Bi-Decomposition of Discrete Function SetsRM

1
Bi-Decomposition of Discrete Function Sets

• Bernd Steinbach , Christian Lang , and
• Marek A. Perkowski
• Freiberg University of Mining and Technology
• Institute of Computer Science, Freiberg
  (Sachs.), Germany
• Portland State University, Department of
  Electrical and
• Computer Engineering, Portland (Oregon), USA

2
Outline

• Introduction
• Function Sets
• Bi-Decomposition
• Decomposition Strategy
• EXOR-Decomposition of Function Sets
• Results
• Conclusion

3
Introduction

• Incompletely specified functions (ISFs) are a
  generalization of Boolean functions.
• There are many multi-stage design algorithms for
  ISFs.
• We propose function sets as a generalization of
  ISFs to improve many of these design algorithms.
• We demonstrate our method on the example of
  EXOR-bi-decomposition.

4
Function Sets I

• There are two ways to interpret ISFs
• incompletely specified function
• set of 2s fully specified functions,
• s number of dont cares
• Example F f1, f2, f3, f4

5
Function Sets II

• The functions of an ISF and the AND and OR
  operation form a lattice, a special type of
  Boolean algebra
• If f1, f2 Î F, then f1 Ù f2 Î F and f1 Ú f2 Î F
• Example f3 Ù f4 f1 Î F, and f3 Ú f4 f2 Î F

6
Function Sets III

• There are function sets that are lattices, but
  not ISFs Rf1, f2 ¹ F f1, f2, f3, f4
• There are function sets that are not lattices
• Sf3, f4, f3 Ù f4 f1 Ï S

Rf1,f2
Sf3,f4
7
Bi-Decomposition
Bi-Decomposition for Binary Circuits Structure
A
g(A, C)
f(A, B, C)
o
C
B
h(B, C)
f(A,B,C) g(A,C) Ú h(B,C)
OR- Bi-Decomposition AND- Bi-Decomposition EXOR- B
i-Decomposition
f(A,B,C) g(A,C) Ù h(B,C)
f(A,B,C) g(A,C) Å h(B,C)
8
Decomposition Strategy I

• An ISF F(A, B, C) is bi-decomposed into function
  sets G(A, C) and H(B, C)
• This decomposition is recursively repeated.
• More functions in G(A, C) means fewer functions
  in H(B, C) and vice versa.
• Our strategy Include into G(A, C) as many
  functions as possible, design G(A, C), then make
  the same with H(B, C).

9
Decomposition Strategy II
Example
design(F) G bi_decompose(F) g design(G) H
compute_h(F, g) h design(H) return
bi_compose(g, h)
10
Decomposition Strategy III
Example
design(F) G bi_decompose(F) g design(G) H
compute_h(F, g) h design(H) return
bi_compose(g, h)
g
a
F
b
c
?
d
11
Decomposition Strategy IV
Example
design(F) G bi_decompose(F) g design(G) H
compute_h(F, g) h design(H) return
bi_compose(g, h)
g
a
F
b
c
H
d
F(a,b,c,d)
g(a,b)
0
1
0
0
0
1
0
F
H(c,d)
12
Decomposition Strategy V
Example
design(F) G bi_decompose(F) g design(G) H
compute_h(F, g) h design(H) return
bi_compose(g, h)
g
a
f
b
c
h
d
f(a,b,c,d)
0
1
0
0
h(c,d)
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Function Sets in Bi-Decomposition

• Pass as many decomposition functions to the next
  stage of decomposition as possible.
• For OR and AND decomposition ISFs are sufficient
  to describe all decomposed functions.
• In EXOR decomposition ISFs describe only a small
  fraction of all possible subfunctions
• G(A, C) and H(B, C).

14
EXOR-Decomposition of Functions

• A function f(A, B, C) is EXOR-decomposable if its
  decomposition chart consists of two types of
  columns one being the negation of the other.
• There are two decomposition functions g(A, C),
  the first column and its negation.
• Example

f(a,b,c,d)
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EXOR-Decomposition of ISFs I

• An ISF F(A, B, C) can consists of independent
  parts.
• Each independent part consists of horizontally or
  vertically connected cares in the decomposition
  chart.

F(a,b,c,d)
cd
00
01
11
10
ab
0
1
F
F
00
1
0
F
F
01
F
F
0
0
11
F
F
0
0
10
16
EXOR-Decomposition of ISFs II

• Each independent part has an ISF Gi(A, C) and its
  negation as decomposition functions.
• All decomposition functions gi are combinations
  of the Gi or their negation (/Gi).
• Function set is not an ISF new data
  structure

F(a,b,c,d)
G1(a,b)
G2(a,b)
cd
00
01
11
10
ab
0
1
F
F
00
0
F
1
0
F
F
01
1
F
F
0
0
11
F
F
0
F
0
10
F
0
F
0
17
Combinational ISFs (C-ISF)

• A C-ISF is a set of functions specified by a set
  of component ISFs with disjoint care sets.
• The cares of each component ISF may be negated.
• A C-ISF contains 2component ISF functions.

F1(a,b)
F2(a,b)
FltF1,F2gt f1, f2, f3, f4
ab
0
F
00
1
F
01
F
0
11
F
0
10
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Decomposition of C-ISFs I

• Each component ISF Fi of a C-ISF FltF1,¼,Fngt can
  be negated.
• The decomposability of the function depends on
  the pattern of negations of its component ISFs .
• A large number of component ISFs is possible.
• We propose a greedy algorithm that successively
  adds the component ISFs to the resulting ISF.

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Decomposition of C-ISFs II

• Example Decomposition of FltF1, F2, F3gt

F1
c
EXOR-decomposable relating a, b - c not
EXOR-decomposable relating a, b - c
00
01
ab
1
F
00
0
1
01
F
F
11
F
F
10
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Multi-Valued Bi-Decomposition

• Function sets of Boolean functions can be
  generalized to function sets of multi-valued
  functions.
• EXOR-decomposition can be extended to MODSUM-
  (sum modulo n) decomposition.
• AND- and OR-decompositions correspond to MIN- and
  MAX-decompositions

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Results I

• Decomposition of machine-learning benchmarks
• Selection of type of decomposition
• F modsum (max (G1max, G2max) , H )
• Comparison of complexity of G using ISFs and
  C-ISFs

A1
G1max
G
ISF vs. C-ISF
max
A2
G2max
F
mod sum
B
H
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Results II

• inp - Number of Inputs

DFC(F)mo(mi1mi2...min) DFC discrete
function cardinality
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Conclusion

• Function sets are a generalization of ISFs.
• C-ISFs are a particular class of function sets
  that describe the decomposed functions of
  EXOR-decomposition
• C-ISFs can be efficiently bi-decomposed
• Better decompositions can be found using C-ISFs
  instead of ISFs

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Further Work

• Applications
• Ashenhurst and Curtis decompositions
• finite state machine design (set of functions
  is used to represent
• set of state encodings)
• Extensions
• function sets for multiple output functions
• a new concept of sets of relations