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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 ) +

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Maitra Cascade Minimization

Voudouris Dimitrios

  • Dr. Papakonstantinou George

National Technical University of Athens

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Maitra Cascade Minimization - National Technical University of Athens

  • Basic Definitions(1)

Complex Maitra Term [1]

Constant 0(1) Boolean Function Literal

  • MiMaitra Term

aliteral GArbitrary 2 variable switching function (~Maitra Cell)

Reversible Wave Cascade expression (Maitra

expression) [1]

  • ∑

=

=

m 1 i i

M Q

) M G(a, M

i 1 i

=

+

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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)

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Maitra Cascade Minimization - National Technical University of Athens

  • Reversible Wave Cascade Cellular

Architecture

R11 Rn1 R12 Rn2 R1m Rnm X1 Xn F(x) Maitra (Complex) Term Maitra Cells XOR Collector row Generalized (n+1)x(n+1) Toffoli gate

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Maitra Cascade Minimization - National Technical University of Athens

  • Boolean Decompositions

Shannon, Negative

Davio, Positive Davio decompositions

New Decompositions

1 2 2 1

) ( ) ( ) ( f xf X f f f x X f xf f x X f ⊕ = ⊕ = ⊕ =

2 1 2 1 1 2

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( f f x X f f x f x X f f x f x X f f x f x X f f x f x X f ⊕ + = ⊕ ⊕ + = ⊕ ⊕ = ⊕ + = ⊕ ⊕ + = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

2 1 1 2 1 2

f x f x X f f x f x X f f x f x X f f f x X f ⊕ ⊕ + = + ⊕ = ⊕ ⊕ = ⊕ + =

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Maitra Cascade Minimization - National Technical University of Athens

  • Cascade Merging

Lemma 1: Relation: holds iff: Example:

) 6 , 6 , 6 ( ), 5 , 6 , 5 ( ), 6 , 5 , 5 ( ), 4 , 4 , 4 ( ), 3 , 3 , 3 ( ), 2 , 4 , 2 ( ), 4 , 2 , 2 ( ), 1 , 3 , 1 ( ), 3 , 1 , 1 ( ) , , (

2 1

= r r r

2 1 2 1 2 1 2 1

, ), , ( ) , ( ) , (

2 1

y y y y y y x F y x F y x F

r r r

≠ ≠ ⊕ = ⊕

2 ) 125 123 ( 1252 1234 )]} ) [(( ] ) {[( ] ) ) [(( ] ) [(

4 3 2 1 3 2 1 4 3 2 1 4 3 2 1

⊕ = ⊕ ⇔ + ⊕ + ⊕ + = + ⊕ + ⊕ + x x x x x x x x x x x x x x x

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Maitra Cascade Minimization - National Technical University of Athens

  • Maitra Cell Classes

Cell has non-constant

inputs:

Cell with a constant

input (cascaded 0)

6 5 3 4 2 2 3 1 1 Cell Index 2

(representative)

Cell Index 1 Cell Class 6 2 2 1 1 Cell Index 2 Cell Index 1

(representative)

Cell Class

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Maitra Cascade Minimization - National Technical University of Athens

  • Relative Complex Terms (1)

Two complex terms

with Ri, Qi maitra cells are relatives if Ri and Qi, i≤n belong to the same maitra cell class.

n 2 1 2 n 2 1 1

...Q Q Q P , ...R R R P = =

n n

x P P x P P P P ⊕ ⊕ , , ,

The following complex terms are relatives:

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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: are equivalents since they both represent the same switching function and furthermore terms: 1234 & 2412 are relatives and terms 6661 & 6662 are relatives. ) 6662 ( ) 2412 ( ), 6661 ( ) 1234 (

2 1

⊕ = ⊕ = Q Q

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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 is a complex term.

Example

2 1, P

P

2 1

P P ⊕

) 2414 ( ) 6661 ( ) 1234 ( ) 241 ( ) 123 ( 666155 ) 241466 ( ) 123455 ( = ⊕ = = ⊕

(123455) and (241466) are relatives (123) and (241) are complements (6661)=x4 and

) 2414 ( ) ) ( ( ) ( ) 1234 (

4 3 2 1 4 4 3 2 1 4

= + = ⊕ + = ⊕ x x x x x x x x x x

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Maitra Cascade Minimization - National Technical University of Athens

  • Relative Complex Terms (4)

Lemma 3: If P,P1,M,M1 are complex terms and:

  • P, P1 relatives

M, M1 have

  • cells of same class from CIL(M1)+2 until the last cell

Then: and P,P2 are relative complex terms Example

1

P P M = ⊕

) ( ) (

1

> ≥ M CIL M CIL

2 1

P P M = ⊕

) 2441 ( ) 6611 ( ) 1243 ( & ) 2243 ( ) 6123 ( ) 1243 ( = ⊕ = ⊕

(6123) & (6611) have CIL=1 & 2 respectively and (2243) & (2441) are relatives

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Maitra Cascade Minimization - National Technical University of Athens

  • Relative Complex Terms (5)

Lemma 4: If P1, P2 are complex terms and:

  • cells of the same class from CIL(P2)+2 until the last

cell

Then is a complex term and is relative P1. Example

) ( ) (

2 1

P CIL P CIL < <

2 1

P P ⊕

) 661413 ( ) 661433 ( ) 666613 ( = ⊕

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Maitra Cascade Minimization - National Technical University of Athens

  • Complex Term Splitting

Lemma 5 (Term splitting):

P1,Q1,Q2 complex terms

  • cells
  • f the same class
  • cells of different cell

class.

Example Lemma 6:

P1,P2 relative complex

terms

P1,P2 are split at position i:

  • P11,P21 and P21,P22 are

relatives. Example

2 2 2 2 1 1 1 1 2 1 2 1 1

... ... ... ... ... ...

2 1 2 1

n n n i

p q p p p q p p Q Q p p p p P ⊕ = ⊕ = =

2 1 2 1

, , , , , , q q p i j n j p p p

i j j j

≠ ≤

) 1235 ( ) 2411 ( ) 1232 ( ) 1244 ( ) 1264 ( ) 1234 ( ⊕ = ⊕ =

22 21 2 12 11 1

, P P P P P P ⊕ = ⊕ =

) 1415 ( ) 1414 ( ) 1411 ( ) 1236 ( ) 1234 ( ) 1233 ( ⊕ = ⊕ =

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Maitra Cascade Minimization - National Technical University of Athens

  • Minimization Theorems (1)

Theorem 1: Each minimal expression of f can be

expressed as:

  • Proof

XOR-sum minimal form of f: Comparison with Lemma 1.

) , ( ) , ( ) , ( ) , ( ) , ( ) , (

1 1 1 1 1 1

g x F z x F y x F f z x F y x F f y x F f

r q p q p p

⊕ ⊕ = ⊕ = =

) , ( ... ) , ( ) ,..., (

1 1 1 1 1 n rn r n

y x F y x F x x f ⊕ ⊕ =

  • Constant subfunction

Equiv. forms

  • (p,q)=(3,4),(3,6),(4,6)

and y,z subfunctions

  • (p,q,r)=(3,4,6)

z y g z g y f f f ⊕ ⊕ ⊕ = , , , ,

2 1

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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 from the minimal expressions of its subfunctions. Proof: 3 cases (Theorem 1):

  • Theorem 1 Constant subfunction.

Expr(f) produced from the minimal expression of non constant

subfunctions

) ,..., , (

2 1 n

x x x f ) , ( ) ( y x F f Expr

p

=

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  • Minimization Theorems (3)
  • , y,z subfunctions.

w(f)=w(y)+w(z) Expr(f) produced by the minimal expressions of subfunctions.

  • ,

w(y) ≤ w(z) ≤ w(g) w(f) ≤ 5 w(y) = 1

w(f) = 3 w(y) = w(z) = w(g) = 1 w(fi) = w(fj) = 2 1 common

term

w(f) = w(y) + w(z) + w(g) = m = 4,5 w(y) = 1 m ≤ w(f0) + w(f2) ≤

2 * w(y) + w(g) + w(z) = m + 1

w(f0) + w(f2) = m. Proof follows that of previous case. w(f0) + w(f2) = m + 1

(minimal exprs of these subfunctions for this particular case), At least one minimal expr of f0 and f2 with one common term (y) to be merged.

) , ( ) , ( ) , ( ) ( g x F z x F y x F f Expr

r q p

⊕ ⊕ = ) , ( ) , ( ) ( z x F y x F f Expr

q p

⊕ = z y g z g y f f f ⊕ ⊕ ⊕ = , , , ,

2 1

z y f g y f ⊕ = ⊕ =

2

,

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Maitra Cascade Minimization - National Technical University of Athens

  • Minimization Theorems (4)

Theorem 3: Let Q1 be a minimal expression of f, produced by the minimal expressions Fi1, Fj1 of fi, fj

  • respectively. An equivalent to Q1 expression Q2
  • f f can be obtained from two other minimal

expressions Fi2, Fj2 of fi, fj which are equivalents to Fi1, Fj1.

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  • Minimization Theorems (5)

Proof:

If w(f) = w(fi) (i=0,1,2) then the proof is trivial. If w(f) = w(fi) + w(fj) then the proof is trivial. If w(f) = w(fi) + w(fj) – 1. Common term: Pi11 = Pj11

Pi11, Pj11, Pi21, Pj21 relatives.

It holds:

r j j j j q i i i i

P P P F P P P F Q

1 12 11 1 1 12 11 1 1

... , ... ⊕ ⊕ ⊕ = ⊕ ⊕ ⊕ = ←

′ ⊕ ⊕ ⊕ ′ ⊕ ⊕ ⊕ ′ ′ = ′ ⊕ ⊕ ⊕ ′ ⊕ ⊕ = ⊕ ⊕ ⊕ = ⊕ ⊕ ⊕ = ←

2 1 12 1 1 12 3 11 2 1 11 1 1 11 1 2 22 21 2 2 22 21 2 2

) ... ( ) ... ( ) ... ( ) ... ( ... , ... k P P k P P k P k P P k P P Q P P P F P P P F Q

r j j q i i i r j j q i i r j j j j q i i i i

2 21 11 2 22 1 12 2 1 1 21 11 2 22 1 12 2 1

) ... ( ) ... ( ) ... ( ) ... ( M P P P P P P F F M P P P P P P F F

j j r j j r j j j j i i q i i q i i i i

= ⊕ = ⊕ ⊕ ⊕ ⊕ ⊕ ⇔ = ⊕ = ⊕ = ⊕ ⊕ ⊕ ⊕ ⊕ ⇔ = ⊕

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Maitra Cascade Minimization - National Technical University of Athens

  • Minimization Theorems (6)

Furthermore: and M1,M2 will have cells of the same cell class from position MAX(CIL(M1),CIL(M2))+2 until the last cell (Lemma 2). Without loss of generality: CIL(M2) ≤ CIL(M1).

CIL(M2) < CIL(M1) M is relative to M2 (Lemma 4).

and

r ≤ 3 M merges with at most 2 terms

CIL(M2) = CIL(M1) M will have cells of the same class (with M1 & M2)

from position x (> CIL(M2) + 2) until the last cell, while the rest of them will represent a literal. The proof follows that of previous case (Lemma 3).

M P P M M

j i

= ⊕ = ⊕

21 21 2 1

) ... ( ) ... (

2 22 2 1 12 r j j r j j

P P M P P ⊕ ⊕ ⊕ = ⊕ ⊕

2 2 22 1 2 22 3 21 2 2 2 22 1 2 22 2 22 2 21 3 21 2 2 2 22 1 2 22 2 22 1 21 2

) ... ( ) ... ( ) ... ( ) ... ( ) ( ) ... ( ) ... ( ) ( k P P M k P P k P Q k P P k P P k P k P k P Q k P P k P P k P k P Q

r j j r i i i r j j r i i i i i r j j r i i i i

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ = ⇔ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ = ⇔ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ =

x x ⊕ ⊕ ,

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Maitra Cascade Minimization - National Technical University of Athens

  • Heuristic Algorithm outline

Input : a switching function in minterm formulation Decomposition (recursively) using ETDDs (Shannon and

Davio expansions).

Composition (recursively): Produce expressions for f

from the minimal expressions of its subfunctions.

If complex terms with the same generator are found between two

such expressions (Fi, Fj):

Merge them to produce a by-product complex term. Try to merge that by-product with the rest of the terms in Fi, Fj.

If this by-product is merged, then the weight of the function is

reduced by 1.

At the end keep those expressions with the least number

  • f complex terms.
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  • Example
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Maitra Cascade Minimization - National Technical University of Athens

  • Experimental results

Lee’s Algorithm: G.Lee, R.Drechsler: ETDD-Based Synthesis of term-based FPGAs for incompletely specified boolean functions, ASP-DAC 1998 Minict: N. Song, M. Perkowski: Minimization of exclusive sums of multi- valued complex terms for logic cell arrays, ISMVL 98, p 32

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  • Conclusions and Future work

Contribution

New boolean decompositions are presented. An algorithm has been described for producing

minimal reversible wave cascades for switching functions up to 5 variables and near minimal for functions with more variables, without the need to calculate equivalent expressions.

Future work

Multi-output switching functions. Minimal solution for functions of up to 6 variables.

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Maitra Cascade Minimization - National Technical University of Athens

  • Bibliography(1)
  • K.K. Maitra: Cascaded switching networks of two-input flexible cells,

IRE Trans. Electron. Comput., pp, 136-143, 1962.

  • R. C. Minnick: Cutpoint cellular logic, IEEE Trans. Electron. Comput.,
  • vol. EC-13. pp. 685-698, Dec,1964.
  • J. T. Butler: Restricted Cellular Networks, IEEE Trans. Computers

25(11): 1139-1142, 1976.

  • A. Mishchenko, M. Perkowski: Logic Synthesis of Reversible Wave

Cascades, International Workshop on Logic And Synthesis 2002, New Orleans, Louisiana, June 4-7, 2002.

  • A. Sarabi, N. Song, M. Chrzanowska-Jeske, M. Perkowski: A

comprehensive approach to logic synthesis and physical design for two-dimensional logic arrays, DAC 1994, 321-326.

  • I. Schaefer, M. Perkowski, H. Wu: Multilevel logic synthesis for cellular

FPGAs based on orthogonal expansions, Proc, IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion in Circuit Design, Hanburg, Germany, pp. 42-51, Sept. 1993.

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  • Bibliography (2)
  • G. Lee: Logic synthesis for celullar architecture FPGA using BDD,

ASP-DAC 97, pp 253-258 Jan 1997.

  • G. Lee, R. Drechsler: ETDD-based Synthesis of term-based FPGAs for

incompletely specified boolean functions, ASP-DAC 1998.

  • P. Lindgren, R. Drechsler, B. Becker: Look-up table FPGA synthesis

from minimized multi-valued pseudo kronecker expressions, ISMVL 98.

  • G. Papakonstantinou, F. Gritzali: Modulo-2 expressions of switching

functions, Electronic Letters, 13(1977).

  • G. Papakonstantinou: Synthesis of cutpoing cellular arrays with

exclusive-OR collector row, Electronic Letters, 13(1977).

  • J. Preskill: Lecture notes in quantum computing,

http://www.Theory.caltech.edu/~preskill/ph229

  • G. Papakonstantinou: Cascade Transformation, IEEE Transactions on

computers, Jan 1976.

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  • Bibliography (3)
  • N. Song, M. Perkowski: Minimization of exclusive sums of multi-valued

complex terms for logic cell arrays, ISMVL 98, p 32.

  • N. Song, M. Perkowski: A new approach to and/or/exor factorization

for regular arrays, Proc. 1998 Euromicro, pp. 269-276, Vasteras, Sweden, August 25-27, 1998.

  • C. Bennet: Logical Reversibility of Computation, IBM Journal of

Research and Development, 17, 1973, pp. 525-532.

  • D. Voudouris, S. Stergiou, G. Papakonstantinou: Minimization of

reversible wave cascades, IEICE, under revision.

  • S. Stergiou, D. Voudouris, G. Papakonstantinou: Exact and Heuristic

MVESOP Minimization Algorithms, IEICE Trans. on Fundamentals, vol.E87-A,No 1,Jan. 2004.