Symmetry Detection for Large Boolean Functions Using Circuit Representation, Simulation and Satisfia

1
Symmetry Detection for Large Boolean Functions
Using Circuit Representation, Simulation and
Satisfiability

• Jin S. Zhang, Portland State University
• Alan Mishchenko, UC Berkeley
• Robert Brayton, UC Berkeley
• Malgorzata Chrzanowska-Jeske, Portland State
  University

Supported by Maseeh Fellowship from PSU, NSF
grant CCR-9988402 and CCR 0312676, SRC contract
1361.001, and by the California Micro program
with our industrial sponsors, Altera, Intel,
Magma, and Synplicity.
2
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

3
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

4
Importance of Classical Symmetries

• Widely used in logic synthesis and verification
• BDD minimization
• Decomposition
• Boolean matching
• Constructive logic synthesis
• Placement routing
• Formal verification

5
Prior Work on Symmetry Computation

• BDD-based methods Möller ICCAD93, Panda ICCAD94,
  Tsai TCAD96, Mishchenko TCAD03
• BDD construction takes majority of run time
• Cannot be applied to large designs
• Incomplete circuit-based methods performing
  structural analysis Wang, ICCD03
• Detect classical and higher-order symmetry
• Complete circuit-based method using simulation
  and ATPG Pomeranz, TCAD94
• Does not rely on recent improvements in SAT

6
Our Contributions

• Compute all classical two-variable symmetries for
  large Boolean functions

7
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

8
Definitions

• Completely specified Boolean function
• F (x1, ,xn) where x1, xn, F range over 0, 1.
• Support of F all the variables that F depends
  on.
• Cofactors
• Cofactors w.r.t. variables xi and xj
• F00 F xi ? 0, xj ? 0,
• F01, F10, F11 are defined similarly.

9
Representations of Boolean Functions

• Binary Decision Diagram (BDD)

• And-Inverter Graph (AIG)

10
Classical Symmetries
For a pair of variables
F (,x, y,) F (,y, x,)
F 01 ? F 10 0
Non-skew non-equivalent
11
Classical Symmetries
For a pair of variables
F (,x, y,) F (,y, x,)
F 01 ? F 10 0
Non-skew non-equivalent
12
Transitivity of Symmetry
Variables a, b, c
a
b
c
Form larger symmetry group
13
Simulation

• Computes the values of the internal signals and
  POs from the value of the PIs
• Random / Guided
• Control of simulation rounds
• Static / Dynamic

14
Boolean Satisfiability (SAT)

• Proves that a given Boolean formula has a
  satisfying assignment
• F (xy)(y ?z ?), satisfying assignments x
  1, y 0
• SAT vs. BDDs
• SAT involves search, heuristics can improve
  performance
• BDDs preprocess the search space
• By building canonical form of Boolean functions
• Hard or impossible for large functions

15
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

16
Overview of the Algorithm
Derive network AIG
Call SAT for a remaining pair
17
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

18
Structural Analysis Detect Symmetric Variable
Pairs

• Convert AIGs to Implication Supergates (ISs)
• Introduced in Chang DAC00
• Compute local symmetries for each Implication
  Supergates
• Propagate global symmetries from primary inputs
  to primary outputs

19
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

20
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

21
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

22
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
23
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
24
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
25
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
26
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
27
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

AIG
e
a
d
b
c
Represents 2-input AND gate
28
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

Implication supergate
AIG
s1
e
a
b
c
e
s2
a
d
b
c
Represents 2-input AND gate
Represents multi-input AND gate
29
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached

Implication supergate
AIG
s1
e
a
b
c
e
s2
d
a
d
b
c
Represents 2-input AND gate
Represents multi-input AND gate
30
AIG ? Implication Supergates

• Expand the AND gate until a PI or complemented
  edge is reached
• Recover lost structural symmetry in AIG

Implication supergate
AIG
s1
e
a
b
c
e
s2
d
s3
a
d
b
c
b
c
Represents 2-input AND gate
Represents multi-input AND gate
31
Structural Symmetry Detection
IS

• Divide the fanins of each IS into three
  categories
• - Positive PIs
• - Negated PIs
• - Other ISs

s1
e
a
b
c
s2
d
s3
b
c
32
Structural Symmetry Detection
IS
2. Identify initial candidate symmetric pairs
for each IS - Pair up variables in positive and
negative PIs separately
s1
e
a
b
c
s2
d
s3
b
c
33
Structural Symmetry Detection
IS
3. Keep a symmetric pair if the variables -
not in the support of other IS or - symmetric
for each fanin in other IS
s1
e
a
b
c
s2
d
s3
b
c
34
Structural Symmetry Detection
IS
4. Propagate the symmetric pairs to its parent
IS if - the symmetry holds for at least one
fanin of the other IS, and - the variables are
not in the support for other fanins
s1
e
a
b
c
s2
d
s3
b
c
35
Summary Structural Analysis

• Detect symmetry in one sweep over the circuit
• Discover most easy structural symmetries
• Simpler, faster and more scalable than Wang
  ICCD03
• We only compute classical symmetries

36
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

37
Simulation Detect Non-symmetric Variable Pairs

• Random guided simulation
• Performance random simulation until saturation
• Guided simulation uses SAT counter examples and
  their distance-1 patterns
• E.g. Distance-1 patterns of 011 are 111,
  001, 010

38
Why Distance-1 Patterns?

• For a realistic and sparse function
• Realistic appear in practical circuit
• Sparse many 0s and fewer 1s or vice versa

39
Why Distance-1 Patterns?

• For a realistic and sparse function
• Random simulation often fails

40
Why Distance-1 Patterns?

• For a realistic and sparse function
• Random simulation often fails
• SAT counterexamples reach unlikely values

41
Why Distance-1 Patterns?

• For a realistic and sparse function
• Random simulation often fails
• SAT counterexamples reach unlikely values
• Distance-1 patterns represent its neighborhood,
  contain other unlikely values

42
SimulationDetect Non-symmetric Variable Pairs

• Random guided simulation
• Bit-Parallel computation
• Simulating one AIG node involves bit-wise
  operations
• Simulate 32 or 64 bits simultaneously

1010
43
SimulationDetect Non-symmetric Variable Pairs

• Random guided simulation
• Bit-Parallel computation
• One vector targets all variable pairs
  simultaneously

44
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
ab
cd
00
01
11
10
00
0
0
0
1
01
0
0
1
1
(4) Simulate the above patterns
11
0
0
1
1
10
0
0
0
1

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
45
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
cd
00
01
11
10
00
0
0
0
1
01
0
0
1
1
(4) Simulate the above patterns
11
0
0
1
1
10
0
0
0
1

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
46
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
1
cd
00
01
11
10
00
0
0
0
1
01
0
0
1
1
(4) Simulate the above patterns
11
0
0
1
1
10
0
0
0
1

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
47
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
1
cd
00
01
11
10
1
00
0
0
0
1
01
0
0
1
1
(4) Simulate the above patterns
11
0
0
1
1
10
0
0
0
1

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
48
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
1
cd
00
01
11
10
1
00
0
0
0
1
0
01
0
0
1
1
(4) Simulate the above patterns R 0110
11
0
0
1
1
10
0
0
0
1

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
49
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
1
cd
00
01
11
10
1
00
0
0
0
1
0
01
0
0
1
1
(4) Simulate the above patterns R 0110
11
0
0
1
1
10
0
0
0
1
(5) Check variable pairs with the same polarity
in P a, c, a, d, c, d

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
50
Simulation Example
F(a, b, c, d) ab d (ac bc)
(3) Flip the bits on the diagonal
0011 1111 1001
1010
0
ab
1
cd
00
01
11
10
1
00
0
0
0
1
0
01
0
0
1
1
(4) Simulate the above patterns R 0110
11
0
0
1
1
10
0
0
0
1
(5) Check variable pairs with the same polarity
in P a, c, a, d, c, d

• Suppose the vector P is 1011.

(2) Create the bit matrix a b c d
1 0 1 1 1 0 1 1 1 0
1 1 1 0 1 1
(6) non-symmetric pairs a, c, c, d
51
Summary Simulation

• Employ both random and guided simulation
• More effective than earlier approaches
• Random simulation detects easy non-symmetric
  pairs
• Guided simulation detects hard-to-find
  non-symmetric pairs
• Target all variable pairs simultaneously and use
  bit-parallel computation
• More efficient than earlier approaches

52
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

53
Transitivity AnalysisDetect Symmetry/Non-symmetr
y

• Infer new symmetric/non-symmetric status based on
  existing symmetry information
• Bit-wise operations on the rows and columns of
  the bit-matrix representing symmetry information

54
Transitivity Analysis Rule 1

• Infer new symmetric pairs

x1, x2 variables A, B sets of variables
55
Transitivity Analysis Rule 2

• Infer new non-symmetric pairs

x1, x2 variables A, B sets of variables
56
Transitivity Analysis Rule 3

• Infer new non-symmetric pairs

x1, x2 variables A, B sets of variables
57
Summary Transitivity Analysis

• Performed each time a symmetry/non-symmetry
  status is obtained
• Infer additional symmetric and non-symmetric
  pairs

58
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

59
Boolean Satisfiability (SAT)Detect
Symmetry/Non-symmetry

• Check if F01 F10
• Equivalent, variables are symmetric
• Not equivalent, not symmetric, return
  counter-examples
• The combinational equivalence checker (CEC) is
  based on Improvements to combinational
  equivalence checking
• Mishchenko, IWLS06/ICCAD06
• Significant performance improvement over other
  SAT solvers

60
Reducing SAT Calls

• SAT is expensive, even with improvements

Structural Analysis
Random Guide Simulation
61
Summary SAT

• The underlying CEC is very efficient
• Many techniques used to reduce the number of
  needed SAT calls

62
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• SAT
• Transitivity analysis
• Experimental results
• Conclusions

63
Experiments Setup

• Implemented in ABC A system for sequential
  synthesis and verification
• Brayton, Mishchenko, et al.
• Ran on a Pentium 4 computer
• 1.6Ghz CPU, 1Gb Ram

64
Goals for Experiments

• Analyze the effectiveness of different techniques
• Demonstrate the benefit of simulation and SAT
  integration
• Illustrate the performance improvement

65
Exp 1 Contribution of Different Techniques
66
Exp 1 Contribution of Different Techniques

• Of all symmetric variable pairs in MCNC suite
• Structural analysis 70
• Transitivity analysis 21
• SAT 9
• Of all non-symmetric variable pairs in MCNC suite
• Simulation Transitivity analysis 99.7
• SAT 0.3

67
Exp 2 Effectiveness of tight integration
Call SAT for a remaining pair
Derive network AIG
Perform guided Simulation
Y
Our Algorithm
68
Exp 2 Effectiveness of tight integration
Call SAT for a remaining pair
Derive network AIG
All variable pairs processed?
N
Vanilla flow
69
Experimental Results Effectiveness of SIM/SAT
integration
70
Experimental Results Effectiveness of SIM/SAT
integration
71
Exp 3 Performance Comparison Proposed vs. BDDs

• Out of 24 large benchmarks
• BDDs could not be constructed for 5 benchmarks
• Proposed method was faster than BDDs Mishchenko,
  TCAD03 in 10 benchmarks
• 3.6x faster
• Proposed method was slower in 9 benchmarks
• 1.6x slower

72
Outline

• Motivations
• Background
• Our approach to symmetry computation
• Structural analysis
• Simulation
• Transitivity analysis
• SAT
• Experimental results
• Conclusions

73
Conclusions

• Enabled complete symmetry computation onlarge
  designs
• Structural analysis and simulation were keyto
  its performance
• Simulation and SAT integration improved
  performance an order of magnitude
• Lay the foundation for computing other functional
  properties using similar approach
• Linear cofactor relationships
• Higher-order symmetries

74
Future Work

• Extend the current hybrid approach to other
  functional properties
• Detect functional properties of sequential
  circuits
• Develop efficient methods for NPN-equivalence
  checking of large Boolean functions without BDDs

75
Thank you!