Abstract: A new method is presented for determining all minimal dependence sets, irredundant normal forms, and irredundant normal forms of minimal dependence sets of a Boolean function f. The method reduces the above problems into those of determining all minimal positive dependence sets, irredundant positive normal forms, and irredundant positive normal forms of minimal positive dependence sets, respectively, of a Boolean function f* corresponding to f. For each problem a corresponding cover formula is developed such that the prime implicants of it are in one-to-one correspondence to all possible solutions.

Abstract: This paper describes a technique for generating the minimal cuts from the minimal paths, or vice versa, for s-coherent systems. The process is a recursive 2-stage expansion based upon de Morgan's theorems; ie, it is the inversion of a Boolean polynomial having all common-valued (either all 0 or all 1) components, so that the inverse also has only common-valued components of the opposite sign. There are procedural short cuts and Quine-type absorptions; absorptions put the polynomial into its minimalized form. The number of stages of recursion is equal to the number of terms (minimal states) in the starting polynomial. The minimal states of the inverse form are the terms of the inverse polynomial after minimalization. Since the system is s-coherent and all components are common-valued in either the original or Inverse minimal forms, the lists of minimal states are unique.

Abstract: This paper describes an algorithm for minimizing an arbitrary Boolean function. The approach differs from most previous procedures in which first all prime implicants are found and then a minimal set is then determined. This procedure imposes a set of conditions on the selection of the next prime implicant in order to obtain a near minimal sum-of-products realization. Extension to the multiple output and incompletely specified function cases is given. An important characteristic of the proposed procedure is the relatively small amount of computer time spent to solve a problem, as compared to other procedures. The MINI algorithm may give better results for a large number of inputs and outputs if relatively few product terms are needed. This procedure is also well suited to find a solution for programmable logic arrays (PLA's) which internally implement large Boolean functions as a sum-of-products.

Abstract: A BRIEF exposition of the theory of local algorithms for simplifying the disjunctive normal forms of Boolean functions is presented. A proof of the essential nature of all the conditions used in the construction of one of the algorithms is presented, and its majorantness is proved.

Abstract: A new approach is presented for the minimization of the word length of the control store in microprogrammed computers. Given the set of microinstructions and the set of microcommands of a microprogrammed computer a minimal subset of microcommands are determined so that every other microcommand not contained in this minimal subset to be generated from it by a single AND or OR gate. The problem of finding such minimal subsets of microcommands is formulated as a Boolean function simplification problem and is based upon an extension of the notion of the minimal dependence sets of a Boolean function.

Abstract: Boolean functions can be realized using multiplexer elements by exploiting Shannon's expansion theorem. The cost of the realization often depends on the choice of the expansion variables versus the residue variables. A method employing Ashenhurst's decomposition charts [6] is described which assists in visualizing the residue functions. The method is effective for six variable functions or less.

Abstract: An extension of a methodology based on Boolean equations to fault detection in modular combinational networks is described. The resulting method allows for the use of cataloged module tests, and it is applicable under the single-faulty-module constraint. The conceptual and computational framework is rather simple, and it is independent of any particular data structure for representing Boolean functions.

Abstract: Fan-out-free networks of AND, OR, NOT, EXOR, and MAJORITY gates are considered. Boolean functions for which such networks exist are defined to be fan-out free. The paper solves the following problems regarding the fan-out-free networks and functions.

Abstract: Transform methods and dyadic groups have been used for the classification of Boolean functions as well as for prime implicant determination. In a recent paper a prime implicant extraction method, based on Walsh-Hadamard transform methods, was presented. It processes the true minterms of the function separately, one at a time. In this paper this transform method is applied to the covering problem. Taking the prime implicants as binary variables a slight modification of the prime-implicant extraction method allows one to identify all complete covers by inspecting the elements of an inverse transform. Redundant forms can be detected and rejected easily. Another method for the determination of all irredundant covers classifies the 2m elements of the dyadic group of element length m as incomplete, redundant or irredundant covers, m beingthe number of prime implicants. A version for hand-worked problems is given, as well as a computer-oriented version.

Abstract: The derivation is given of a simple compact form for evaluating the spectra for the sum and product of two functions in terms of their individual spectra alone. The method avoids the reintroduction of the transform. The results are extended to include the exclusive-OR function and are easily applied for the computation of the spectra of more complex Boolean functions in terms of their individual spectra.

Abstract: This correspondence considers the efficiency of some algorithms for the evaluation of monotonic Boolean functions. It is shown that algorithms based on the criterion of maximizing the local information gain about the Boolean function with n variables may sometimes require a number of computational steps which is n/log n times the computational steps of the optimal algorithm.

Abstract: Circuit size and depth are two important complexity measures for a Boolean function. Uniform hierarchies are shown to exist with respect to each of these measures.

Abstract: The minimization algorithm of Gimpel realizes a minimal TANT network for any Boolean function under a NAND gate cost criterion. A TANT network is a three-level network composed of AND-NOT (i.e., NAND) gates, having only true (i.e., uncomplemented) input variables.

Abstract: The construction of a decimal grouping table and its use to determine essential and nonessential prime implicants for the minimisation of an n-variable Boolean function are presented in the paper. In existing tabular methods, e.g. the Quine-McCluskey technique, each fundamental product is represented by a row of binary 1s and 0s and the finding of a set of prime implicants necessitates the formation of successive tables of binary characters, and only after an exhaustive search in the tables can one discover any prime implicants. Dealing with binary characters is rather tedious, and searching through several tables to establish a prime implicant is time consuming. The proposed grouping table offers the convenience of using decimal minterm numbers and the advantage of using one table in the search for prime implicants. In the grouping table the decimal equivalent of function terms appear in a column and the entries to a row corresponding to a function term N are the decimal equivalent of product terms related to N by one change of variable. From these entries, only those terms which appear in the function under investigation are selected and only these need to be considered for the minimisation of the problem. Thus, unlike other tabular methods, the grouping table provides all possible combinational terms for each fundamental product term as its row terms and also the facility of at-a-glance comparison of all function terms by referring to the same table. In the paper, a method of minimising Boolean functions with the aid of grouping table is illustrated with examples.

Abstract: This paper contains two results: (1) an algebraic characterization of the lattice of faces of the n-cube, based upon axioms independent of the dimension n. The resulting algebraic structure is suited for application to synthesis problems for Boolean functions; (2) explicit construction of the partition of the lattice of faces of the cube into a minimal number of chains, based upon a new bracketing algorithm.

Abstract: THE PROBLEM of finding a monotonic Boolean function best approximation a specified partial (not defined everywhere) Boolean function, is solved by a flow algorithm. Among the monotonic functions giving the best approximation, the function possessing the simplest disjunctive normal form is chosen.

Abstract: MONOTONIC Boolean functions possessing in some layer a fixed number of unit values are considered. An algorithm for finding the maximum upper zero of such functions is presented. It is compared with an algorithm optimal for the whole set of monotonic functions. It is known that many extremal problems reduce to the problem of searching for the maximum upper zero of a monotonic Boolean function. In [ 11 this problem was solved in the Shannon formulation. Namely, first a monotonic Boolean function of n variables was presented for which it is impossible to solve this problem in less than Ck” “I + 1 steps. Secondly, an algorithm Br is constructed, which for an arbitrary Boolean function of n variables finds its maximum upper zero after not more than CLn”’ + 1 steps. Since in the general case this problem requires lengthy calculations, it appeared of interest to distinguish some subclasses of monotonic Boolean functions for which it can be solved more quickly. In this paper we consider a number of such subclasses and present an algorithm, which on operating with these subclasses guarantees a smaller number of steps than the algorithm B1 described in [ 11. We consider the set Ez n of all binary numbers of length n. We say of two numbers &= (a,, , a,)

Abstract: An algorithm for the generation of an optimal code applicable to a parsed Boolean expression is presented. The algorithm generates the code during parsing itself.

Abstract: A theory relating certain labeled directed graphs and Boolean expressions over propositional variables is seen to have several interesting applcations in computer science, most notably to programming, compiling, and switching theory. The applications arise from the capability of realizing any Boolean expression by a member of the class of digraphs here considered, those we shall call "atomic."

Abstract: Under the premise that there is a tradeoff between the amount of protection within a computer system and the system cost, this paper investigates some theoretical measures of protection within systems. A system simply consists of any number of active elements, called subjects, that make use of passive elements, called objects. The investigation is restricted to an analysis and comparison of access mechanisms defined by a family of boolean functions. Some definitions are stated, and some theorems are proved that are valid for all mechanisms within the family considered. Algorithms are presented for the optimal assignment of access codes to subjects are objects for unstructured systems and for several types of structured systems. It is proven that for a very general class of systems, the optimal assignment will still allow at least (n/2)(γ- 1) unauthorized accesses to objects, wheren is the number of subjects, and γ is the largest integer not greater than the quantityn divided by the number of access classes.

Abstract: A simple algorithm for finding multiple fault test sets in combinational logic networks is presented. The method is based on finding boolean differences and hence test sets using augmented boolean matrices. An example is given which illustrates the procedure.

Abstract: An algorithm is presented which evaluates an arbitrary boolean expression in terms of a small instruction set and which often generates fewer instructions than other methods the author has seen. The algorithm tells what code to generate whenever a boolean operator, a branch test, a boolean assignment statement, or a conditional expression is encountered. One's complement arithmetic is assumed throughout.