Abstract: The automatic synthesis of Boolean switching functions by adaptive tree networks is discussed. The concept of heuristic responsibility, by means of which parts of a tree become specialized to certain subsets of input vectors, is explained. Applications to pattern recognition and optical character recognition (OCR) problems are described.

Abstract: An extension of a result by Grigoryev is used to derive a lower bound on the space-time product required for integer multiplication when realized by straight-line algorithms If S is the number of temporary storage locations used by a straight-line algorithm on a random-access machine and T is the number of computation steps, then we show that (S+1)T ⩾ Ω(n2) for binary integer multiplication when the basis for the straight-line algorithm is a set of Boolean functions

Abstract: In this paper, we consider the size of combinational switching networks required to synthesize monotone Boolean functions using only operations from the functionally incomplete set of primitives {disjunction, conjunction}. A general methodology is developed which is used to derive Q(n log n) lower bounds on the size of monotone switching circuits for certain bilinear forms (including Toeplitz and circulant matrix-vector products, and Boolean convolution), certain routing networks (including cyclic and logical shifting), and sorting and merging. A homomorphic mapping technique is also given whereby the lower bounds derived on the sizes of monotone switching networks for Boolean functions can be extended to a larger class of problem domains.

Abstract: An efficient technique is presented for inverting the minimal paths of a reliability logic diagram or fault tree, and then minimizing to obtain the minimal cuts, or else inverting the minimal cuts for the minimal paths. The method is appropriate for both s-coherent and s-noncoherent systems; it can also obtain the minimized dual inverse of any Boolean function. Inversion is more complex with s-noncoherence than with s-coherence because the minimal form (m.f.) is not unique. The result of inversion is the dual prime implicants (p.i.'s). The terms of a dual m.f., the dual minimal states, are obtained by a search process. First the dual p.i.'s are obtained; then a m.f. is found by an algorithmic search with a test for redundancy, reversal-absorption (r.a.). The dual p.i.'s are segregated into the ``core'' p.i.'s [8,9] essential for every m.f. and the ``noncore'' p.i.'s, by r.a. Then a m.f. is found by repeatedly applying r.a. to randomized rearrangements of the noncore terms. Examples are included, adapted from the fault-tree literature.

Abstract: Given a monotone (nondecreasing) switching function F(x 1 ,···,x n ), its prime implicants are the minimal infeasible points, i.e., the minimal solutions to F(x) = 1. A monotone F is regular ifany "right shift" of a feasible point is again feasible. The roofs of a regular function F are those prime implicants al ofwhose right shifts are feasible. The set of these roofs completely determines F. An algorithm is presented to compute the roofs of the dual Boolean function Fd= F(x) This computation is needed, for example, in the synthesis problem ofthreshold logic. The algorithm "scans" all the 2npoints in lexicographical order, skipping over intervals which are clearly roof-free. The amount of this work is proportional to the number of prime implicants of F. Encouraging computational experience is reported.

Abstract: Formula size and depth are two important complexity measures of Boolean functions. We study the tradeoff between those two measures: We give an infinite set of Boolean functions and show for nearly each of them: There is no monotone formula computing it optimal with respect to both measures. We give a lower and upper bound on the product of size and depth of monotone formulae computing our functions. That implies, moreover, a logarithmic lower bound on circuit depth.

Abstract: Recently Delobel and Parker have shown that Multivalued dependencies (MVDs) may be represented as Boolean switching functions, in much the same way as Functional dependencies (FDs) can be represented as Boolean implications. This permits all FD and MVD inferences to be made as logical (Boolean) inferences, a significant plus because the FD/MVD inference axioms are fairly complex. This paper reviews some of the basic implications of this result and outlines new applications in FD/MVD membership testing, generation of dependency closure, cover, and keys, and testing for lossless and independent decompositions.

Abstract: Synchronous combinational complexity, a measure of the size of logic circuits without races, is investigated in this paper. The first author has presented a method for obtaining an $O(n\log n)$ lower bound to synchronous combinational complexity and has shown that this bound applies to “almost all” Boolean functions in n variables. However, he could not constructively exhibit functions to which the lower bound applied (although Wolfgang Paul did produce an example). In this paper we weaken and extend the hypothesis of the lower bound so that a larger class of functions satisfies it and apply it to the determinant and marriage functions of $GF(2)$.

Abstract: In this central chapter I show how a system amenable to boolean description can be analyzed in terms of a set of logical equations. Each equation relates, for any time, the values of a function a, b, c,...(associated with the state, on or off, of a gene, of a chemical reaction, etc.), to the values of input variables, and of memorization variables α, β, γ,...(associated with the presence or absence of the product of a gene, of a chemical reaction, etc.). Time is present in a similar way as in differential equations; in fact, in our logical equations a, b, c,... play essentially the role of the time derivatives of α, β, γ. From the set of logical equations describing a system, one can derive its stable steady states (if any), the pathways (temporal sequences of states) and the conditions determining which pathway will be followed.

Abstract: The stochastic behavior of digital combinational circuits is analyzed by the use of Walsh functions. An n-input Boolean function is represented as a Walsh series and the error caused by noise is measured in terms of a distance which is the fraction of the time that the system output due to noise-corrupted signal differs from that due to signal alone. It is shown that the error can be expressed as the sum of two parts: one part depends only on noise statistics, and the other on both signal and noise. Some interesting properties of both parts are discussed and typical examples are given.

Abstract: This paper presents methods for iterative evaluation of Boolean expressions suitable, e. g., in sequential search and matching when the input variables appear in sequence. The sum-of-product form is assumed and the clue is to partially compute the values of all products partially for each new variable being defined. It is shown that the additive use of an expression for the complementary function can result in a faster evaluation. The method proves to be extendable to multilevel expressions.

Abstract: A network N considered here is a graph (G,E) whose edge e is weighted by I(e) = [α(e), β(e)]. An admissible flow is a flow f satisfying α(e) ⊆ f (e) ⊆ β(e), for all e ⊂ E. A necessary and sufficient condition for existence of admissible Boolean flows in N, and characterization of maximum Boolean flows are presented. Then a property of maximum Boolean flows which are the terminal capacities of two-terminal Boolean flows satisfy a Boolean triangular inequality is shown. Finally the concept of γ-relativeness of a symmetric Boolean mapping m for any threshould γ⊆ S which induces a cluster decomposition of the node set V, by introducing Boolean network metrics is presented.

Abstract: A learning program produces, as its output, a boolean function which describes a concept. The function returns true if and only if the argument is an object which satisfies the logical expression in the body of the function. The learning program's input is a set of objects which are instances of the concept to be learnt. A compiler/interpreter has been written which performs the reverse of the learning process. The concept description is regarded as a program which defines the set of objects which satisfy the given conditions. The interpreter takes as its input, a predicate and produces as its output, an object which belongs to the set.

Abstract: The existing methods for the simplification of Boolean functions are usually classified according to their mathematical background (e.g. consenus method) or their procedural tools (e.g. diagram methods) or are named in honour of scientists (e.g. Quine-McCluskey method). To make the methods more transparent, with respect to the judgment of their effectiveness in solving practical problems, it is suggested that they be classified according to the strategy of prime implicant generation. It is shown that the known methods can be assigned to one of the following basic strategies for the generation of prime implicants: (a) building-up from smaller implicants; (b) expansion of the function along literals or variables; (c) separation of the vector space into 1-subspaces and O-subspaces; (d) systematic inspection of all possible implicants; (e) heuristic methods. The strategy is shown to be a governing factor for the effectiveness of minimisation procedures, particularly if, owing to the complexity of the problems, only approximately minimal solutions can be obtained.

Abstract: A theorem of Bade proves that for a complete Boolean sublattice S£ of 3>(3if) the following holds: SS {PeS£"; P is orthogonal projection operator} We prove that this theorem does not hold for the physically interesting class of non-Boolean propositional systems embedded in a 3>(3if); we derive however a necessary and sufficient condition under which the theorem does hold This condition is automatically satisfied if the propositional system is Boolean

Abstract: The development of Boolean calculus for its application to developing digital system design methodologies that would reduce system complexity, size, cost, speed, power requirements, etc., is discussed. Synthesis procedures for logic circuits are examined particularly asynchronous circuits using clock triggered flip flops.

Abstract: Boolean algebra is the basic mathematic used in the analysis and synthesis of binary systems. Technological advances have led to an increasing interest in multivalued logic systems where more than two logical values are used. In the application of multivalued logic, each logical value could often be represented by a Boolean vector, ie a vector with binary components (0 or 1). Therefore, it is quite important to have a thorough understanding of properties embedded in the algebraic structures using Boolean vectors as basic operands. This gives the motivation for this study. Boolean vector operations are introduced and two major modes associated with the complement are characterized. Then we define a Boolean vector E-algebra and its major features are given. Finally, some applications are discussed and illustrated.

Abstract: Two Boolean functions, f ( x 1 ,…, x n ) and g ( x 1 ,…, x n ) are said to be affine equivalent if g can be written as: g = f o σ + l , where: (1) σ is a permutation and/ or complementation of the input variables of f ; and (2) l is a linear combination (or the negation of a linear combination) of some of the input variables of f . Affine equivalence is an equivalence relation. A general method for counting the number of affine equivalence classes (affine families) on n variables is presented, providing new results for n = 5 and n = 6, and verifying existing results for values of n ⩽ 4.

Abstract: The spectral representation of a Boolean function is analysed with particular concern for the distribution of the information required in testing for the existence of certain fundamental types of symmetry. It is found that this distribution is not even, the high-order end of the spectrum being examined more frequently than the low-order end. This observation gives some basis for the view that the spectrum of a function may characterise the complexity of the function's realisation.

Abstract: An overlapping equation written with the exclusive-OR operator is used for the determination of the irredundant mod-2 sums-of-products expressions of a logical function

Abstract: Recently, an algorithm has been developed for deriving optimal NAND array realizations of complete Boolean functions [1]. The algorithm has two defects. It often requires a cumbersomely large amount of computations and does not handle incomplete functions. The short-cut method presented here is free from those defects.