Presented By Anna Fariha Roll : SN – 213 MS, 1 st Semester Session: Boolean Matching

Dept of CSE, University of Dhaka2 Presentation Outline  Equivalence of Functions  Negation of Input Variables  Permutation of Input Variables  Negation of Output  Classes of Functional Equivalence  Boolean Matching  Boolean Matching Algorithms  Boolean Signature  Signature of Function  Unateness Property  Symmetry Property  Size of On-Set  Signature of Variables  Cofactor Signature  Cofactor Statistics and Component Signature  Single Fault Propagation Weight Signature and Partner Patterns  Flow Chart for Boolean Matching Algorithm using Boolean Signature  Application of Boolean Matching

Dept of CSE, University of Dhaka3 Considered under three cases:  Negation of Input Variables.  Permutation of Input Variables.  Negation of Output. 3 Equivalence of Functions

Dept of CSE, University of Dhaka4 n X = {x 1, x 2, x 3,…, x n }  Set of Boolean input variables. n f (X) and g (X)  Denotes two Boolean functions of X. n  (X)  Maps each x i to itself or its complement. n If g (X) = f (  (X))  Then we say f (X) and g (X) are equivalent functions. 4 Negation of Input Variables (1/2)

Dept of CSE, University of Dhaka5 Example:  X = {x 1, x 2, x 3 }  f (X) = x 1 + x 2 + x 3  g (X)= x 1 + x 2 + x 3   Mapping: x 1 x 2 x 3  g (X) = f (  (X) ), so f (X) and g (X) are equivalent functions.

Dept of CSE, University of Dhaka6 6 Permutation of Input Variables (1/2) n X = {x 1, x 2, x 3,…, x n }  Set of Boolean input variables. n f (X) and g (X)  Denotes two Boolean functions of X. n  (X)  Maps each x i to any x i. n If g (X) = f (  (X))  Then we say f (X) and g (X) are equivalent functions.

Dept of CSE, University of Dhaka7 Permutation of Input Variables (2/2) Example:  X = {x 1, x 2, x 3 }  f (X) = x 1 x 3 + x 2 + x 3 x 1  g (X)= x 3 x 2 + x 1 + x 2 x 3   Mapping: x 1 x 3 x 2 x 1 x 3 x 2  g (X) = f (  (X) ), so f (X) and g (X) are equivalent functions.

Dept of CSE, University of Dhaka8 8 Negation of Output n X = {x 1, x 2, x 3,…, x n }  Set of Boolean input variables. n f (X) and g (X)  Denotes two Boolean functions of X. n If g (X) = f (X) or g (X) = f (X)  Then we say f(X) and g(X) are equivalent functions. n Example:  X = {x 1, x 2, x 3 }  f (X) = x 1 x 2 x 3  g (X)= x 1 + x 2 + x 3  g (X) = f (X), so f (X) and g (X) are equivalent functions.

Dept of CSE, University of Dhaka9  N-equivalent  Equivalent under input Negation.  P-equivalent  Equivalent under input Permutation.  NP-equivalent  Equivalent under input Negation, input Permutation.  NPN-equivalent  Equivalent under input Negation, input Permutation, output Negation. 9 Classes of Functional Equivalence

Dept of CSE, University of Dhaka10 Boolean matching is defined as the problem of determining whether a Boolean function can be functionally equivalent to another one under a permutation of its inputs and complementation of some of its inputs or complementation of itself.

Dept of CSE, University of Dhaka11 n X = {x 1, x 2, x 3,…, x n }  Set of Boolean input variables. n Y = {y 1,y 2, y 3,…, y n }  Set of Boolean input variables. n f (X)  Denotes Boolean function of X. n g (Y)  Denotes Boolean function of Y. n  (X) (mapping from X to Y)  Maps each x i to a unique y j or y j. g (Y) = f (  (X) ) (or f (  (X) ) 11 Boolean Matching (2/2) Total number of mappings 2 n x n! x 2

Dept of CSE, University of Dhaka12  Canonical Forms  Burch et al. -“Efficient Boolean Function Matching”.  Wu et al. -“Efficient Boolean Matching Algorithm for Cell Libraries”.  Debnath et al. - “Fast Boolean Matching Under Permutation Using Representative”.  Ciric et al. - “Efficient Canonical Form for Boolean Matching of Complex Functions in Large Libraries”.  Hinsberger et al. - “Boolean Matching for Large Libraries”.  Based on  Truth table of the function.  Table look-up.  Drawbacks  Space complexity.  Applicable for library cells with seven or fewer input variables.

Dept of CSE, University of Dhaka13  Boolean Signatures  Mohnke et al. - “Permutation and Phase Independent Boolean Comparison”.  Based on  Signature of function.  Signature of variable.  Signatures invariant to  Input permutation.  Input negation.  Output negation.  Signature used as necessary condition for equivalence.  Drawbacks:  Fails to conclude a unique correspondence of variables of the benchmark circuits.

Dept of CSE, University of Dhaka14  Signature Based Canonical Form  Abdollahi – “Symmetry Detection and Boolean Matching Utilizing a Signature-Based Canonical Form of Boolean Functions”.  Based on  Generalized signatures.  Variable symmetries.  Handles combinational functions.  No limitation on the number of input variables.  Spectral Methods  Clarke et al. – “Spectral Transforms for Large Boolean Functions With Applications to Technology Mapping”.  Miller - "A Spectral Method for Boolean Function Matching“.

Dept of CSE, University of Dhaka15  Signature of function.  Signature of variable.  Signatures invariant to  Input permutation.  Input negation.  Output negation.  Signature used as necessary condition for equivalence. 15 Boolean Signature

Dept of CSE, University of Dhaka16 l Unateness Property. l Symmetry Property. l Size of On-Set. 16 Signature of Function

Dept of CSE, University of Dhaka17 Example g(X) = x 1 x 2 x 4 + x 1 x 2 x 5 + x 1 x 3 x 6 + x 1 x 3 x 7  Total variables  n = 7 (x 1, x 2, x 3, x 4, x 5, x 6, x 7 ).  Binate variables (having both complemented and uncomplented values)  b = 3 (x 1, x 2, x 3 ).  Unate variables (having either of the complemented or uncomplented values)  u = 4 (x 4, x 5, x 6 x 7 ).  u = n – b. 17 Unateness Property (1/2)

Dept of CSE, University of Dhaka18  Matched functions must have the same number of unate variables.  Speed-up of phase assignment  Transforming negative unate variables to positive ones by adding inverters.  All unate variables being positive unate  Phase assignment required for binate variables only.  Speed-up of permutation  Unate variables not being associated to binate variables. 18 Unateness Property (2/2)

Dept of CSE, University of Dhaka19 Symmetry Property (1/2)  Symmetry Set  A set of variables that are pair wise interchangeable without affecting the logic functionality.  Symmetry Class  A symmetry class is an ensemble of symmetry sets with the same cardinality.  Matched functions having the same symmetry classes.  Variables in different symmetry sets not being associated.

Dept of CSE, University of Dhaka20 Symmetry Property (2/2) Example g (X) = x 1 x 2 x 3 + x 4 x 5 + x 6 x 7  3 Symmetry Sets 1.{x 1, x 2, x 3 } 2.{x 4, x 5 } 3.{x 6, x 7 }  2 Symmetry Classes 1.C 3 = {{x 1, x 2, x 3 }} 2.C 2 = {{x 4, x 5 }, {x 6, x 7 }}  Signature [|C 1 |, |C 2 |, |C 3 |, |C 4 |, |C 5 |, |C 6 |, |C 7 |] = [0,2,1,0,0,0,0]

Dept of CSE, University of Dhaka21  Set of minterms for which a function f (X) evaluates to 1.  Matched functions must have the same size of on - set.  Size of on-set, onsize (f), is denoted by |f|.  Computed on Ordered Binary Decision Diagram (ROBDD).  Output complementation changes the satisfy count of an n-input function f from |f| to 2 n - |f|. 21 Size of On-Set

Dept of CSE, University of Dhaka22  Cofactor Signature.  Cofactor Statistics and Component Signature.  Single Fault Propagation Weight Signature (SFP) and Partner Patterns. 22 Signature of Variables

Dept of CSE, University of Dhaka23  Cofactor signature of a variable x t for function f : onsize (x t  f x t ) : 0-branch : 1-branch xtxt x t  f x t Cofactor Signature

Dept of CSE, University of Dhaka24 l A sorted list of onsize (x t x j  f x t x j ) for all x t in the input set xtxjfxtxjxtxjfxtxj 01 xtxt xjxj 24 Cofactor Statistics and Component Signature

Dept of CSE, University of Dhaka25 l (onsize (x t  f x t  f x t ), onsize (x t  f x t  f x t ), onsize (f x t  f x t ) ) xtxt xtxt xtxt xtxt Single Fault Propagation Weight Signature and Partner Patterns

Dept of CSE, University of Dhaka26 Flow Chart for Boolean Matching Algorithm using Boolean Signature Function f and g Compute Signatures of Functions Equal? Not Matched No Compute Signatures of variables Yes Equal? Equivalence of f and g ? Matched Not Matched Not Matched Yes No

Dept of CSE, University of Dhaka27  Technology Mapping  Matching of complex gates.  Exploiting implicit don’t care.  Logic Verification  When input correspondence is not given.  Verify correctness of a logic circuit.  Logic Synthesis  Cell library binding process. 27 Application of Boolean Matching

Dept of CSE, University of Dhaka28 Boolean matching is based on exact equivalence checking techniques that find a match whenever possible. Various kinds of algorithms exist for Boolean matching. An overview of Boolean Signature based algorithm is discussed.

Dept of CSE, University of Dhaka29 [1] L. Benini and G. De Micheli, “A survey of Boolean matching techniques for library binding,” ACM Trans. Design Automation of Electronic Systems, vol. 2, no. 3, pp. 193–226, July [2] J. R. Burch and D. E. Long, “Efficient Boolean function matching,” in Proc. Int. Conf. on Computer-Aided Design, pp. 408–411, Nov [3] Q. Wu, C. Y. R. Chen, and J. M. Acken, “Efficient Boolean matching algorithm for cell libraries,” Proc. IEEE Int. Conf. on Computer Design, pp. 36–39, Oct [4] D. Debnath and T. Sasao, “Fast Boolean matching under permutation using representative,” Proc. ASP Design Automation Conf., pp. 359–362, Jan [5] J. Ciric and C. Sechen, “Efficient canonical form for Boolean matching of complex functions in large libraries,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 22, no. 5, pp. 535–544, May [6] D. Debnath and T. Sasao, “Efficient computation of canonical form for Boolean matching in large libraries,” Proc. ASP Design Automation Conf., pp. 591–596, Jan [7] U. Hinsberger and R. Kolla, “Boolean matching for large libraries,” Proc. Design Automation Conf., pp. 206–211, [8] J. Mohnke and S. Malik, “Permutation and phase independent Boolean comparison,” Integration - the VLSI journal, 16: pp , [9] E.M. Clarke et al., "Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping", Proc. Design Automation Conference, pp , [10] D.M.Miller, "A Spectral Method for Boolean Function Matching", Proc. European Design and Test Conference, pp. 602, [11] A. Abdollahi and M. Pedram, "Symmetry Detection and Boolean Matching Utilizing a Signature-Based Canonical Form of Boolean Functions,“ Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions, pp 1128 – 1137, on June [12]

Dept of CSE, University of Dhaka30 ?