Maitra Cascade Minimization Voudouris Dimitrios Dr. - PowerPoint PPT Presentation

Maitra Cascade Minimization � Voudouris Dimitrios � Dr. Papakonstantinou George National Technical University of Athens

Basic Definitions(1) � Complex Maitra Term [1] � Constant 0(1) Boolean Function � Literal � = M G(a, M ) + i 1 i � M i � Maitra Term � a � literal � G � Arbitrary 2 variable switching function (~Maitra Cell) � Reversible Wave Cascade expression (Maitra expression) [1] m ∑ � = Q M i = i 1 � Maitra Cascade Minimization - National Technical University of Athens

Basic Definitions(2) � Minimal (Exact) expression: � Least number of complex terms (weight) � Cascade realizable function(f) � w(f)=1 � Restricted Maitra Cascade � G implements a complete set of 6 functions (Table 1) � Maitra Cascade Minimization - National Technical University of Athens

Reversible Wave Cascade Cellular Architecture Maitra (Complex) 0 0 0 Term R 11 R 12 R 1m X1 Maitra Cells R n1 R n2 R nm Xn XOR Collector row 0 Generalized (n+1)x(n+1) F(x) Toffoli gate � Maitra Cascade Minimization - National Technical University of Athens

Boolean Decompositions � Shannon, Negative = ⊕ f ( X ) x f xf 0 1 Davio, Positive Davio = ⊕ f ( X ) x f f 2 0 decompositions = ⊕ ( ) f X xf f 2 1 � New Decompositions = + ⊕ = + ⊕ ⊕ f ( X ) ( x f ) f f ( X ) ( x f ) ( x f ) 2 1 2 1 = + ⊕ = ⊕ ⊕ ( ) ( ) ( ) f ( X ) ( x f ) ( x f ) f X x f x f 2 1 0 1 = ⊕ + = ⊕ ⊕ f ( X ) ( x f ) ( x f ) f ( X ) ( x f ) ( x f ) 0 1 2 0 = + ⊕ ⊕ f ( X ) ( x f ) ( x f ) = + ⊕ ⊕ ( ) ( ) ( ) f X x f x f 2 0 0 1 = + ⊕ ( ) ( ) f X x f f 2 0 � Maitra Cascade Minimization - National Technical University of Athens

Cascade Merging Lemma 1: ⊕ = ⊕ ≠ ≠ F ( x , y ) F ( x , y ) F ( x , y y ), y y , y y Relation: r 1 r 2 r 1 2 1 2 1 2 1 2 holds iff: = ( , , ) ( 1 , 1 , 3 ), ( 1 , 3 , 1 ), ( 2 , 2 , 4 ), ( 2 , 4 , 2 ), ( 3 , 3 , 3 ), ( 4 , 4 , 4 ), ( 5 , 5 , 6 ), ( 5 , 6 , 5 ), ( 6 , 6 , 6 ) r r r 1 2 Example: + ⊕ + ⊕ + = + ⊕ + ⊕ + [( x x ) x x ] [(( x x ) x ) x ] {[( x x ) x ] [(( x x ) x )]} x 1 2 3 4 1 2 3 4 1 2 3 1 2 3 4 ⇔ ⊕ = ⊕ 1234 1252 ( 123 125 ) 2 � Maitra Cascade Minimization - National Technical University of Athens

Maitra Cell Classes � Cell has non-constant Cell Class Cell Index 1 Cell Index 2 inputs: (representative) 1 1 3 2 2 4 3 5 6 Cell Class Cell Index 1 Cell Index 2 � Cell with a constant (representative) input (cascaded 0) 1 1 2 2 6 � Maitra Cascade Minimization - National Technical University of Athens

Relative Complex Terms (1) � Two complex terms = = P R R ...R , P Q Q ...Q 1 1 2 n 2 1 2 n with R i , Q i maitra cells are relatives if R i and Q i , i ≤ n belong to the same maitra cell class. � The following complex terms are relatives: , P P ⊕ P , P x n ⊕ P , P x n � Maitra Cascade Minimization - National Technical University of Athens

Relative Complex Terms (2) � Maitra Generator Class. � Maitra expressions with relative corresponding terms. � Equivalent maitra expressions � Represent the same switching function. � Belong to the same generator class. Example Expressions: = ⊕ = ⊕ Q ( 1234 ) ( 6661 ), Q ( 2412 ) ( 6662 ) 1 2 are equivalents since they both represent the same switching function and furthermore terms: 1234 & 2412 are relatives and terms 6661 & 6662 are relatives. � Maitra Cascade Minimization - National Technical University of Athens

Relative Complex Terms (3) CIL: Number of cells with one constant input 0. Lemma 2: relative complex terms � P ⊕ 1 , P P P is a 1 2 2 complex term. Example ⊕ = ( 123455 ) ( 241466 ) 666155 (123455) and (241466) are relatives (123) and (241) are complements = ( 123 ) ( 241 ) (6661)=x 4 and ⊕ = ( 1234 ) ( 6661 ) ( 2414 ) ⊕ = + ⊕ = ( 1234 ) x ( x x ) x x x 4 1 2 3 4 4 + = ( ( x x ) x ) x ( 2414 ) 1 2 3 4 �� Maitra Cascade Minimization - National Technical University of Athens

Relative Complex Terms (4) Lemma 3: If P,P 1 ,M,M 1 are complex terms and: ⊕ = � M P P 1 � P, P 1 relatives � M, M 1 have ≥ > CIL ( M ) CIL ( M ) 0 � 1 � cells of same class from CIL(M 1 )+2 until the last cell ⊕ = Then: and P,P 2 are relative complex terms M P P 1 2 Example ⊕ = ( 1243 ) ( 6123 ) ( 2243 ) (6123) & (6611) have CIL=1 & 2 respectively & and (2243) & (2441) are relatives ⊕ = ( 1243 ) ( 6611 ) ( 2441 ) �� Maitra Cascade Minimization - National Technical University of Athens

Relative Complex Terms (5) Lemma 4: If P 1 , P 2 are complex terms and: � < < 0 CIL ( P ) CIL ( P ) 1 2 � cells of the same class from CIL(P 2 )+2 until the last cell Then is a complex term and is relative P 1 . P ⊕ P 1 2 Example ⊕ = ( 666613 ) ( 661433 ) ( 661413 ) �� Maitra Cascade Minimization - National Technical University of Athens

Complex Term Splitting Lemma 5 (Term splitting): Lemma 6: = = P p p ... p ... p � P 1 ,P 2 relative complex 1 1 2 i n terms ⊕ = Q Q 1 2 � P 1 ,P 2 are split at position i: 1 1 1 2 2 2 ⊕ p p ... q ... p p p ... q ... p 1 n 2 n 1 2 1 2 = ⊕ = ⊕ P P P , P P P � P 1 ,Q 1 ,Q 2 complex terms 1 11 12 2 21 22 1 2 ≤ ≠ p , p , p , j n , j i P 11 ,P 21 and P 21 ,P 22 are � cells � j j j relatives. of the same class p , q , q cells of different cell � i 1 2 class. Example Example = ⊕ ( 1233 ) ( 1234 ) ( 1236 ) = ⊕ ( 1234 ) ( 1264 ) ( 1244 ) = ⊕ ( 1411 ) ( 1414 ) ( 1415 ) = ⊕ ( 1232 ) ( 2411 ) ( 1235 ) �� Maitra Cascade Minimization - National Technical University of Athens

Minimization Theorems (1) � Theorem 1: Each minimal expression of f can be expressed as: = f F ( x , y ) � •Constant subfunction p 1 •(p,q)=(3,4),(3,6),(4,6) � = ⊕ Equiv. f F ( x , y ) F ( x , z ) p 1 q 1 and y,z subfunctions � = ⊕ ⊕ forms f F ( x , y ) F ( x , z ) F ( x , g ) •(p,q,r)=(3,4,6) p 1 q 1 r 1 = ⊕ Proof f , f , f y g , 0 1 2 ⊕ , ⊕ z g y z XOR-sum minimal form of f: = ⊕ ⊕ f ( x ,..., x ) F ( x , y ) ... F ( x , y ) 1 n r 1 1 1 rn 1 n Comparison with Lemma 1. �� Maitra Cascade Minimization - National Technical University of Athens

Minimization Theorems (2) Theorem 2: At least one minimal expression of a switching function with less than 6 variables can be obtained f ( x , x ,..., x ) 1 2 n from the minimal expressions of its subfunctions. Proof: 3 cases (Theorem 1): = � Expr ( f ) F ( x , y ) p � Theorem 1 � Constant subfunction. � Expr( f) produced from the minimal expression of non constant subfunctions �� Maitra Cascade Minimization - National Technical University of Athens

Minimization Theorems (3) � , y,z subfunctions. = ⊕ Expr ( f ) F ( x , y ) F ( x , z ) p q � w(f)=w(y)+w(z) � Expr(f) produced by the minimal expressions of subfunctions. , � = ⊕ ⊕ = ⊕ ⊕ ⊕ Expr ( f ) F ( x , y ) F ( x , z ) F ( x , g ) f , f , f y g , z g , y z p q r 0 1 2 � w(y) ≤ w(z) ≤ w(g) � w(f) ≤ 5 � w(y) = 1 � w(f) = 3 � w(y) = w(z) = w(g) = 1 � w(f i ) = w(f j ) = 2 � 1 common term � w(f) = w(y) + w(z) + w(g) = m = 4,5 � w(y) = 1 � m ≤ w(f 0 ) + w(f 2 ) ≤ 2 * w(y) + w(g) + w(z) = m + 1 � w(f 0 ) + w(f 2 ) = m. Proof follows that of previous case. � w(f 0 ) + w(f 2 ) = m + 1 � = ⊕ = ⊕ (minimal exprs of these f y g , f y z 0 2 subfunctions for this particular case), At least one minimal expr of f 0 and f 2 with one common term (y) to be merged. �� Maitra Cascade Minimization - National Technical University of Athens

Minimization Theorems (4) Theorem 3: Let Q 1 be a minimal expression of f, produced by the minimal expressions F i1 , F j1 of f i , f j respectively. An equivalent to Q 1 expression Q 2 of f can be obtained from two other minimal expressions F i2 , F j2 of f i , f j which are equivalents to F i1 , F j1 . �� Maitra Cascade Minimization - National Technical University of Athens