Structural Functionality as a Fundamental Property of Boolean Algebra and Base for Its Real-Valued Realizations

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1 Structural Fuctioality as a Fudametal Property of Boolea Algebra ad Base for Its Real-Valued Realizatios Draga G. Radojević Uiversity of Belgrade, Istitute Mihajlo Pupi, Belgrade draga.radojevic@pupi.rs Abstract. The value of the complex Boolea fuctio ca be calculated directly o the basis of its compoets value. It is a priciple kow as the truth fuctioality. Properties of the Boolea algebra have idifferet values. The truth fuctioal priciple is take as a valid priciple i geeral case i the covetioal geeralizatio: multi-valued ad/or real-valued realizatios (fuzzy logic i the broad sese). This paper presets that truth fuctioality is ot valued idifferet property of the Boolea algebra ad it is valid oly i twovalued realizatio, ad thus it caot be the basic of the value geeralizatio. The value geeralizatio (real-valued realizatios) eables icomparably more descriptiveess tha the two-valued classical Boolea algebra, so that the fiite Boolea algebra is eough for ay real applicatio. Each fiite Boolea algebra is atomic. Every Boolea fuctio (the elemet of the aalyzed fiite Boolea algebra) ca be preseted uiquely as disjuctio of the relevat atoms disjuctive caoical form. Which atoms are ad which are ot icluded i the aalyzed Boolea fuctio is defied by its structure: 0-1 vector which dimesio matches the umber of atoms (i the case of idepedet variables, the umber of atoms is 2 ). Atom correspods uiquely to each vector structure positio ad value 0 meas that the adequate atom is ot icluded i the aalyzed fuctio, ad 1 meas that it is icluded. The priciple of the structural fuctioality is: the structure of the complex Boolea fuctio is defied directly o the basis of its compoets structure. The truth fuctioality is a value image of the structural fuctioality oly i the case of two-valued realizatio. Each isistig o the truth fuctioality, such as i the case of covetioal multi-valued logic ad fuzzy logic i geeral sese, is ujustified from the poit of the Boolea cosistecy. Keywords: Boolea algebra, atomic Boolea fuctios, disjuctive caoical form, Boolea fuctio structure, structural fuctioality, truth fuctioality, geeralized value realizatio of the Boolea fuctios. 1 Itroductio The truth fuctioality is a property of the two-valued realizatio of Boolea algebra. The value of complex Boolea fuctio ca be calculated directly based o its A. Lauret et al. (Eds.): IPMU 2014, Part II, CCIS 443, pp , Spriger Iteratioal Publishig Switzerlad 2014

2 Structural Fuctioality as a Fudametal Property of Boolea Algebra 29 compoets values. The priciple of the truth fuctioality is take as a basis of geeralizatio i regular (covetioal) value geeralizatio of the theories based o the Boolea algebra: multi-valued ad/or real-valued logic (fuzzy logic i geeral sese). Practically, there is o explaatio why the truth fuctioality is take as a basis of the valued geeralizatio except ``because it is usual ad techically useful`` (obviously ot mathematical motivatio!). This paper presets that truth fuctioality is valid oly i two-valued realizatio of the Boolea algebra. Every fiite Boolea algebra is atomic. The Boolea algebra geerated with idepedet variables cotais 2 atoms. Atoms are the simplest elemets from the poit of the valued realizatio (i the classical case for the free set of 0-1 idepedet variables values, oly oe atom has value 1 ad all the others 0). The importat atom property of the aalyzed Boolea algebra is the fact that they do ot have aythig i commo ad/or that cojuctio of two differet atoms is idetical to 0 elemet of the Boolea algebra i.e. it has value realizatio idetical to 0. Each Boolea fuctio (elemet of the aalyzed Boolea algebra) ca uiquely be preseted as a uio of relevat atoms disjuctive caoical form. Which atoms are ad which are ot icluded is defied with 0-1 vector dimesio 2 by the structural fuctio. Atom uiquely correspods to each vector positio ad value 0 meas that the appropriate atom is ot icluded i the aalyzed fuctio, ad 1 meas that it is icluded. The structure of the complex Boolea fuctio is defied directly o the basis of the structure of its compoets the priciple of the structural fuctioality. This priciple is value idifferet algebra ad it must be saved i all Boolea cosistet value realizatios. The truth fuctioality is a value image of the structural fuctioality ad oly i the case of two-valued realizatio kow truth table, for example. Each isistig o the truth fuctioality i geeral case (multi-valued ad/or real-valued realizatio) has as a result impossibility of simultaeous keepig all properties of Boolea algebra. So, the aswer o the questio stated i the paper title is that the structural fuctioality is the basic of the Boolea cosistet geeralizatio of the fiite Boolea algebra valued realizatio i geeral case. The Boolea cosistet geeralizatio eables direct geeralizatio of all theories based o the classical fiite Boolea algebra. From the poit of the Boolea cosistecy, it is completely ujustified to isist o the truth fuctioality such as i the case of the covetioal multi-valued logic ad fuzzy logic i geeral sese. 2 Fuzzy Logic i Boolea Frame Fuzzy logic, realized i the Boolea frame, meas that all Boolea axioms ad theorems are valid i the most geeral case i.e. i the real-valued case. Sice the appropriate classical techiques are based o the Boolea algebra, precisely, o its two-valued realizatio, the cosistet geeralizatio should be based o the real-valued realizatio of the Boolea algebra. The real-valued realizatio of the fiite or atomic Boolea algebra [4, 5 ad 6] is described here i detail. The mai problem of the covetioal approaches (the mai stream of the usual realizatio) is fact that they are based o the priciple of the truth fuctioality, which

3 30 D.G. Radojević is take from the classical logic based o the two-valued realizatio. From the poit of the Boolea algebra, this priciple is adequate or correct oly i two-valued case. The reaso is simple: the Boolea fuctio has the vector ature i geeral case, but i the classical case the attetio is draw oly to oe compoet (which is defied with 0-1 values of the idepedet variables). I geeral case, whe the values of the idepedet variables are ot oly 0 or 1 but also iclude everythig i betwee, it is ecessary to iclude more compoets i calculatio ad i the most geeral case all compoets of the vector immaet to the aalyzed Boolea fuctio i the aalyzed Boolea algebra. I order to illustrate the mai idea, we will use the Boolea fuctio of two idepedet variables x ad y, from the famous Boolea paper from 1848 [2]: ( xy, ) ( 1,1) xy ( 1, 0) x( 1 y) ( 0,1)( 1 x) y ( 0, 0)( 1 x)( 1 y) φ = φ + φ +φ +φ. (1) Actually, this equatio ca be treated also as a special case of the Boolea polyomial [4]: ( xy, ) ( 1,1) x y ( 1,0)( x x y) φ( )( y x y) +φ( )( x y+ x y) φ =φ +φ + 0,1 0, 0 1. The idepedet variables i geeral case take the value from the uit real iter- xy, 0,1. val [ ] is geeralized product [4] or t-orm with the followig property: ( x+ y ) x y ( x y) max 1,0 mi,. (2) 2.1 Boolea Polyomial The real-valued realizatio of the fiite (atomic) Boolea algebra is based o the Boolea polyomials. The free Boolea fuctio ca be uiquely trasformed ito the appropriate Boolea [6]. Example 1: Usig the equatio (2) of the equivalece relatio, exclusive disjuctio ad implicatio, respectively ( ) def ( ) ( ) ( ) ( ) a. φ x,y = x y φ 11, = 1; φ 10, = 0; φ 01, = 0; φ 0, 0 = 1; x y = 1 x y+ 2x y.

4 Structural Fuctioality as a Fudametal Property of Boolea Algebra 31 ( ) def ( ) ( ) ( ) ( ) b. φ x,y = x y φ 11, = 0; φ 10, = 1; φ 01, = 1; φ 0, 0 = 0; x y = x+ y 2x y. ( ) def ( ) ( ) ( ) ( ) c. φ x,y = x y φ 11, = 1; φ 10, = 0; φ 01, = 1; φ 0, 0 = 1; x y = 1 x+ x y. The fiite (atomic) Boolea algebra geerated by the set of idepedet va- Ω= x,...,, 1 x is ΒΑ( Ω ) =Ρ( Ρ( Ω )), where: Ρ( Ω ) is a set of all sub- riables { } sets Ω. The atomic elemets of the aalyzed Boolea algebra ΒΑ( Ω) are [6]: ( S)( x x ) x x S ( ) α,..., =, Ρ Ω. 1 (3) i j xi S xj Ω\ S The atomic Boolea polyomial α ( )(,..., 1 ) atomic elemet α ( )( ) S x x uiquely correspods to the S x,...,, 1 x ad it is defied by the followig equatio [6]: C ( S)( x1,..., x) ( 1 ) xi; S ( ). (4) α = Ρ Ω C Ρ( Ω\ S) xi C S Example 2: The atomic Boolea polyomials for the Boolea algebra geerated with Ω= { x,y} are: ({ }) ({}) ({ }) ({}) 1 α x,y = x y; α x = x x y; α y = y x y; α = x y+ x y. The values of the atomic polyomials i the real-valued case are ega- α S x 1,...,x 01, S Ρ( Ω ), ad their sum is idetically equal to 1. I x,y 01, : tive ( )( ) [ ] the case of the example described by the equatio (2) for [ ]

5 32 D.G. Radojević ( ) ( ) ( ) x y+ x x y + y x y + 1 x y+ x y 1. The classical two-valued case is just a special case which satisfies this fudametal equity, sice the value of oly oe atom is equal to 1 ad all others are idetically equal to 0. The free Boolea fuctio, the appropriate elemet of the aalyzed Boolea algebra φ( x,..., 1 x ) ( ), ΒΑ Ω ca uiquely be preseted i disjuctive caoical form as a disjuctio of relative atomic elemets: ( ) ( ) ( )( ) 1,..., φ 1,...,. (5) S Ρ( Ω) φ x x = S α S x x Where: φ ( ), ( Ρ( Ω) ) atom α ( S)( x x ) i the aalyzed Boolea fuctio ( ),..., 1 the followig way: S S the relatio of iclusio of correspodig φ x,...,, 1 x is defied i ( S) def ( S ( xi ) i ) = φ χ = 1,...,, φ 1, xi S χ S ( xi) = def, ( S Ρ( Ω) ). 0, xi S (6) The relatio of iclusio determies which atoms ( Ρ( Ω) ) aalyzed Boolea fuctio: S are icluded i the ( S)( x1 x) ( x1 x) ( S)( x x ) φ( x x ) 1, α,..., φ,..., φ ( S ) = 0, α 1,..., 1,...,, (6.1) (The values of the correspodig relatio of iclusio are equal to 1) ad which are ot icluded (the values of the relatio of iclusio are equal to 0). The Boolea polyomial uiquely correspods to the aalyzed Boolea fuctio as Figure (5): φ ( x1,..., x) = φ ( S) α ( S)( x1,..., x). (7) S Ρ( Ω) The Boolea polyomial (7) ca be preseted as a scalar product for two vectors: ( x,..., x ) ( x,..., x ), x,..., x [ 0,1] φ 1 =φα 1 1 (8)

6 Structural Fuctioality as a Fudametal Property of Boolea Algebra 33 φ = φ( S) S Ρ( Ω) is a structure of the aalyzed Boolea fuctio φ ( x,..., 1 x ) ΒΑ ( Ω ), i.e. a vector of the relatio of iclusio of atomic fuctios i the aalyzed fuctio. T α ( x1,..., x) = α( S)( x1,..., x) S Ρ( Ω) is a vector of atomic polyomials of the aalyzed fiite (atomic) Boolea algebra ΒΑ( Ω ). Example 2: Structures of the aalyzed Boolea fuctios from the example 1: x y = , x y = , = x y [ ] [ ] [ ]. ad atomic polyomials vectors for two idepedet variables: ( x, x ) x1 x2 x x x. 1 x1 x2 + x1x α 1 2 = x2 x1x2 3 The Structural Fuctioality The priciple of the structural fuctioality: The structure of the free complex Boolea fuctio ca be calculated directly o the basis of its compoets usig the followig idetities: φ ψ =φ ψ φ ψ =φ ψ = = 1. φ φ φ The kow priciple of the truth fuctioality is just a image of the structural fuctioality at the value level ad thus oly i the case of two-valued realizatio. I geeral case (multi-valued ad/or real-valued realizatios), the priciple of the truth fuctioality is ot able to keep all Boolea algebra properties. That is the reaso why fuzzy approaches, based o the priciple of the truth fuctioality (covetioal fuzzy logic i wider sese), caot be i the Boolea frame ad/or they are ot the Boolea cosistet geeralizatios. Structures as algebra properties of the Boolea fuctios keep all Boolea laws [6]:

7 34 D.G. Radojević 3.1 Laws of Mootoicity Associativity Commutatively φ ψ ξ = φ ψ ξ, (9) =. ( ) ( ) ( ) ( ) φ ψ ξ φ ψ ξ φ ψ =ψ φ, φ ψ =ψ φ; (10) =, =. φ ψ ψ φ φ ψ ψ φ Distributive Idetity φ ψ ξ = φ ψ φ ξ, (11) =. ( ) ( ) ( ) ( ) ( ) ( ) φ ψ ξ φ ψ φ ξ φ 0 =φ, φ 1= 1; (12) 0= 0, 1 =. φ φ φ Idempotet =, =. φ φ φ φ φ φ (13) Absorptio =, =. ( ) ( ) φ φ ξ φ φ φ ξ φ (14) 3.2 No Mootoicity Laws Complemetarity = 0, = 1. φ φ φ φ (15) De Morga Laws =, =. ( ) ( ) φ ψ φ ψ φ ψ φ ψ (16)

8 Structural Fuctioality as a Fudametal Property of Boolea Algebra 35 4 Coclusio This approach based o the applicatio of the structural fuctioality keeps all Boolea algebra laws i all possible value realizatios (from the classical two-valued to the most geeral real-valued realizatio) idepedetly from the selected operators of the geeralized product. If you iterpret the geeral case cocretely (real-valued realizatio), the eve the special case (two-valued realizatio) is treated i a differet way. Excluded middle ad o-cotradictio are defied for two-valued realizatio case (i logic, for example, the statemet is either truth or utruth ad it caot be both truth ad utruth) ad thus they are ot adequate for geeral case. It seems that i the case of the covetioal approaches to the geeralizatio, the excluded middle may eve ot be valid. I geeral case oe statemet ca be partially truth but the it is utruth with complemetary itesity. From the poit of the structure, the complemetary fuctio correspods to each Boolea fuctio so that it cotais all atoms which the aalyzed fuctio does ot cotai ad whereat they do ot have a sigle commo atom. The cosequece is that the disjuctio of the free Boolea fuctio ad its complemetary fuctios cotai all atoms ad /or it is idetical to the Boolea costat 1, i.e. it has a value idetically equal to 1 the excluded middle. Also, its cojuctio does ot iclude a sigle atom ad/or it is idetical to the Boolea costat 0, i.e. it has a value idetically equal to 0 o-cotradictio. However, these fudametal laws are valid i all cosistet value realizatios ad kow defiitios are valid oly i classical two-valued case. Actually, excluded middle ad o-cotradictio uiquely defie the complemetary property of the aalyzed property ad thus these two laws are fudametal ad importat for cogitio i geeral. This ca be illustrated i a simple example, such as a glass filled with water. The classical two-valued case treats oly full or empty glass. Empty is complemet of full ad vice versa. I geeral or real case, the glass ca be partially full ad, at the same time, with the rest it is empty with complemetary itesity, so that the sum of the itesity full ad itesity empty is idetically equal to 1. It is obvious that, besides the fact that properties full ad empty do ot have aythig i commo, they are both simultaeously i the same glass ad thus the uio itesity is equal to 1 ad itersectio itesity to 0. The free classical theory, based o the fiite Boolea algebra usig the real-valued realizatio of the Boolea algebra, ca be geeralized directly [6]. This is very importat for may iterestig examples which are logically complex such as: artificial itelligece, mathematical cogitio, prototype theory i psychology, cocept theory, etc. The structural fuctioality sheds a ew light o the relatioship betwee sytax ad sematics i the classical logic which will be preseted i the ext paper.

9 36 D.G. Radojević Refereces 1. Zadeh, L.A.: From Circuit Theory to System Theory. I: Proc. of Istitute of Ratio Egieerig, vol. 50, pp (1962) 2. Boole, G.: The Calculus of Logic. The Cambridge ad Dubli Mathematical Joural 3, (1848) 3. Radojevic, D.: New [0,1]-valued logic: A atural geeralizatio of Boolea logic. Yugoslav Joural of Operatioal Research - YUJOR 10(2), (2000) 4. Radojevic, D.: There is Eough Room for Zadeh s Ideas, Besides Aristotle s i a Boolea Frame. I: 2d Iteratioal Workshop o Soft Computig Applicatios, SOFA 2007 (2007) 5. Radojevic, D.: Iterpolative Realizatio of Boolea algebra as a Cosistet Frame for Gradatio ad/or Fuzziess. I: Nikravesh, M., Kacprzyk, J., Zadeh, L.A. (eds.) Forgig New Frotiers: Fuzzy Pioeers II. STUDFUZZ, vol. 218, pp Spriger, Heidelberg (2008) 6. Radojevic, D.: Real-valued realizatio of Boolea algebra is atural frame for cosistet fuzzy logic. I: Seisig, R., Trillas, E., Moraga, C., Termii, S. (eds.) O Fuzziess, A Homage to Lotfi Zadeh. STUDFUZZ, vol. 2, pp Spriger, Heidelberg (2013)