# Detection of Disjuncts of Switching Functions and Multi-level Circuit Design†

TL;DR: A method of detection of component functions of fewer number of variables, disjunct of which forms a given switching function, which yields bettor economy in several cases with regard to number of logical circuit elements.

Abstract: This paper describes a method of detection of component functions of fewer number of variables, disjunct of which forms a given switching function. The multi-level electronic circuit synthesized from the component functions yields bettor economy in several cases with regard to number of logical circuit elements compared with the case when the same function is synthesized from the ‘ minimal’ two-stage forms.

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TL;DR: A major revision of the paper "Minimal Boolean Expressions with ~[ore than Two Levels of Sums and Products," presented at the Third Annual Symposium, Chicago, October 1962 is presented.

Abstract: An approach to the problem of multilevel Boolean minimization is described. The conventional prime implicant is generalized to the multilevel case, and the properties of multilevel prime implicants are investigated. A systematic procedure for computing multilevel prime implicants is described, and several examples are worked out. I t is shown how ,'absolutely minimal\" forms can be obtained by carrying out multilevel minimization to a sutSciently large number of levels. 1. [ntroduclion Boolean forms cart be classified by the number of levels of sums and products they contain; e.g. a sum of products or a product of sums has two levels, whereas a sum of products of sums has three. There are a number of well-known methods for finding minimal two-level forms for a given function. However, there has been comparatively little progress in the development of methods for finding minimal forms with more than two levels; some references to prior work are appended to this paper. This paper describes an approach to the problem of finding minimal N-level forms and \"absolutely minimal\" forms. The techniques described are applicable to any and all Boolean functions, and are suitable for automatic computat ion. The approach generalizes two-level minimization procedures, and it is assumed that the reader is familiar with these methods. 2. Definitions and Notation Attention is restricted to Boolean forms in which only individual letters are complemented. Definition 1. (1) Expressions composed of single literals, e.g. \"x,\" \"2,\" \"y,\" a~ld \"~,\" are the only zero-level forms. (2) If q~l, ¢2, . • • , Op (P >1 ) are N-level forms, then q~l V 02 V .. ~ / ~ p is a sum of N-level forms and ( ~ ) /x (¢~) /~ • . . /~ (q~p) is a product of N-level forms. (3) Sums of N-level forms and products of N-level forms are the only (N + 1)-level forms. For brevity, we refer to N-level forms as N-forms, and sums of N-level forms a~d products of N-levels forms as sN-forms and pN-forms. I t is very important This constitutes a major revision of the paper, \"Minimal Boolean Expressions with ~[ore than Two Levels of Sums and Products,\" presented at the Third Annual Symposium ~r~ Switching Circuit Theory and Logical Design, AIEE Fall General Meeting, Chicago, October 1962. The revision was supported by Air Force Contract 33(657)-7811. t Information Systems Laboratory, Department of Electrical Engineering. 283 Journal of the Association for Computing Machinery, Vol. 11, No. 3 (July, 1964}, pp. 283-295 284 EUGENE L. LAWL:EI{ to :':ole tha% by defir~itio~, every N-forrn is an (N@l)-form, being both an s.V.,form m~d a pN-fonn. Examph:s: gX).forms: ~.~, v.gw. dl~forms: u, u V w V 2 . plof(: ~sn.s: ~q ui)w, \"u 'fl w V 2, ~x(v ',./ ~'), (u V O)(v V @). :,£..forms: u, uf~v, u V w ' , /2, u(v V 0), (u V ~)(v V go), l~oolca:'~ fu::ctio:m are deplored by the letters f, g, and h, and canonical ~rn~s of fimctions by their dechnal equivalents. (The numbering of the eanonicai tf~rms is :~lch that ~,):;c is assigned the number 1. ) The function having i, j, • . . , ]c a:~,~ em~onical terms is dermted by the letter f, g, or h with the supe~cript~ i, j, , • , I.', i.e. f<~\"'~, gc.i,...,k, or h < ~ ' .k Boobm~ hmctiol~s are also defined by Boolean forms. For exampl% the function f defin~i by the form c \\ / ~ is specified by writing f ( z , y ) = z V ~. To disgi:guish between %rms and functions, the function defined by a given Boolean form ~h is de::oted by t ¢ I. Thus, f = I:~Vgl. (Occ~skm~dly we abuse this notation by omitting the absolute value sig:~ if the :mm,~i~~g i~ obvkms,) A~ ineompt(dely spec{fied Boolean function, or incomplete hmetion, is representM by a~ ft, all minimal aN-forms for (f0, h) are contained in P. (2) P is a necessary set of prime sN-implieants for (f0, ft) if and only if for every h ~ f~, at least one minimal aN-form for (f0, h) is contained in P (if such a form exists). THEOREM 2'. Let P be a su~icient set of prime s( N -1) irnplicants for (fo , f~). Then every minimal pN-form for (fo , fl ) is a product of forms contained in P. THEOREM 3'. Let P be a necessary set of prime s( N -1)-implicants for (fo , fl ). Then there exists at least one minimal pN-form for (fo , f~) that is a product of forms contained in P. 4. Computing Necessary and Sufficient Sets of Prime Implicants The proposed approach to minimization is recursive: 2-level minimization is the basis for 3-level minimization, 3-level minimization is the basis for 4-level minimization, and so forth. I t is apparent that this recursive approach will be impracticable if it is necessary to carry out pN-minimization for all possible (h, f~) to compute either a necessary or a sufficient set of prime pN-implicants for (fo, f~). To avoid this difficulty, we shall make use of the following lemmas, which follow immediately from the definitions. LB

88 citations

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07 Oct 1962

TL;DR: This paper presents a new approach to the problem of multi-level Boolean minimization, extended to minimal expressions with more than three levels and to "absolute" minimal expressions.

Abstract: This paper presents a new approach to the problem of multi-level Boolean minimization. Conventional two-level minimization methods are embedded in a process that makes extensive use of the properties of incompletely specified functions. Particular attention is given to the problem of obtaining minimal sums-of-products-of-sums. Then the approach is extended to minimal expressions with more than three levels and to "absolute" minimal expressions. Several examples are worked out.

5 citations

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TL;DR: In this paper, a simple and straightforward procedure for finding absolute minimal third-order expressions (in the sum-ofproduct-of-sum) of a special class of Boolean functions called unate functions is suggested.

Abstract: A simple and straightforward procedure for finding absolute minimal third-order expressions (in the ‘ sum-of-product-of-sum’ forms) of a special class of Boolean functions called unate functions is suggested in the paper. The central idea developed through the procedure involves a decomposition of the assigned Boolean function first into a group of sub-functions called maximal uniliteral sub-functions (MTJL's) each of which is realizable in a minimal second-order ‘ product-of-sum ’ form and then a selection of an appropriate sub-set of these maximal uniliteral sub-functions or MUL's (or their sub-functions) in order to cover all the prime implicants of the function minimally.

2 citations

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TL;DR: In this article, a map technique for identifying symmetrizable functions is presented, which greatly reduces the work in ascertaining symmetricity, and it is unique in being also applicable to completely or incompletely specified functions which are the complement of a function of type (i).

Abstract: This paper presents a new map technique for identifying symmetrizable functions. The technique greatly reduces the work in ascertaining symmetricity, and it is unique in being also applicable to completely or incompletely specified functions which: (i) Contain imbedded symmetrizable function(s). (ii) Are the complement of a function of type (i). (iii) Contain an imbedded function of type (ii). Discussion of the technique and its extensions is included.

1 citations

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