Showing papers on "Boolean function published in 1969"

Practical problems in the use of computers in medical diagnosis

Abstract: For the computer to assist in the medical diagnosis and treatment processes, a series of systematic mathematical algorithms must be developed that correspond in some sense to the reasoning used by the diagnostician. In terms of the propositional calculus of symbolic logic, the patient presents a symptom profile, G, which is a Boolean function of possible symptoms, either present or absent or undetermined. The diagnosis, which is also a Boolean function, f, of possible diseases, is found from the fundamental formula of medical diagnosis, E → (G → f), where E is the Boolean function representing medical knowledge. An example of an elementary computation is given to illustrate the process of solving the fundamental formula. In cases where complete symptom information is unavailable, the computer can apply Bayes' formula, which correlates the conditional probability of having a disease complex given a certain symptom profile with that of having the given symptom profile given the disease complex and the total probability of the patient's having the disease complex. A simple illustrative example is given of the application of Bayes' formula. An "uncertainty principle" exists in the collection of statistics for determining the total probability, due to the time lag in processing data. A method is given for handling data from patients presenting incomplete symptom profiles. When medical knowledge is so voluminous as to make impractical the recording of all possible symptom-disease combinations, then techniques developed in pattern-recognition studies can be utilized. The least-probable symptom complex will be omitted, and the loss can be compensated for by comparing the patient's given symptom profile with "weighted" possible symptom complexes. Again, an illustrative example is given. After the probabilities for possible alternative diagnoses have been determined, the treatment plan must be determined. We distinguish three basic kinds of problem: 1) treatment decisions under certainty, which involve simple value-theory considerations and linear programming; 2) treatment decisions under risk, where the probabilities of alternative diagnoses are known, which involve optimization of a mathematical expectation; and 3) treatment decisions under uncertainty, where the probabilities of the alternative diagnoses are unknown, which are amenable to the application of "game theory." Such decisions all involve value determinations, which can be broken down into tangible and intangible values. When alternative diagnoses remain, the response of the patient to the treatment chosen is a further symptom, which can be used in reevaluating the diagnosis. In such cases the techniques of dynamic programming may be applied. The problem of determining the transition probabilities of a patient's moving from one state of health to another during a certain course of treatment presents difficulties, and a method for alleviating the problem is presented, utilizing Bayes' formula to process current patient charts instead of relying on past record searches. Conditional probabilities can be further utilized in the comparison of two treatments. Finally, a brief survey is given of applications of computer-diagnosis aids as reported in the literature.

The Characterization and Properties of Cascade Realizable Switching Functions

Abstract: This paper is a study of switching functions realizable by a single cascaded switching network composed of two-input, one-output elements where a distinct variable is applied to one input and the other receives the output of the previous element. A special type of Boolean formula called a standard cascade form is introduced with the property that all cascade realizable functions, and only these, can be written in this form. This characterization leads to a strong necessary condition on such functions: there is no consensus for any pair of its prime implicants which are, therefore, all core. It also leads to a new, efficient procedure for testing an arbitrary function f for cascade realizability. The test operates on the prime implicants of f, and yields a realization employing a particular complete set of cell types from which it is especially easy to derive any other realization using any other complete set.

Logical Instruments for Teaching Logical Design

Abstract: A way of teaching combinatorial logic is presently based on experimentation with physical models of logical relations ("logical instruments"). The Marquand chart [7] originated in 1881 is described and discussed. A graphical representation of a Boolean function on that chart is processed by experimental application of "implication masks." As an example of such a processing, the paper describes the construction of a minimal ??-form of a given Boolean function (by respecting DON'T-CARE conditions). The main part of the minimization algorithm is based on a sufficient condition of extension (of the expression (9) towards the minimal), which is satisfied if a certain "necessary condition of extension" does hold and if a certain "sufficient condition of exclusion" does not. To save processing time, a weight is assigned to each point on the chart where the Boolean function must be 1 (true). Then the minimization algorithm is applied, with a preference to points with smaller weights. Logical punch cards are used as a logical instrument for experimental evaluation of weights.

An Approach for the Realization of Threshold Functions of Order r

Abstract: In this paper we shall study the problem of nonlinear separation. As usual, a Boolean function F of n binary variables, x l ,..., x n , x i Σ{1,0}, i=1,..., n, will be represented by a set of vertices C in the n-dimensional Euclidean space Enwhere each vertex has n binary-valued components, {1,0}.

Multigate Synthesis of General Boolean Functions by Threshold Logic Elements

Abstract: A decomposition and reconstruction approach for synthesizing an arbitrary Boolean function with a minimum number of threshold logic elements connected by feedforward paths only is presented. Attention is mainly focused on cascade-type realizations. The approach has the advantage that near-minimal solutions are readily derived. An estimate of how closely the minimality has been approached is obtainable in this method. The method has been successfully applied by the authors to Boolean functions of 5 and 6 variables.

On Mod-2 Sums of Products

Abstract: An algorithm is provided for obtaining any Boolean function as a modulo-2 sum of products containing only uncomplemented variables. The proofs, which verify the algorithm and show the uniqueness of the results, are simple and free of special mathematical notation.

Unateness Test of a Boolean Function and Two General Synthesis Methods Using Threshold Logic Elements

Abstract: This paper presents an elaboration of the unateness test of a Boolean function based on the definition of the so-called simple zero or unit transition of arguments of the given Boolean function. It is proved that the unateness of the given Boolean function is subject to the necessary and sufficient condition that there must be no occurrence of both simple unit transition and zero transition at the same time. Whether this condition is satisfied or not can easily be seen from the truth table of the given function.

Realization of Boolean functions with a large number of unspecified states by a single threshold logic element

Abstract: A systematic method of testing and realization of Boolean functions with a large number of unspecified states by a single threshold logic element has been presented in this paper. By this method, it is possible to determine many different threshold realizations of the given incomplete function. The method can be applied to switching functions containing any number of variables.

Approximate Methods of Solving Logical Problems

Abstract: In the formulation of logical problems it is frequently required not only that a solution be obtained, but also that this solution be the very best solution, in some sense or other. For example, in synthesizing a logical circuit one might be required to obtain a scheme which not only realizes a given system of Boolean functions, but which at the same time contains a minimal number of elements; in seeking proofs to theorems, one wishes to find not only correct proofs, but also the simplest of the correct proofs; in analyzing chess positions one wishes not only to find a winning sequence of moves, but to choose the most elegant of the winning sequences. In each such problem, it is possible to introduce some quality function, and then seek an exact solution which corresponds to an extremum of the quality function thus introduced.

Minimizing the Number of Arguments of Boolean Functions

Abstract: Let f(X) be a Boolean function, X ≡ {x0, x1, ..., xn−1} the set of its arguments, and X the set of all subsets of X.

Efficient realization of boolean functions by pruning nand(nor) trees

Abstract: : A combinatorial tree structure composed entirely of NAND(NOR) blocks is pruned in a non-exhaustive fashion to yield minimal or near-minimal networks. It is assumed that complemented variables are not available and that there are no fan-in or fan-out limitations. The cost of a network is taken as being primarily determined by the number of logic blocks with the number of inputs and logic levels as secondary factors. The pruning algorithm lends itself to both hand methods and machine computation, although the synthesis procedure has not been programmed. Of the 68 nondegenerate functional equivalence classes of 3 variables, the minimum number of blocks results in 63 cases; only one more block in excess of the minimum is required in each of the other 5 cases.

Special class of symmetric Boolean functions: proof and comment

Abstract: The existence of a special class of totally symmetric functions has been reported in an earlier letter. In this letter, a formal proof substantiating this existence is presented. It is also pointed out that in each ncube there are two such functions which are complements of each other.

Coincidence between Boolean products and its application to third-order simplification†

Abstract: The second-order expressions of Boolean functions can have either sum-of-product or product-of-sum forms For a Boolean function specified in the irredundant sum-of-product form as the disjunction of a number of prime implicants or p terms, groups of these p terms can sometimes be more economically realized in the minimal product-of-sum forms than in the sum-of-product forms To know whether a group of p terms in the irredundant sum-of-product form of the function has a more economic realization in the product-of-sum form, the concept of coincidence between the p terms of the function is introduced in the paper and a number of interesting properties of the function in relation to coincidence are established The coincidence between a pair of p terms in a function is defined as the number of literals occurring as mutually common in their algebraic representations It is next shown that the study of the properties of Boolean functions in relation to coincidence also aids in readily obtaining the economic th

Automation techniques for Large Scale Integration system design

Abstract: Integrated Circuit technology has placed many new demands on logic packaging techniques. In order for Large Scale Integration to succeed, additional computer design concepts must be developed. Designers must move away from unit logic and towards dense packages where many integrated circuits are contained in extremely small areas. This implies optimum utilization of input/output connections, since the package size is strongly dependent on the number of I/O's. This paper describes highly-automated computer techniques which may be applied to selected logical graphs to generate the Boolean functions representing the logical graphs. These functions are cataloged into discrete classes. A set of logic functions are then selected from the complete list of discrete classes. Each logical function can cover one or more of the classes in the list. Thus, a small set of logical functions may be utilized to represent a large list of discrete classes obtained from the original partitioning. These functions can be automatically imposed on the original logic graph for evaluation. The automated system is set up to allow rapid evaluation of parameter changes such as function size, and number of function types. The evaluation of functions leads to a set that may be used as the building blocks for implementation of a machine. The designer is free to implement these building blocks in a way that best fits the given technology. This is so because the automated system did not impose any circuit constraints on the logical design. The variable parameters which are included in the system are the number of logical circuits per function, number of I/O's per function, the logic function size allowed, and true and complement available or not available at the output. Reimplementing the machines with the selected function set the packaging parameters that may be varied include the number of functions in a package, the total number of logical circuits in a package, and the number of input and output ports in that assembly. With these freedoms allowed, a designer is in a position to help determine the technology that should be used for his machine design. He can evaluate the amount of redundancy he is willing to allow, and evaluate whether the redundancy actually saves him money or costs him more. He is able to evaluate the effect of the functional density on the wiring complexity since the computer generates the interconnection count between the packaged functions. In this paper, a large high-performance machine and small low-performance machine were analyzed. A function set was chosen for each machine. Results are shown in terms of function usage, logic circuit usage, and wiring complexity.

Loop-Free Threshold Element Structures

Abstract: The paper deals with the problem of "compound" and "cascade threshold" element "synthesis" of an arbitrary Boolean function from the "multithreshold weight threshold vector" (MTWTV). The above synthesis procedure is presented to reveal a unique feature of the multithreshold weight threshold vector, from which several realizations of threshold element nets can be obtained from one of the multithreshold weight threshold vectors.

Synthesis of Error Correcting Threshold Gate Networks

Abstract: Methods are presented for designing threshold gate networks with error-correcting capabilities. The particular method presented is an extension of the tree method for realizing a Boolean function. A primary realization is a realization for some function on the tree such that either the separating function or the gaps are assigned without knowledge of any other realizations on the tree. It is shown that a realization obtained by the tree method corrects t errors of gates in the system if and only if all primary realizations are selected so that they correct t errors of gates in the primary realizations. Relations are then presented for selecting the primary realizations. By this method, an error can be corrected by the gate immediately following the occurrence of the error or by any other gate whose output is affected by the error.

Threshold logic: A simplified synthesis by a recursive method

Abstract: Despite the numerous threshold logic synthesis methods developed in the past decade, the logic designer is still handicapped by the technological independent characterization of these methods Hence, it has long been the desire for the logic designer to have a simple, efficient threshold-logic synthesis method, analogous to conventional Boolean synthesis using ANDs, ORs, or combinations of ANDs and ORs The scope of this paper is to describe the theoretical basis of a direct and efficient synthesis method for a threshold function(s) from its Boolean expression A set of recursive formulas of weight and threshold is thus derived This method is capable of handling a relatively large number of Boolean variables, whether the Boolean expression is a single threshold function or compound threshold functions The minimal realization of the Boolean threshold function can also be obtained Another noted advantage of this recursive synthesis method is to give a clearer picture by permitting the investigator to examine every intermediate step while the final solution is being attained

Synthesis of Optimal Algorithms for Recognizing Boolean Functions

Abstract: What we understand here as a pattern is a collection of objects with some common property, possessed only by the given collection and distinguishing it from others. An object belonging to some pattern is called a realization of this pattern. In other words, a realization is a concrete embodiment of the pattern. The objects we shall be considering are characterizable by a set of binary characters, R = {r0, r1, ..., rn−1}.

Studies on some aspects of the problem of three-level NAND network synthesis†

Abstract: In this paper some aspects on the realization of a Boolean function by a three-level NAND network have been studied. It has been shown that the problem is an involved one and usually the most economic network is obtained only when suitable redundant prime implicants are allowed along with the prime implicants present in any cover. Since the number of covers associated with any function is large, it is not possible to obtain the most economic network without exhaustive testing. However, it is observed that the problem becomes simpler if one proceeds to obtain the network using the minimum number of gates in the AND level. For most of the problems this gives the absolute minimum network. In this paper a method has been suggested to obtain such a network.

The logical complexity of geometric properties in the plane

Abstract: This paper studies the logical complexity of geometric properties in the plane. These properties are characterized according to the length of formulas necessary to express them. In this study a portion of the plane X will be considered as a collection of squares {x1, ..., xn} arranged into a fine grid. A pattern P is a subset of X and the grid is assumed to be fine enough so that approximations to geometrical figures can be obtained. P can be defined by mapping m: X →{0,1} where for x e X, m(x) = 1 iff x e P. A set of patterns or a geometrical property ℘ can be expressed as a Boolean function f on the n variables x1, ..., xn such that the value of f is 1 under the assignment m iff P e ℘.

The random net which has basic organs realizing parity Boolean functions

Abstract: We briefly review the fundamental concept of pattern recognition by random net proposed in [1] (See [1] for detail). / is the set of n \"input points\". Denoting by π(Λ) the number of elements belonging to a finite set A, we have π(I)=n. Any subset fa I will be called (input) pattern which may be interpreted as a binary 0, 1 sequence of length n. The 2 possible patterns constitute the (input) pattern space F. To say that there are given K categories on F is to say that K probability distributions