Logic Synthesis for Switching Lattices

Abstract

This paper studies the implementation of Boolean functions by lattices of four-terminal switches. Each switch is controlled by a Boolean literal. If the literal takes the value 1, the corresponding switch is connected to its four neighbors; else it is not connected. A Boolean function is implemented in terms of connectivity across the lattice: it evaluates to 1 iff there exists a connected path between two opposing edges of the lattice. The paper addresses the following synthesis problem: how should one assign literals to switches in a lattice in order to implement a given target Boolean function? The goal is to minimize the lattice size, measured in terms of the number of switches. An efficient algorithm for this task is presented-one that does not exhaustively enumerate paths but rather exploits the concept of Boolean function duality. The algorithm produces lattices with a size that grows linearly with the number of products of the target Boolean function in ISOP form. It runs in time that grows polynomially. Synthesis trials are performed on standard benchmark circuits. The synthesis results are compared to a lower-bound calculation on the lattice size.

Synthesizing Optimal Switching Lattices

“…We call such a sequence a TB-path. As observed by Altun and Riedel [2012], there is no 4-connected on path between T and B if and only if there is a sequence of 8-connected off cells (Figure 1(d)) that connect L and R. We call such a sequence an LR-path. Figure 2 shows an implementation of (x 1 ∧ x 4 ) ∨ (x 2 ∧ x 3 ).…”

“…The DP construction [Altun and Riedel 2012] computes ISOP formulas for the function ϕ and its dual ϕ , then constructs a lattice with one column for each term in ϕ, and one row for each term in ϕ . While this construction is general, there are drawbacks.…”

“…In this context a "switch" is ideally a very simple device, such as a four-terminal switch [Altun and Riedel 2012]. The four terminals of the switch link to the four neighbours of a lattice cell, so that these are either all connected (when the switch is on) or disconnected (when the switch is off).…”

“…A switching lattice can similarly be equipped with "left" and "right" plates and left-to-right connectivity can be associated with a Boolean function [Altun and Riedel 2012]. This way, a single lattice can implement two different functions, although not simultaneously.…”

“…Altun and Riedel [2012] propose a synthesis method for switching lattices-henceforth referred to as the dual-product construction, or DP-as follows. For the target function ϕ, find an irredundant sum-of-products (ISOP) form ψ of ϕ.…”