J. Paul Roth

Minimization over Boolean graphs

This paper presents a systematic procedure for thed esign of gate-type combinational switching circuits without diredted loops. Each such circuit (Boolean graph) is in correspondence with a sequence of decompositions of the Boolean function which it realizes. A general approach to functional decomposition i s given and, in terms oaf convenient positional representation, efficient tests for the detection of decompositions are derived. These results are employed in the development of an alphabetic search procedure for determining minimum-cost Boolean graphs which satisfy any given design specifications.

Algebraic topological methods for the synthesis of switching systems, part III: minimization of nonsingular Boolean trees

An algorithm is given for solving a general problem in combinational switching-circuit minimization theory. The circuits considered consist of a disjunction (OR-ing together) of trees of any set of logical elements, with the restriction that in any given tree no input appears more than once. To each logical element is attached a positive cost. A method is presented for designing a minimum-cost circuit of this variety for any given logical function. Two parallel treatments are given, one viewing it as an abstract mathematical problem, the other considering it as an engineering problem.

Algorithms for logical design

A series of “extraction algorithms” are presented that are applicable to the design of a wide class of combinational circuits. A special operation, the #-product, is introduced for carrying out these algorithms. Complete algorithms for the design of minimal and near-minimal 2-level AND-OR circuits are given, utilizing the extraction algorithms and the #-product. (These algorithms have been programmed for the International Business Machines Corporation (IBM) 704 and are available through the SHARE Library. Data are shown on the usage of these programs by IBM circuit designers.) Indication is also given as to how the extraction algorithms may be applied to a broader class of problems, namely to the design of disjunctions of nonsingular circuits.