Incompletely Specified Operations and their Clones Jelena Coli c - PowerPoint PPT Presentation

Incompletely Specified Operations and their Clones Jelena ˇ Coli´ c Oravec University of Novi Sad Progress in Decision Procedures: From Formalizations to Applications Belgrade, March 30, 2013 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 1 / 23

Previously on this subject... Jelena ˇ Coli´ c, Hajime Machida and Jovanka Pantovi´ c. Clones of Incompletely Specified Operations. ISMVL 2012 , pages 256–261. Jelena ˇ Coli´ c, Hajime Machida and Jovanka Pantovi´ c. One-point Extension of the Algebra of Incompletely Specified Operations. Multiple-Valued Logic and Soft Computing , to be published in 2013. Jelena ˇ Coli´ c. On the Lattice of Clones of Incompletely Specified Operations. Conference on Universal Algebra and Lattice Theory, Szeged 2012 . Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 2 / 23

What is an IS operation? 1 IS clones 2 Compositions Definitions of IS clone IS operations vs. hyperoperations IS operations via a one-point extension 3 Extended IS operations Algebra of extended IS operations Future work 4 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 3 / 23

What is an IS operation? What is an IS operation? Total operation: 0 1 OR 0 0 1 1 1 1 Let h ( x 1 , x 2 ) = OR ( g ( x 1 ) , x 2 ) Partial operation: OR ( g ( x 1 ) , 1 ) undefined if g ( x 1 ) is undefined Incompletely specified operation: OR ( g ( x 1 ) , 1 ) = 1 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 4 / 23

What is an IS operation? How to define it formaly? Let A be a finite set and k �∈ A . Partial operation: f : A n → A ∪ { k } , k − undefined Incompletely specified operation: f : A n → A ∪ { k } , k − unspecified I A - set of all IS operations on A Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 5 / 23

IS clones Compositions Composition of total and hyperoperations The composition of f ∈ O ( n ) and g 1 , . . . , g n ∈ O ( m ) is an m -ary A A operation defined by f ( g 1 , . . . , g n )( x 1 , . . . , x m ) = f ( g 1 ( x 1 , . . . , x m ) , . . . , g n ( x 1 , . . . , x m )) . The composition of f ∈ H ( n ) and g 1 , . . . , g n ∈ H ( m ) is an m -ary A A hyperoperation defined by � f ( g 1 , . . . , g n )( x 1 , . . . , x m ) = f ( y 1 , . . . , y n ) ( y 1 , . . . , y n ) ∈ A n y i ∈ g i ( x 1 , . . . , x m ) 1 ≤ i ≤ n Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 6 / 23

IS clones Compositions New composition Definition Let f ∈ I ( n ) and g 1 , . . . , g n ∈ I ( m ) . The i-composition of f and g 1 , . . . , g n A A is an m-ary IS operation defined by � f [ g 1 , . . . , g n ]( x 1 , . . . , x m ) = f ( y 1 , . . . , y n ) ( y 1 , . . . , y n ) ∈ A n y i ⊑ g i ( x 1 , . . . , x m ) 1 ≤ i ≤ n where � x 1 , if x 1 = x 2 = . . . = x l , � { x i : 1 ≤ i ≤ l } = k , otherwise. ⊑ = { ( x , x ) : x ∈ A ∪ { k }} ∪ { ( x , k ) : x ∈ A } Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 7 / 23

IS clones Compositions Example A = { 0 , 1 } composition of partial operations 0 1 g 1 g 2 OR ( g 1 , g 2 ) OR 0 0 1 0 1 2 ⇒ 0 2 1 1 1 1 0 0 1 0 OR ( g 1 , g 2 )( 0 ) = OR ( g 1 ( 0 ) , g 2 ( 0 )) = 2 i-composition of IS operations 0 1 g 1 g 2 OR [ g 1 , g 2 ] OR ⇒ 0 0 1 0 1 2 0 1 1 1 1 1 0 0 1 0 OR [ g 1 , g 2 ]( 0 ) = OR ( g 1 ( 0 ) , g 2 ( 0 )) = OR ( 1 , 0 ) ⊓ OR ( 1 , 1 ) = 1 ⊓ 1 = 1 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 8 / 23

IS clones Definitions of IS clone IS clone e n , A ( x 1 , . . . , x i , . . . , x n ) = x i is an i -th n -ary projection. i Definition A set C ⊆ I A is called a clone of incompletely specified operations (or IS clone) if C contains all projections and C is closed with respect to i-composition. Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 9 / 23

IS clones Definitions of IS clone IS clone (second definiton) for f ∈ I ( 1 ) let ζ f = τ f = ∆ f = f ; A for f ∈ I ( n ) A , n ≥ 2 , let ζ f , τ f ∈ I ( n ) and ∆ f ∈ I ( n − 1 ) be defined as A A ( ζ f )( x 1 , x 2 , . . . , x n ) = f ( x 2 , . . . , x n , x 1 ) ( τ f )( x 1 , x 2 , x 3 , . . . , x n ) = f ( x 2 , x 1 , x 3 , . . . , x n ) (∆ f )( x 1 , x 2 , . . . , x n − 1 ) = f ( x 1 , x 1 , x 2 . . . , x n − 1 ) for f ∈ I ( n ) and g ∈ I ( m ) let f ⋄ g ∈ I ( m + n − 1 ) be defined as A A A � ( f ⋄ g )( x 1 , . . . , x m + n − 1 ) = f ( y , x m + 1 , . . . , x m + n − 1 ) y ∈ A y ⊑ g ( x 1 , . . . , x m ) Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 10 / 23

IS clones Definitions of IS clone Example A = { 0 , 1 } 0 1 g OR 0 0 1 0 0 1 1 1 1 2 Let h ( x 1 , x 2 ) = OR ( g ( x 1 ) , x 2 ) . For partial operations: h 0 1 0 0 1 1 2 2 h ( 1 , 1 ) = OR ( g ( 1 ) , 1 ) = 2 For IS operations: h 0 1 0 0 1 1 2 1 h ( 1 , 1 ) = OR ( g ( 1 ) , 1 ) = OR ( 0 , 1 ) ⊓ OR ( 1 , 1 ) = 1 ⊓ 1 = 1 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 11 / 23

IS clones Definitions of IS clone IS clone (second definiton) I A = ( I A ; ⋄ , ζ, τ, ∆ , e 2 , A 1 ) full algebra of IS operations Theorem C ⊆ I A is an IS clone if and only if C is a subuniverse of the full algebra of IS operations. Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 12 / 23

IS clones IS operations vs. hyperoperations IS operations vs. hyperoperations λ : H A → I A , f �→ f is � f ( x 1 , . . . , x n ) , | f ( x 1 , . . . , x n ) | = 1 f is ( x 1 , . . . , x n ) = k , otherwise Theorem (i) For | A | = 2 , λ is an isomorphism from ( H A ; ◦ , ζ, τ, ∆ , e 2 , A 1 ) to ( I A ; ⋄ , ζ, τ, ∆ , e 2 , A 1 ) . (ii) For | A | ≥ 3 , λ is a homomorphism from ( H A ; ζ, τ, ∆ , e 2 , A 1 ) to ( I A ; ζ, τ, ∆ , e 2 , A 1 ) . (iii) For | A | ≥ 3 , there exist f , g ∈ H A satisfying λ ( f ◦ g ) � = λ ( f ) ⋄ λ ( g ) . Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 13 / 23

IS clones IS operations vs. hyperoperations Example A = { 0 , 1 , 2 } (ii) λ is not injective f is = g is f g 0 { 0 } 0 { 0 } 0 0 ⇒ 1 { 1 } 1 { 1 } 1 1 2 { 0 , 1 } 2 { 0 , 2 } 2 3 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 14 / 23

IS clones IS operations vs. hyperoperations Example A = { 0 , 1 , 2 } (iii) ( f ◦ g ) is � = f is ⋄ g is ( f ◦ g ) is f 0 1 2 g 0 1 2 0 { 1 } 0 { 0 } 0 1 ⇒ { 0 , 1 } { 0 } 1 1 1 1 { 1 } { 0 , 2 } 2 2 2 1 f ◦ g ( 2 , 1 ) = f ( 0 , 1 ) ∪ f ( 2 , 1 ) = { 1 } f is ⋄ g is f is g is 0 1 2 0 1 2 0 1 0 0 0 1 ⇒ 1 3 1 0 1 1 2 1 2 3 2 3 f is ⋄ g is ( 2 , 1 ) = f is ( 0 , 1 ) ⊓ f is ( 1 , 1 ) ⊓ f is ( 2 , 1 ) = 3 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 15 / 23

IS operations via a one-point extension Extended IS operations One-point extension Let us define the mapping I A → O A ∪{ k } : f �→ f + , as follows: � f + ( x 1 , . . . , x n ) = f ( y 1 , . . . , y n ) ( y 1 , . . . , y n ) ∈ A n , ( y 1 , . . . , y n ) ⊑ ( x 1 , . . . , x m ) F + = { f + : f ∈ F } ⊆ O A ∪{ k } for F ⊆ I A . Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 16 / 23

IS operations via a one-point extension Extended IS operations Example (one-point extension) A = { 0 , 1 } Partial operation: OR + 0 1 2 0 0 1 2 1 1 1 2 2 2 2 2 OR + ( 2 , 1 ) = 2 Incompletely specified operation: OR + 0 1 2 0 0 1 2 1 1 1 1 2 2 1 2 OR + ( 2 , 1 ) = OR ( 0 , 1 ) ⊓ OR ( 1 , 1 ) = 1 ⊓ 1 = 1 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 17 / 23

IS operations via a one-point extension Algebra of extended IS operations Algebra of extended IS operations Mapping I A → I + A : f �→ f + , is not an isomorphism from A ; ◦ , ζ, τ, ∆ , e 2 , A ∪{ k } ( I A ; ⋄ , ζ, τ, ∆ , e 2 , A 1 ) to ( I + ) . 1 A is closed w.r.t. ζ, τ and e 2 , A ∪{ k } I + : 1 � + = e 2 , A ∪{ k } e 2 , A � 1 1 ( ζ f ) + = ζ ( f + ) ( τ f ) + = τ ( f + ) I + A is not closed w.r.t. ∆ and ◦ : (∆ f ) + � = ∆( f + ) ( f ⋄ g ) + � = f + ◦ g + Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 18 / 23

IS operations via a one-point extension Algebra of extended IS operations Example (∆ f ) + � = ∆( f + ) (∆ f ) + ∆ f f 0 1 0 0 ⇒ ⇒ 0 0 0 0 0 1 0 1 2 0 1 0 2 0 f + ∆( f + ) 0 1 2 0 0 0 0 0 0 ⇒ 1 2 0 2 1 0 2 2 0 2 2 2 Jelena ˇ Coli´ c Oravec (University of Novi Sad) IS Operations and their Clones Belgrade 2013 19 / 23