Fig. 2: Conjunctive normal form gain computation.
a Considered 4th-degree CNF Boolean function, i.e., 4-SAT problem. b–e The main idea of the gain computing shows (b) crossbar memory array implementation and (c–e) three in-memory computing operations. The approach is similar to the polynomial pseudo gradient computation. Specifically, sums of clause literals are first computed similarly to the inverted monomial approach (Supplementary Fig. S4a) in the forward vector-by-matrix multiplication pass, as shown in panel (c). The sum values are compared to clause orders in backward steps (d) and (e) to identify break and make type clauses. Unit inputs, that can be scaled by clause factor when clauses in CNF are weighted (e.g., used in weighted SAT problems), are applied for the identified break and make clauses in the backward vector-by-matrix multiplication. The results are then properly gated at the periphery to compute make and break values for each variable. The corresponding variable gains (i.e., negative pseudo partial derivatives) are found by subtracting the outputs of two backward passes, as shown in panel (a). Values shown in blues correspond to x1 = 1, x2 = 0, x3 = 1, x4 = 0, i.e., the same example of assignment as in Fig. 1. Also note that the considered 4-SAT problem is equivalent to the polynomial function in Fig. 1a when assuming a1 = 1, a2 = −1, a3 = 1, and a4 = − 1.
