 2003. p295-307 [16] Zhang, W. and Zhang, L., “Soundness and Completeness of a 4-Valued Bipolar Logic.” Multiple-Valued Logic and Soft Computing. Vol. 9, 2003, pp241-256. [17] Zhang, W., “Bipolar Logic and Bipolar Fuzzy Logic.” Information Sciences, in press. [18] Zhang, W., “Equilibrium Energy and Stability Measures for Bipolar Fuzzy Control and Global Regulation.” International Journal on Fuzzy Systems. Vol. 5, No. 2, June 2003, pp114-122. [19] Zhang, W., “Bipolar Fuzzy Sets and Fuzzy Equilibrium Relations for Bipolar Information Fusion and Visualization.” To appear.


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TABLE V F OUR FUZZY EQUILIBRIUM CLASSES E(US) [18].

Fuzzy Equilibrium Class E(US) from Table III if R(US,China) = (-0.9,0.9) Fuzzy Equilibrium Class E(US) if R(US,China)=(0,0) Fuzzy Equilibrium Class E(US) if R(US,China)=(0,1) Fuzzy Equilibrium Class E(US) if R(US,China)=(-1,0)

{US(-0.8 1), NATO-Allies(-0.7 0.9), Japan(-0.7 0.8), Taiwan(-0.4 0.6), South-Korea(-0.7 Vietnam(-0.2 0.2), Russia(-0.8 0.8), East-European-Countries(-0.5 0.6), China(-0.9 0.9), North-Korea(-0.7 0.5), Cuba(-0.9 0.7)} {US(-0.2 1), NATO-Allies(-0.2 0.9), Japan(-0.4 0.8), Taiwan(-0.2 0.6), South-Korea(-0.1 Vietnam(-0.2 0.2), Russia(-0.5 0.6), East-European-Countries(-0.1 0.6), China(-0.4 0.7), North-Korea(-0.7 0.1), Cuba(-0.9 0.1)} {US(-0.4 1), NATO-Allies(-0.3 0.9), Japan(-0.6 0.8), Taiwan(-0.5 0.6), South-Korea(-0.3 Vietnam(0.2,0.2), Russia(-0.5 0.9), East-European-Countries(-0.2 0.6), China(-0.4 1), North-Korea(-0.7 0.3), Cuba(-0.9 0.3)} {US(-0.7 1), NATO-Allies(-0.8 0.9), Japan(-0.8 0.8), Taiwan(-0.4 0.6), South-Korea(-0.6 Vietnam(-0.2 0.2), Russia(-0.9 0.6), East-European-Countries(-0.6 0.6), China(-1 0.7), North-Korea(-0.7 0.4), Cuba(-0.9 0.6)}

0.9), 0.9), 0.9), 0.9),

TABLE VI E QUILIBRIUM ENERGY ANALYSIS FOR THE FOUR W HAT-I F OPTIONS [18].

(ε− , ε+ )(E(U S)) if if if if

(US,China) = (-0.9,0.9) R(US,China) = (0,0) R(US,China) = (0,1) R(US,China) = (-1,0)

(-7.3, (-3.9, (-5.0, (-7.6,

7.9) 6.5) 7.3) 7.3)

|ε|(E(U S)) = |ε− | + ε+ 15.2 10.4 12.3 14.9

εimp (E(U S))

Stability

1.4 4.5 4.5 2.1

(15.2-1.4)/15.2 (15.2-4.5)/15.2 (15.2-4.5)/15.2 (15.2-2.1)/15.2

= = = =

0.908 0.704 0.704 0.862