Symmetry Detection in Boolean Functions

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Symmetry Detection in Boolean Functions

• Stacey Balamucki
• Dan Steffy
• Advisor Dr. Debnath

2
Outline

• Problem Description
• Applications
• Background
• Method
• Implementation
• Results
• Conclusions

3
Project Objective

• Our goal was to develop a system to detect
  symmetry between variables in Boolean functions

4
Applications

• More efficient algorithms for circuit design
• Applications in Boolean matching
• Building BDDs (Binary Decision Diagrams)
• Digital Circuit Testing
• Circuit Verification

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What is Symmetry?

• A Boolean function is said to be totally
  symmetric if it remains invariant under any
  permutation of its variables
• A Boolean function is said to be partially
  symmetric around some subset of its variables if
  it is invariant with respect to any permutation
  of those variables

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Checking for Symmetry

• Let f be a Boolean function with inputs x and y
• To check for symmetry between x and y in f,
  create a new function g which is identical to f
  except that x and y are switched
• If we find that f g then we may conclude that f
  is symmetric with respect to x and y

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Equality of Boolean Functions

• Given two n-variable Boolean functions f and g,
  it is not easy to check if f g
• There are 2n different inputs, so checking each
  one is impossible for large values of n
• This is where we can turn to Boolean
  Satisfiabilty for help

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Exclusive-OR

• Let us define the operator ? as Exclusive-OR,
  which means P ? Q is true if and only if P or Q
  is true, but P and Q are not both true

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Using XOR (cont.)

• If f g, then f ? g will always be false since
  (true ? true) (false ? false) false
• If f ? g then there must exist some input X such
  that f (X) ? g (X), in which case f (X) ? g (X)
  would be true
• Therefore f ? g will always be false if and only
  if f g

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Fixing Variable Assignments

• For some variables x,y, let f10(X) represent
  f(X) with x,y fixed as 1,0
• For x,y, let f01(X) represent f(X) with x,y
  fixed as 0,1
• For example
• f01(X) f ( ,0,1,)
• f10(X) f ( ,1,0,)

x y
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Looking for Symmetry

• Checking to see if f10(X) f01(X) is sufficient
  to detect symmetry
• In order to use this method we must be able to
  check equality of Boolean functions efficiently

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SAT Solver and Exclusive-OR

• We may now use the Exclusive-OR function with a
  SAT solver to find symmetry
• Send f10? f01 to the SAT solver
• The function f10? f01 is unsatisfiable if and
  only if f10(X) f01(X)
• This is true if and only if f is symmetric with
  respect to x,y

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SAT Solver Specifications

• We use zChaff, currently one of the best SAT
  solver according to the SAT 2003 Competition
• zChaff requires that input is in a special form
  known as CNF form (conjunctive normal form)

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File Formats

• Benchmarking samples and the problems that we
  tested were not in CNF form
• We developed a method to convert general Boolean
  functions in blif format into CNF form
• We have implemented our method in C

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Blif to CNF

• Converting the blif file to CNF is a somewhat
  difficult task
• Our program successfully converts a blif file
  into CNF format

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Blif Format

• This is an example of a blif file
• .model ABC
• .inputs a b c
• .outputs d e
• .names a b d
• 10 1
• - 1 1
• .names b c e
• 11 1
• 0 - 1
• .end

• Each line that starts with .names starts another
  logic block

a
b
d
a/a
b
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CNF Format

• CNF format consists of several literals ORed
  together inside clauses that are ANDed together
• f (a,b,c,d) (a d)(c d)(a b c)

clause
literal
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Reading Blif File

• The program uses two buffers (Word, Buffer) to
  grab one line at a time in the file
• We do some comparisons with the first word of the
  line to see what kind of line it is (e.g. input,
  output, names)
• When we find a match we send it to the
  corresponding IF statement

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Compare Word Buffer
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Storing Variables

• We store all the variables in a list and organize
  it between input, output, and temp variables.

A (INPUT)
B(OUTPUT)
C (TEMP)
Variable is inserted into the array as either an
input, output or temp variable
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Converting to CNF

• The logic blocks within the BLIF file represent
  logical relationships between the variables
• These logical relationships such as equality,
  AND, and OR can be converted into CNF form
• Each block is looked at one by one and converted
  into CNF representation

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CNF Formulas for Simple Gates
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Example Blif to CNF

• Each logic block of the BLIF file represents a
  disjunction of conjunctions
• y (ab b)
• First we assign a temporary variable for each
  conjunction
• y x1 x2
• Where
• x1 ab
• x2 b
• Then convert each simple statement into CNF form

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Example (cont.)

• Next we use the conversion formulas on the
  previous table and make clauses
• y x1 x2
• (x1 x2 y)(x1 y)(x2 y)
• x1 ab
• (a b x1)(a x1)(b x1)
• x2 b
• (b x2)(b x2)

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Exclusive-OR

• Once our function is in CNF form we must generate
  another CNF function to represent f10? f01 for
  each pair of variables that need to be checked
• Once in this format, f10? f01 can be sent to
  zChaff to test if it is satisfiable

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Generating f10? f01

• Method for generating f10? f01
• Create CNF formula to represent f10 and f01 by
  making two copies of f and fixing x, y in each of
  them as 1,0 and 0,1
• Use an Exclusive-OR gate on the outputs of each
  of these functions
• Convert the Exclusive-OR gate into CNF form
• p ? q ( p q ) ( p q )
• Combine everything into one CNF function

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Generating f10? f01
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Calling zChaff

• Once we have generated f10? f01 we can send it to
  zChaff to check for Satisfiabilty
• zChaff requires CNF format
• The standard format used to represent CNF
  functions is called DIMACS format

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Transitivity of Symmetry

• Symmetry of variables in Boolean functions is
  transitive
• This means that if a function f is symmetric with
  respect to variables x,y, and it is also
  symmetric with respect to variables y,z then we
  can conclude it is also symmetric with respect to
  x,z

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Use of Transitivity

• We may use the property of transitivity to our
  advantage when checking pairs of variables
• For example, if we have already determined that f
  is symmetric with respect to x,y and y,z then
  we need do not check x,z for symmetry

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Advantage of Using Transitivity

• Suppose the function f has n variables and is
  totally symmetric
• If we take advantage of transitivity we only need
  to check n-1 pairs of variables for symmetry to
  find out if the function is totally symmetric
• Checking all possibilities would take
  n(n-1)/2 tries which could be much larger than n-1

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Experimental Results
For more than one output we took the sizes of
the first output
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Comparison to GRM Results
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Conclusions

• Use of Exclusive-OR and Satisfiabilty is a
  promising method for detecting symmetry
• Initial results are comparable on many instances
  with leading methods
• Further refinement of this method and combination
  with other methods could solve previously
  unsolvable instances

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Future Work

• Optimization of code and integration with other
  SAT solvers
• Use of search reduction techniques to reduce the
  number of variables checked for symmetry
• Improved methods of converting blif to CNF form
  that will generate less clauses and temporary
  variables
• Improved methods of generating f10? f01 using
  less clauses and variables

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Acknowledgements

• National Science Foundation (NSF)
• REU Program
• Dr. D. Debnath
• Dr. I. K. Sethi, Dr. F. Mili, Dr. J. Li
• Mr. B. Clark
• Dr. L. Zhang

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References

• T.H. Cormen, C. E. Leiseron, and R. L. Rivest,
  Introduction to Algorithms. The MIT Press, 1998.
• S. R. Das, and C. L. Sheng, On Detecting Total
  or Partial Symmetry of Switching Functions,
  IEEE Transactions on Computers, pp.352-355, March
  1971.
• B.- G. Kim and D.L. Dietmeyer, Multilevel Logic
  Synthesis of Symmetric Switching Functions, IEEE
  Trans. Computer-Aided Design, vol. 10, No. 4,
  Apr. 1991.
• T. Larrabee, Test Pattern Generation using
  Boolean Satisfiabilty, IEEE Trans.
  Computer-Aided Design, vol. 11, no. 1, Jan. 1992.
• J. P. Marques-Silva, and K. A. Sakallah, Boolean
  Satisfiabilty in Electronic Design Automation,
  ACM/IEEE Proc. Design Automation Conference, pp.
  675-680, 2000.

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References (cont.)

• M.W. Moskewicz, C.F. Madigan, Y. Zhao, L. Zhang,
  and S. Malik, Chaff Engineering an Efficient
  SAT Solver, 39th Design Automation Conference,
  Las Vegas, June 2001.
• C.- C. Tsai, and M. Marek-Sadowska, Generalized
  Reed-Muller Forms as a Tool to Detect Symmetry,
  IEEE Trans. Computers, vol. 45, no. 1, Jan. 1996.
• T. Sasao, Switching Theory for Logic Synthesis.
  Kluwer Academic Publishers, 1999.