Fuzzy Topological Systems

F uzzy T op ological Systems ∗ Ap ostolos Syrop oulos Greek Molecular Computing Group Xan thi, Greece asyropoulos@gmail.com V aleria de P aiv a Sc ho ol of Computer Science Univ ersit y of Birmingham, UK valeria.depaiva@gmail.com Abstract Dialectica categories are a v ery versatile categorical model of linear logic. These ha ve been used to mo del many seemingly differen t things (e.g., Petri nets and Lambek’s calculus). In this note, we expand our previous work on fuzzy p etri nets to deal with fuzzy top ological systems. One basic idea is to use as the dualizing ob ject in the Dialectica categories construction, the unit real interv al I = [0 , 1], which has all the prop erties of a line ale . The second basic idea is to generalize Vick ers’s notion of a topological system. 1 In tro duction F uzzy set theory and fuzzy logic hav e b een inv en ted by Lotfi ali Asker Zadeh. This is a theory that started from a generalization of the set concept and the notion of a truth v alue (for an ov erview, for example, see [8]). In fuzzy set theory , an elemen t of a fuzzy subset belongs to it to a degree, whic h is usually a num b er b etw een 0 and 1. F or example, if w e hav e a fuzzy subset of white colors, then all the gray-scale colors are white to a certain degree and, th us, b elong to t his set with a degree. The following definition by Zadeh himself explains what fuzzy logic is: 1 Definition F uzzy logic is a precise system of reasoning, deduction and computation in which the ob jects of discourse and analysis are asso ciated with information whic h is, or is allo w ed to b e, imprecise, uncertain, incomplete, unreliable, partially true or partially p ossible. Categories, which were inv ented by Samuel Eilenberg and Saunders Mac Lane, form a very high-lev el abstract mathematical theory that unifies all branches of mathematics. Category theory pla ys a central role in mo dern mathematics and theoretical computer science, and, in addition, it is used in mathematical ph ysics, in softw are engineering, etc. Categories ha v e b een used to mo del and study logical systems. In particular, the Dialectica categories of de Paiv a [4] are categorical mo del of linear logic [7]. These categories ha v e b een used to mo del P etri nets [2], the Lambek Calculus [12], state in programming [3], and to define fuzzy p etri nets [6]. Using some of the ideas in our previous w ork on f uzzy petri nets, w e w an ted to dev elop the idea of fuzzy topological systems, that is, the fuzzy counterpart of Vick ers’s [13] top ological systems. In this note, we presen t fuzzy top ological systems and discuss some of their prop erties. ∗ This pap er was accepted for presen tation and it was read at the 8th Panhel lenic L o gic Symp osium , July 4–8, 2011, Ioannina, Greece. 1 The definition was posted to the bisc-group mailing list on 22/11/2008. 1 2 The category Dial I ( Set ) The Dialectica categories construction (see for example [5]) can b e instantiated using any lineale and the basic category Set . As discussed in [11], the unit interv al, since it is a Heyting algebra, has all the prop erties of a lineale structure. Recall that a lineale is a structure defined as follo ws: Definition The quintuple ( L, ≤ , ◦ , 1 , ( ) is a lineale if: • ( L, ≤ ) is p oset, • ◦ : L × L → L is an order-preserving m ultiplication, suc h that ( L, ◦ , 1) is a symmetric monoidal structure (i.e., for all a ∈ L , a ◦ 1 = 1 ◦ a = a ). • if for any a, b ∈ L exists a largest x ∈ L suc h that a ◦ x ≤ b , then this element is denoted a ( b and is called the pseudo-complement of a with resp ect to b . No w, one can prov e that the quintuple (I , ≤ , ∧ , 1 , ⇒ ), where I is the unit interv al, a ∧ b = min { a, b } , and a ⇒ b = W { c : c ∧ a ≤ b } ( a ∨ b = max { a, b } ), is a lineale. Let U and X b e nonempty sets. A binary fuzzy relation R in U and X is a fuzzy subset of U × X , or U × X → I. The v alue of R ( u, x ) is interpreted as the de gr e e of membership of the ordered pair ( u, x ) in R . Let us now define a category of fuzzy relations. Definition The category Dial I ( Set ) has as ob jects triples A = ( U, X , α ), where U and X are sets and α is a map U × X → I. Th us, each ob ject is a fuzzy relation. A map from A = ( U, X , α ) to B = ( V , Y , β ) is a pair of Set maps ( f , g ), f : U → V , g : Y → X such that α ( u, g ( y )) ≤ β ( f ( u ) , y ) , or in pictorial form: U × Y id U × g - U × X ≥ V × Y f × id Y ? β - I α ? Assume that ( f , g ) and ( f 0 , g 0 ) are the following arro ws: ( U, X , α ) ( f ,g ) − → ( V , Y , β ) ( f 0 ,g 0 ) − → ( W , Z, γ ) . Then ( f , g ) ◦ ( f 0 , g 0 ) = ( f ◦ f 0 , g 0 ◦ g ) such that α  u,  g 0 ◦ g  ( z )  ≤ γ   f ◦ f 0  ( u ) , z  . T ensor pro ducts and the internal-hom in Dial I ( Set ) are given as in the Girard-v ariant of the Dialectica construction [4]. Giv en ob jects A = ( U, X, α ) and B = ( V , Y , β ), the tensor product A ⊗ B is ( U × V , X V × Y U , α × β ), where the α × β is the relation that, using the lineale structure of I , tak es the minimum of the membership degrees. The linear function-space or in ternal-hom is giv en by A → B = ( V U × Y X , U × X, α → β ), where again the relation α → β is given by the implication in the lineale. With this structure w e obtain: 2 Theorem 2.1 The c ate gory Dial I ( Sets ) is a monoidal close d c ate gory with pr o ducts and c opr o ducts. Pro ducts and copro ducts are given by A × B = ( U × V , X + Y , γ ) and A ⊕ B = ( U + V , X × Y , δ ), where γ : U × V × ( X + Y ) → I is the fuzzy relation that is defined as follo ws γ  ( u, v ) , z  =  α ( u, x ) , if z = ( x, 0) β ( v , y ) , if z = ( y , 1) Similarly for the copro duct A ⊕ B . 3 F uzzy T op ological Systems Let A = ( U, X , α ) b e an ob ject of Dial I ( Set ), where X is a frame, that is, a p oset ( X , ≤ ) where 1. ev ery subset S of X has a join 2. ev ery finite subset S of X has a meet 3. binary meets distribute ov er joins, if Y is a subset of X : x ∧ _ Y = _ n x ∧ y : y ∈ Y o . Giv en such a triple, we can view A as a fuzzy top olo gic al system , that is, the fuzzy counterpart of Vic k ers’s [13] top olo gic al systems . A top olo gic al system in Vic k er’s monograph[13] is a triple ( U, | = , X ), where X is a frame whose elemen ts are called op ens and U is a set whose elements are called p oints . Also, the relation | = is a subset of U × X , and when u | = x , we say that u satisfies x . In addition, the follo wing must hold • if S is a finite subset of X , then u | = ^ S ⇐ ⇒ u | = x for all x ∈ S . • if S is an y subset of X , then u | = _ S ⇐ ⇒ u | = x for some x ∈ S . Giv en t wo top ological systems ( U, X ) and ( V , Y ), a map from ( U, X ) to ( V , Y ) consists of a function f : U → V and a frame homomorphism φ : Y → X , if u | = φ ( y ) ⇔ f ( u ) | = y . T op ological systems and con tin uous maps b et w een them form a category , whic h we write as T opSystems . In order to fuzzify top ological systems, we need to fuzzify the relation “ | =.” Ho w ev er, the requiremen t imposed on the relation of satisfaction is too sev ere when dealing with fuzzy structures. Indeed, in some reasonable categorical mo dels of fuzzy structures (see, for example [1, 11]), the authors use a weak er condition where the e quiv alence op erator is replaced b y an implication op erator. Thus we suggest that the corresp onding condition for morphisms of fuzzy top ological systems should b ecome u | = φ ( y ) ⇒ f ( u ) | = y . Definition A fuzzy top olo gic al system is a triple ( U, α, X ), where U is a set, X is a frame and α : U × X → I a binary fuzzy relation suc h that: 3 (i) If S is a finite subset of X , then α ( u, ^ S ) ≤ α ( u, x ) for all x ∈ S . (ii) If S is an y subset of X , then α ( u, _ S ) ≤ α ( u, x ) for some x ∈ S . (iii) α ( u, > ) = 1 and α ( u, ⊥ ) = 0 for all u ∈ U . T o see that fuzzy top ological systems also form a category we need to show that given morphisms ( f , F ) : ( U, X ) → ( V , Y ) and ( g , G ) : ( V , Y ) → ( W , Z ), the obvious comp osition ( g ◦ f , F ◦ G ) : ( U, X ) → ( W, Z ) is also a morphism of fuzzy top ological systems. But we kno w Dial I ( Set ) is a category and conditions (i), (ii) and (iii) do not apply to morphisms. Identities are given b y ( id U , id X ) : ( U, X ) → ( U, X ). The collection of ob jects of Dial I ( Set ) that are fuzzy topological systems and the arrows b et w een them, form the category FT opSystems , which is a sub category of Dial I ( Set ). Prop osition 3.1 Any top olo gic al system ( U, X ) is a fuzzy top olo gic al system ( U, ι, X ) , wher e ι ( u, x ) =  1 , when u | = x 0 , otherwise Pro of Consider the first prop ert y of the relation “ | =” u | = ^ S ⇐ ⇒ u | = x for all x ∈ S . This will b e translated to ι ( u, ^ S ) ≤ ι ( u, x ) for all x ∈ S . The inequalit y is in fact an equalit y since whenev er u | = x , ι ( u, x ) = 1. Therefore, w e can transform this condition in to the following one ι ( u, ^ S ) = ι ( u, x ) for all x ∈ S . A similar argumen t holds true for the second prop ert y . The following result is based on the previous one: Theorem 3.2 The c ate gory of top olo gic al systems is a ful l sub c ate gory of Dial I ( Set ) . Ob viously , it is not enough to provide generalization of structures—one needs to demonstrate that these new structures ha v e some usefulness. The following example gives an interpretation of these structures in a “real-life” situation. Example Vic k ers [13, p. 53] giv es an in teresting physical interpretation of top ological systems. In particular, he considers the set U to b e a set of programs that generate bit streams and the op ens to b e assertions ab out bit streams. F or exanple, if u is a program that generates the infinite bit stream 010101010101. . . and “ starts 01010” is an assertion that is satified if a bit stream starts with the digits “01010”, then this is expressed as follo ws: x | = starts 01010 . Assume now that x 0 is a program that pro duces bit streams that lo ok lik e the follo wing one 4 0 1 0 1 0 1 0 1 0 The individuals bits are not identical to either “1” or “0,” but rather similar to these. One can sp eculate that these bits are the result of some interaction of x 0 with its environmen t and this is the reason they are not identical. Then, we can say that x 0 satisfies the assertion “ starts 01010” to some degree, since the elemen ts that make up the stream pro duced by x 0 are not identical, but rather similar. 4 F rom F uzzy T op ological Systems to F uzzy T op ological Spaces It is not difficult to map fuzzy top ological systems to fuzzy top ological spaces (for an o v erview of the theory of fuzzy top ologies see [14]). The following definition shows ho w to map an op en to fuzzy set: Definition Assume that a ∈ A , where ( U, α, X ) is a fuzzy top ological space. Then the extent of an op en x is a function whose graph is given b elo w: n  u, α ( u, x )  : u ∈ U o . Prop osition 4.1 The c ol le ction of al l fuzzy sets cr e ate d by the extents of the memb ers of A c orr e- sp ond to a fuzzy top olo gy on X . Pro of Assume that a and b are opens and let a ( x ) = α ( x, a ), b ( x ) = α ( x, b ), and ψ ( x ) = α ( x, a ∧ b ). Then α ( x, a ∧ b ) ≤ α ( x, a ) and α ( x, a ∧ b ) ≤ α ( x, b ). In different w ords, ψ ( x ) ≤ a ( x ) and ψ ( x ) ≤ b ( x ), whic h implies that ψ ( x ) ≤ min { a ( x ) , b ( x ) } that is ψ = a ∩ b . Similarly , assume that { a i } is a collection of op ens such that a i ( x ) = α ( x, a i ) and φ ( x ) = α ( x, W i a i ). The fact that there is one φ ( x ) ≤ a j ( x ), while for all other a i it holds that φ ( x ) ≥ a j ( x ), implies that φ ( x ) = sup i a i ( x ), that is, φ = S i a i ( x ). Finally , the last conditions generate the sets 1 ( x ) = 1 and 0 ( x ) = 0. So, the op ens form a fuzzy top ology on X . 5 Pro ducts and Sums of F uzzy T op ological Systems In section 2 we describ ed the categorical pro ducts and copro ducts of any tw o ob jects of Dial I ( Set ). Giv en tw o fuzzy top ological systems A = ( U, X, α ) and B = ( V , Y , β ), their top ological pro duct is the space A × B = ( U × V , X + Y , γ ). Since X and Y are frames it is necessary to mo dify the definition of X + Y and, consequen tly , the definition of γ . Definition Assume that A = ( U, X , α ) and B = ( V , Y , β ) are t w o fuzzy top ological systems. Then their top ological pro duct A × B is the system ( U × V , γ , X ⊗ Y ), where X ⊗ Y is the tensor pro duct of the t w o frames X and Y (see [13, pp. 80–85] for details) and γ is defined as follows: γ  ( u, v ) , _ i x i ⊗ y i  = max  α ( u, x ) , β ( v , y )  . Ob viously , the top ological pro duct is not the same as the categorical pro duct. The top ological sum is more straigth tforw ard: 5 Definition Assume that A = ( U, X , α ) and B = ( V , Y , β ) are t w o fuzzy top ological systems. Then their top ological sum A + B is the system ( U + V , γ , X × Y ), where γ is defined as follo ws: γ  z , ( x, y )  =  α ( u, x ) , if z = ( u, 0) β ( v , y ) , if z = ( v , 1) Comparing the top ological sum with the categorical sum reveals that they are iden tical. 6 Conclusions W e ha ve simply started thinking ab out the possibilities of using Dialectica-lik e models in the con text of fuzzy top ological structures. Much remains to b e done, in particular we would lik e to see if a framew ork based on an implicational notion of morphism lik e ours can cope with em b edding sev eral of the other notions of fuzzy sets considered by Ro dabaugh [9]. Also seems likely that we could extend the w ork of Solovy ov [10] on v ariable-basis top ological spaces using similar ideas. References [1] Mic hael Barr. F uzzy mo dels of linear logic. Mathematic al Structur es in Computer Scienc e , 6(3):301–312, 1996. [2] Carolyn Bro wn and Doug Gurr. 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