1. Winter 2014 S. Areibi School of Engineering University of Guelph
ENGG3190 Logic Synthesis “Two Level Logic Synthesis: Heuristics (ESPRESSO)” (Part II)
Winter 2014
S. Areibi
School of Engineering
University of Guelph

2. Outline Optimality Heuristics ESPRESSO
Expand, Reduce, Irredundant Steps
Input Format
Comparison: QM vs. ESPRESSO

3. Exact Algorithm vs. Heuristics

4. Don’t forget that essential prime implicants should be included in every solution!!

5. Global Local Global Optimal versus Local Optimal
By Iterative Improvement we can move from a Local bad solution to yet another Local better solution

6. Local Search

7. Local Search
HOW?

8. ESPRESSO ESPRESSO utilizes modified local search
keep in mind local minimum problem
Composed of three main operations:
Expand, Reduce, Irredundant
Espresso Heuristic (in a nutshell)
Apply Expand and Irredundant operators to:
optimize the current function specification
go down hill  improve the objective function
Uses the Reduce operator to:
get out of local minimum
go up hill  deteriorate the objective function
This is iterated till the solution converges

9. EXPAND
Objective: Take a cube c and make it prime by removing literals.
a). Greedy way

10. Expand Operation Expand
expand essential cubes in F in decreasing size to a prime cube
prime cube - fully expanded against OFF-set
essential cube - contains essential vertex
essential vertex - minterm no other cube covers
remove any covered cubes ON DC
OFF 00 01 11 10 00 01 11 10
Expand

11. EXPAND: Check Validity
Determining the correct
direction of expansion is important!!
Expand abc by removing a (results in bc)
Is it valid? NO

12. EXPAND: Objective

13. REDUCE: Cube Operations
shrink cubes in descending order of size while maintaining cover
smaller cubes can expand in more directions
smaller cubes more likely to be covered by other cubes during expansion ON DC
OFF 00 01 11 10 00 01 11 10
Reduce

14. REDUCE Operator: Validate Solutions

15. REDUCE: Good vs. Bad
Problem: Given a cover F and c  F, find the smallest cube c  c such that
F\{ c } + { c }
is still a cover. c is called the maximally reduced cube of c.
REDUCE is
order dependent on
off
WHY?
Don’t care

16. REDUCE: Order Dependent
Example:
Two orders:
REDUCE is order dependent !

17. REDUCE followed by Expand
Main Idea: Make a prime not a prime but still maintain cover:
{c1 ,…, ci,…, ck}  {c1 ,…,ci,ci+1 ,…,ck }
But
Get out of a local minimum (prime and irredundant is local minimum)
Then have nonprimes, so can expand again in different directions. (Since EXPAND is “smart”, it may know best direction)

18. Reduce + Expand: Example

19. Redundant Prime Implicant (RPI)
If each minterm subsuming a prime implicant (PI) is also covered by other essential prime implicants, then that PI is called a redundant prime implicant (RPI).
Also called redundant prime cube (RPC). A 1
EPI C
RPI B

20. Irredundant Cubes Let F = {c1, c2, …, ck} be a cover for f, i.e.
f = ik=1 ci
A cube ci F is irredundant if F\{ci}  f
Example: f = ab + ac + bc ac bc bc
Not covered ab ac c b
F\{ab}  f a

21. IRREDUNDANT
Problem:
Given a cover of cubes {ci}, find a minimal subset {cik} that is also a cover, i.e.

22. Irredundant: Definitions
Relatively essential set Er
Implicants covering some minterms of the function not covered by other implicants
Important remark: we do not know all the primes!
Totally redundant set Rt
Implicants covered by the relatively essentials
Partially redundant set Rp
Remaining implicants
Find a subset of Rp that, together with Er covers the function.

23. Irredundant: Example Er = {α, ε} Rt = ∅ Rp = {β, γ, δ} α 10 10 11α δ γ β ε b a c
Er = {α, ε}
Rt = ∅
Rp = {β, γ, δ}
Irredundant Cover
Minimum Irredundant Cover

24. Espresso – Irredundant Operator

25. Irredundant & Minimal Covers

26. Espresso – Additional Concerns
We will cover this Later!

27. ESPRESSO: Steps a’b’ a’b ab ab’ x = a’b + ab + a’b’ Expand a’b’ a’b ab
Irredundant
Reduce
a’b’
a’b ab
ab’
a’b’
a’b ab
ab’
Expand
a’b’
a’b ab
ab’
x = a’ + b
a’b’
a’b ab
ab’
Irredundant
Cost Stable

28. ESPRESSO Illustrated

29. Heuristic Logic Minimization
Apply a sequence of logic transformations to reduce a cost function
Transformations
Expand:
Input expansion
Enlarge cube by combining smaller cubes
Reduces total number of cubes
Output expansion
Use cube for one output to cover another
Reduce: break up cube into sub-cubes
Increases total number of cubes
Hope to allow overall cost reduction in a later expand operation
Irredundant
Remove redundant cubes from a cover

30. Example: Input Expansion
Expand: input expansion z y
Redundant! x
On-set member
Off-set member
xyz f
000 1
01-
-11
xyz f
0-0 1
01-
-11
xyz f
0-0 1
-11

31. Example: Output Expansion
Expand: output expansion
Two output functions of three variables each with initial covers shown below z z y y x x f1 f2 f1 f2
xyz
F1F2
0-1 10
1-0
00- 01
-00
-11
xyz
F1F2
0-1 11
1-0 10
00- 01
-00
-11

32. Espresso – Algorithm

33. Example of ESPRESSO Input/Output
ƒ(A,B,C,D) = m(4,5,6,8,9,10,13) + d(0,7,15)
Espresso Input
Espresso Output
.i 4
.o 1
.ilb a b c d
.ob f
.p 10 .e
-- # inputs
-- # outputs
-- input names
-- output name
-- number of product terms
-- A'BC'D'
-- A'BC'D
-- A'BCD'
-- AB'C'D'
-- AB'C'D
-- AB'CD'
-- ABC'D
-- A'B'C'D' don't care
-- A'BCD don't care
-- ABCD don't care
-- end of list
.i 4
.o 1
.ilb a b c d
.ob f
.p 3 .e
ƒ = A C' D + A B' D' + A' B

34. How it Works: Interesting Mechanics
The core representation is PCN cube lists
Most operations are those covered in previous lecture (Unate Recursive Paradigm (URP))
Several heuristic techniques to order cubes (cleaver ordering techniques)
You can easily implement the ESPRESSO heuristic with previous knowledge
The main idea of Expansion is how to avoid expanding in the wrong direction!!
Let’s see the details

35. EXPAND Step : Details Outline: Expand one cube, ci , at a time
Build “blocking” matrix B = B ci
See which other cubes cj can be feasibly covered using B.
Choose expansion (literals to be removed) to cover most other cj .
Note:

36. Expanding is covering more 1’s
Covering more 1’s means removing variables from the cube
Removing variables from the cube is essentially less literals!!
How to find XZ’ or XY starting from XYZ’W’?
We will use “Set Covering” technique

37. Covering problem!!
Cover each element
more than once.
The covering problem is an Integer Linear Programming problem.
Has many applications in industry: airline scheduling, crew scheduling, oil company vehicle scheduling, VLSI Design …
We will turn the EXPAND problem into a covering problem.

38. Set Covering and Partitioning
Min Z = c S
j=1
j j
a S = 1
ij j
i  {1,…m}
Subject to:
Set Covering
> n
Min Z = c S
j=1
j j
a S = 1
ij j
i  {1,…m}
Subject to:
Set Partitioning

39. EXPAND as a Covering Problem
Expand = a Covering Problem on the Blocking Matrix
First: Given function F, build a cube cover of the 0’s in F
Why?: We need to know what our cube cannot touch when it expands!!
How?: URP Complement of the starting cover of the function!
Recall OFF Set, ON Set, DC Set
The Blocking Matrix will help us avoid expansion in the wrong direction.
We will start to expand from the Cube W’XYZ’ (since it is the worst!!)
This is exactly the same step in ESSPRESSO!
URP Complement will give me the grey Cubes

40. Building the Blocking Matrix
Expand = a Covering Problem on the Blocking Matrix
One row for each variable in the cube you are trying to expand
One Column for each cube in the cover of the OFF set.
Put a “1” in the matrix if the cube variable (row) <> polarity of variable in the cube (column) of the OFF cover; else “0”. If don’t care, it’s a “0” (draw blank) 1 1 1 1 1
Once you build the Blocking Matrix it will help us in avoiding the expansion in the wrong direction.
We might not have too many rows, but we might have many columns in real life problems
What do I do with this Blocking Matrix?

41. What does a ‘1’ in a row and column really mean?
For example Row #1, if we attempt to remove W’ from W’XYZ’ the Matrix says “be careful” since we will hit WXZ’ or WY’Z’

42. The Blocking Matrix will assist us in obtaining a legal cube expansion!
If we pick the first row (W’) and the second row (X) then we will cover all the F’ cubes.
So we will end up with W’X and get rid of the Y and Z’
A second solution would be W’Z’ (1st and 4th rows)
Check the K-Map on the left hand side (where is W’X, W’Z’?)

43. The Reduce-Expand-Irredundant Can lead to excellent solutions.
Clever tricks and elegant heuristics.
Lots of usage of covering problems.

46. Besides ESPRESSO there are other similar tools that can produce good solutions and based on heuristic techniques.

47. Comparison Performance Comparison Between Quine McClusky and Espresso
Exact Method for Logic Minimization
Compute Time is prohibitive for Minimizations (Doubly Exponential)
Times-out for medium complexity benchmarks
ESPRESSO
Heuristic Method for Logic Minimization
Based on Unate Recursive Paradigm
Inferior to Quine McClusky in minimization
Some sacrifice in optimization for a large reduction in run time
Relatively unchanged for 20 years.
Performance Comparison Between Quine McClusky and Espresso

48. ESPRESSO :: Conclusion
The algorithm successively generates new covers until no further improvement is possible.
Produces near-optimal solutions.
Used for:
PLA minimization, or
as a sub-function in multilevel logic minimization.
Can process very large circuits.
10,000 literals, 100 inputs, 100 outputs
Less than 15 minutes on a high-speed workstation

50. Extra Slides

51. Need for Iterations in ESPRESSO
Espresso: Why Iterate on Reduce, Irredundant Cover, Expand?
Initial Set of Primes found by
Steps1 and 2 of the Espresso
Method
Result of REDUCE:
Shrink primes while still
covering the ON-set
Choice of order in which
to perform shrink is important
4 primes, irredundant cover,
but not a minimal cover!

52. ESPRESSO Example Espresso Iteration (Continued)
Second EXPAND generates a
different set of prime implicants
IRREDUNDANT COVER found by
final step of espresso
Only three prime implicants!

53. Other operators Reduce Irredundant Future Expand operation Reduce
Identified as redundant; removed
Irredundant

54. Example of an Application of Operators
(12 cubes)
(10 cubes)
-011 10
--11 11
1001 01
1101
-001
-1-0
1010
0110
0010
-100
1000
(9 cubes)
reduce
--11 11
1001 01
1101 10
-001
-1-0
1010
0110
-010
-100
1000
--11 11
1001 01
1101 10
-0-1
-1-0
1010
0-10
-100
10-0
expand
expand
reduce
expand

55. Example of a minimization loop
F = Expand(F, D)
F = Irredundant(F,D)
do {
Cost = |F|
F = Reduce(F,D)
F = Expand(F,D)
F = Irredundant(F,d)
} while (|F| < Cost)
F= Make_sparse(F,D)
Make_sparse reduces output parts of a cube (e.g., from 11 to 10) to remove redundant connections)
Example:
xyz
F1F2
11- 10
-01
1-1 11
0-0 01
xyz
F1F2
11- 10
-01
1-1 01
0-0

56. P S R T Q The goal is to start with this and do some optimization
Optimization involves three steps: Expand, Reduce, Irredundancy

57. The goal is to make cubes become “Prime Cover”, i. e
The goal is to make cubes become “Prime Cover”, i.e., cover as many ones as possible
This operation is called Expand

58. To create new solutions, all redundant cubes are removed
This is called Irredundant Operation
Note: this is still a local optimum solution

59. This is an essential step (Reduce) that might allow us to explore new solutions.
The new solution is far from optimal!! But it can help in next iterations to find a more superior solution.

60. Expanding Cubes again will explore a new solution that might not have been seen before and whose cost is less than previous solutions.
All 1’s in the T Cube have been covered by two other cubes.
The T Cube becomes redundant!!

61. Removing the redundant Cube leads to a better solution now.

62. We managed to get the global minima but this will not always be the case!!
Most combinatorial optimization problems can be solved effectively using heuristics or meta-heuristics

63. Irredundant: Cube Operations
find minimal cover with each cube containing an essential vertex
find relatively essential cubes E
removing them violates cover - keep them
redundant cubes R = F - E
can be individually removed
totally redundant Rt - covered by E+D
remove Rt
partially redundant Rp - R - Rt
new F = E + minimal set of Rp E ON DC
OFF 01 11 00 01 11 10 Rp Rt
Irredundant 00 10 Rp

64. REDUCE Algorithm

65. IRREDUNDANT In ESPRESSO
The algorithm in ESPRESSO uses a modification of the tautology check.
The key idea is the following:
Rather than checking the tautology, we determine the set of cubes that, when removed, prevents the function from being a tautology!!
The tautology of the cofactor can be used to determine containment.
Hence, we can determine the set of cubes that when removed, prevents the containment!!