Lattice-Based Computation of Boolean Functions Mustafa Altun and Marc Riedel University of Minnesota

Switch-based Boolean computation Shannon’s work: A Symbolic Analysis of Relay and Switching Circuits(1938)

1D and 2D switches

A lattice of 2D switches 3 × 3 2D switching network and its lattice form

Boolean functionality and paths  Switches are controlled by Boolean literals.  f L evaluates to 1 iff there exists a top-to-bottom path.  g L evaluates to 1 iff there exists a left-to-right path.

Logic synthesis problem How can we implement a given target Boolean function f T with a lattice of 2D switches? Example: f T = x 1 x 2 x 3 +x 1 x 4

Logic synthesis problem Example: f T = x 1 x 2 x 3 +x 1 x 4 +x 1 x 5 9 TOP-TO-BOTTOM PATHS!

Our synthesis method Example: f T = x 1 x 2 x 3 +x 1 x 4 +x 1 x 5 f T D = (x 1 +x 2 +x 3 )(x 1 +x 4 )(x 1 +x 5 ) f T D = x 1 + x 2 x 4 x 5 + x 3 x 4 x 5  Obtain the dual of f T.  Assign each product of f T to a column.  Assign each product of f T D to a row.  Compute an intersection set for each site.  Arbitrarily select a literal from an intersection set and assign it to the corresponding site.

Our synthesis method

Math behind the method – Theorem 1 Theorem 1 allows us to only consider column-paths. We do not need to enumerate all paths!

Math behind the method – Theorem 2 Theorem 2 explains the relation between intersection sets and column-paths. Each column is for each product!

Our method’s performance Area of the lattice: m×n The time complexity: O(m 2 n 2 ) n and m are the number of products of the target function f T and its dual f T D, respectively.

Future work  We are investigating our method’s applicability to different technologies.  We are studying the applicability of the Theorems to the famous problem of testing whether two given monotone Boolean functions are mutually dual.

Thank you!