ELEC 7770
Advanced VLSI Design
Spring 2007
Logic Equivalence
Vishwani D. Agrawal
James J. Danaher Professor
ECE Department, Auburn University
Auburn, AL 36849
[email protected]
http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr07
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Equivalence Checking
 Definition: Establishing that two circuits are

functionally equivalent.
Applications:
 Verify that a design is identical to specification.
 Verify that synthesis did not change the function.
 Verify that corrections made to a design did not create
new errors.
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Compare Two Circuits
a
a
c
f
b
c
b
f
 Graphs isomorphic?
 Boolean functions identical?
 Timing behaviors identical?
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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ATPG Approach (Miter)
Circuit 1
(Verified design)
Circuit 2
(Sythesized or
modified design)
stuck-at-0
stuck-at-0
 Redundancy of a stuck-at-0 fault, checked by ATPG, establishes
equivalence of the corresponding output pair.
 If the fault is detectable, its tests are used to diagnose the
differences.
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Difficulties with Miter
 ATPG is NP-complete
 When circuits are equivalent, proving

redundancy of faults is computationally
expensive.
When circuits are different, test vectors are
quickly found, but diagnosis is difficult.
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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A Heuristic Approach
 Derive V1, test vectors for all faults in C1.
 Derive V2, test vectors for all faults in C2.
 If the combined set, V1+V2, produces the same

outputs from the two circuits, then they are
probably equivalent.
Reference: V. D. Agrawal, “Choice of Tests for
Logic Verification and Equivalence Checking
and the Use of Fault Simulation,” Proc. 13th
International Conf. VLSI Design, January 2000,
pp. 306-311.
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Example Circuit C1
x1
C1
x2
x3
x4
Tests
C1 = x1 x3 x4 + x2 x3 + x2 x4
x1
1
1
1
1
x3
1
1
x2
1
1
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ELEC 7770: Advanced VLSI Design (Agrawal)
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Example Circuit C2
x1
x2
C2
x3
x4
Tests
C2 = x1 x3 x4 + x2 x3 + x2 x4
x1
1
1
1
1
x3
1
1
x2
1
1
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ELEC 7770: Advanced VLSI Design (Agrawal)
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C1 ≡ C2
Tests
x1
1
1
1
1
x3
Tests
1
1
1
x2
x1
1
1
1
1
x3
1
1
1
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x2
1
1
x4
x4
C1
C2
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C2’: Erroneous Implementation of C2
x1
x2
C2’
x3
x4
Tests
x3
C2’ = x1 x2 x3 x4 + x2 x3 + x2 x4
C2 = x1 x3 x4 + x2 x3 + x2 x4
x1
1
1
1
1
1
1
1
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x2
minterm
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Incorrect Result: C1 ≡ C2’
C1 = x1 x3 x4 + x2 x3 + x2 x4
Tests
x1
1
1
1
1
C2’ = x1 x2 x3 x4 + x2 x3 + x2 x4
x3
Tests
1
1
1
x2
x1
x3
1
1
1
1
1
1
1
1
x4
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s-a-0
Additional Safeguard
C1
(Verified design)
0
s-a-1
C2
(Sythesized or
modified design)
 Simulate V1+V2 for equivalence:
 Output always 0
 No single fault on PI’s detected
 Still not perfect
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Probabilistic Equivalence
 Consider two Boolean functions F and G of the same set



of input variables {X1, . . . , Xn}.
Let f = Prob(F=1), g = Prob(G=1), xi = Prob(Xi=1)
For any arbitrarily given values of xi, if f = g, then F and G
are equivalent with probability 1.
References:
 J. Jain, J. Bittner, D. S. Fussell and J. A. Abraham, “Probabilistic
Verification of Boolean Functions,” Formal Methods in System
Design, vol. 1, pp 63-117, 1992.
 V. D. Agrawal and D. Lee, “Characteristic Polynomial Method for
Verification and Test of Combinational Circuits,” Proc. 9th
International Conf. VLSI Design, January 1996, pp. 341-342.
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Simplest Example
 F = X1.X2,
 G = X1+X2,


f = x1 x2
g = (1 – x1)(1 – x2)
= 1 – x1 – x2 + x1 x2
Input probabilities, x1 and x2, are randomly
taken from {0.0, 1.0}
We make a wrong decision if f = g, i.e.,
x1x2 = 1 – x1 – x2 + x1 x2
or
x1 + x2 = 1
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Probability of Wrong Decision
x2
Randomly
selected
point (x1,x2)
1.0
x1 + x2 = 1
0
1.0
x1
Probability of wrong decision
= Random point falls on line {x1 + x2 = 1}
= (area of line)/(area of unit square)
=0
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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Calculation of Signal Probability
 Exact calculation
 Exponential complexity.
 Affected by roundoff errors.
 Alternative: Monte Carlo method
 Randomly select input probabilities
 Generate random input vectors
 Simulate circuits F and G
 If outputs have a mismatch, circuits are not

equivalent.
Else, stop after “sufficiently” large number of vectors
(open problem).
Spring 07, Feb 8
ELEC 7770: Advanced VLSI Design (Agrawal)
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References on Signal Probability
 S. C. Seth and V. D. Agrawal, “A New Model for

Computation of Probabilistic Testability in
Combinational Circuits,” INTEGRATION, The
VLSI Journal, vol. 7, pp. 49-75, 1989.
V. D. Agrawal and D. Lee and H. Woźniakowski,
“Numerical Computation of Characteristic
Polynomials of Boolean Functions and its
Applications,” Numerical Algorithms, vol. 17, pp.
261-278, 1998.
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ELEC 7770: Advanced VLSI Design (Agrawal)
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More on Equivalence Checking
 Don’t cares
 Sequential circuits
 Time-frame expansion
 Initial state
 Design debugging (diagnosis)
 Reference: S.-Y. Hwang and K.-T. Cheng,
Formal Equivalence Checking and Design
Debugging, Springer, 1998.
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ELEC 7770: Advanced VLSI Design (Agrawal)
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