Abstract: We introduce a property of boolean functions, called transitivity which holds of integer, polynomial, and matrix products as well as of many interesting related computational problems. We show that the area of any circuit computing a transitive function grows quadratically with the circuit's maximum data-rate, expressed in bit/second. This result provides a precise analytic expression of an area-time tradeoff for a wide class of V.L.S.I. circuits. Furthermore, (as shown elsewhere), this tradeoff is achievable. Thus we have matching (to within a constant multiplicative factor) upper and lower complexity bounds for the three above products, in the V.L.S.I. circuits computational model.

Abstract: This paper suggests a technique for generating the minimal cuts from the minimal paths, or vice-versa, for an s-coherent system. In this method, the path polynomial is expressed in a particular form and then complements of only simpler Boolean functions are required. Examples illustrate the method.

Abstract: Making use of arguments from information theory it is shown that a boolean function can represent multivalued dependencies. A method is described by which a hypergraph can be constructed to represent dependencies in a relation. A new normal form called generalized Boyce-Codd normal form is defined. An explicit formula is derived for representing dependencies that would remain in a projection of a relation. A definition of join is given which makes the derivation of many theoretical results easy. Another definition given is that of information in a relation. The information gets conserved whenever lossless decompositions are involved. It is shown that the use of null elements is important in handling data.

Abstract: A method is presented for generating a single formula involving arbitrary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.

Abstract: A method is given for the evaluation of the Rademacher–Walsh spectra of the Boolean sum, product, and EXCLUSIVE-OR of two functions without reintroducing the Rademacher–Walsh transform. The results are derived using a general coding scheme and depend heavily upon the use of a dyadic convolution operation.

Abstract: The first part of this thesis considers the complexity of Boolean functions. The main complexity measures used are the number of gates in combinational networks and the size of Boolean formulas. The case of monotone realizations, using only the operations of AND and OR, of monotone functions is emphasized. For a particular class of monotone functions, the quadratic functions, the worst-case values for the monotone circuit complexity is shown to be proportional to 2 n /log n. The number of -gates necessary to compute any quadratic function is also analyzed. A technique for deriving bounds on monotone circuit size of threshold functions is applied to the "majority" function, (threshold n/2), to establish a lower bound on its monotone circuit complexity of 3n-0(1). For the function "threshold 2", previously known lower bounds on the number of V-gates required are extended in the case of a circuit which has a minimal number of -gates. As a result, it follows that no monotone circuit for "threshold 2" can simultaneously have both a minimal number of -gates and a minimum number of V-gates. The complexity of combinations of functions on disjoint sets of variables is studied, and a gap between the formula and circuit size of a particular function is given. Finally, we study the effect of allowing negation in a formula for monotonic functions. Examples are given both of functions in which using negations allows more succinct expressions, and functions in which it does not. The second part of this thesis describes an algorithm for computing shortest paths in a graph. These results show that an algorithm originally proposed by Spira for this problem can have slow running time. The lacuna in his algorithm is repaired, and it show to have 0(n (logn) ) average running time over wide classes of graphs as Spira originally claimed. As a special case, a transitive closure algorithm with 0(n logn n) average time is also described.

Abstract: A bistellar graph is defined as a graph whose edge-set can be partitioned into stars so that each vertex is incident to at most two stars. A structural characterization of bistellar graphs is given: they are seen to be closely related to “almost monotone” boolean functions, as well as to injective graphs. It is shown that the maximum independent set problem for bistellar graphs is NP-complete. Implications on the computational complexity of quadratic 0-1 optimization problems are discussed.

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 formula over "and", "or", "negation" computing it optimal with respect to both measures. That implies a logarithmic lower bound on circuit depth.

Abstract: Our efforts to develop an automatic database system conversion facility yielded a powerful, yet simple query language which was designed for ease of conversion. The path expression of this query language is a convenient and appealing notation for describing complex traversals with multiple boolean qualifications. This paper describes the path expression, shows how automatic conversions can be done, introduces the boolean functions as part of the basic path expression, offers four extensions (path macros, implied path, path replacement, and path optimization), and discusses some implementation issues.

Abstract: The recognition of “virtually quadratic” 0–1 optimization problems leads to the study of those graphs (quadratic graphs) whose edge-set can be covered by complete bipartite graphs so that each vertex belongs to at most two such complete bipartite graphs.

Abstract: PURPOSE:To realize a high-speed operation in accordance with increment of the input number by generating the signals of the fixed Boolean function through the internal logic circuit and applying the N-bit binary signal to the 1st and 3rd terminals of the 1st-step vertical connection circuit and the N-bit input signal to the 2nd and 4th terminals each. CONSTITUTION:Vertical connection circuits 10-16 possess 4 units of the input terminal, 2 units of the output terminal and 1 unit of the internal logic circuit respectively and then coupled to one another in multiple steps in order to form a tree shape. And signals GK, EK, jK+1 and eK+1 are applied to the 1st, 2nd, 3rd and 4th input terminals respectively. As a result, the internal logic circuit generates the signals featuring the Boolean functions of GK+1=GKVEgK+1 and EK+1=EK eK+1. And N-bit binary signals gO-gN-1 are applied continuously to the 1st and 3rd terminals of 1st-step vertical connection circuits 10-13, and N-bit input signals eO-eN-1 are applied to the 2nd 4th terminals of circuits and 10-13 respectively. In this way, a high-speed operation becomes possible in accordance with increment of the input number.

Abstract: The properties of polynomic functions on Rn are reviewed. Such functions are then used to establish a general synthesis procedure for arbitrary multivalued switching functions. The relationship with threshold logic and Boolean logic is also explored.

Abstract: An asymptotic estimate is established for the number of equivalence classes of Boolean functions of n variables under transformations of the form f(x) \rightarrow f(Ax+b)+ L(x) , where x=(x_{l}, \cdots ,x_{n}), A is a nonsingular n by n matrix, b is a vector, and L(x) is a linear function.

Abstract: The technique of pseudo-Boolean methods which forms the basis for bivalent (0, 1) programming, has been used for many socio-economic and engineering problems in the past. In this chapter, we review these methods and discuss their applications to the quantized coefficient design of digital filters. Some other applications of these methods are also briefly described.

Abstract: A technique is described which uses the Rademacher-Walsh spectrum to obtain the spectra, and consequently the Boolean form, of the clock-steering functions of bistables. Its usefulness over standard methods is derived from the fact that its simplicity is independent of the number of input variables. Being numerically based, it also is particularly relevant for c.a.d. adoption.

Abstract: This paper was stimulated by discussions which arose at the last NATO-ASI in Urbino and by Caldarola’s new definition of coherency [1]. Moreover, the development of a procedure, for the analysis of a fault-tree constituted by AND-OR-NOT gates [2], showed the necessity of identifying the possible logic structures of actual physical systems.

Abstract: The cases, where ~ and 2~ -1 are composite numbers, have not been completely investigated. For 2~ = 16 a covered system of pairs has been constructed independently by Kotzig and Anderson [8] and for 2~ = 28 by Anderson [9]. The covered cyclic system of order 2~ = 16 of the form (2), mentioned above in a table, is not isomorphic to Anderson's example. The first value of the parameter 2~, for which the existence of covered systems of pairs has not been proved, is equal to 36.

Abstract: This paper proposes a nonlinear feedback shift-register system Rn composed by combining a memory with an n-stage shift-register. A new method for generating a nonlinear binary cyclic shift sequence of length T (1 ≤ T ≤ 2n) which can be generated by the proposed system is discussed using a binary graph Gn corresponding to the state transition of the system Rn. As a first step, the number of sequences is briefly discussed. It is then shown that a sequence of length T that can be generated by the system Rn can also be generated by extending the traditional Scholtz algorithm to determine the non-degenerate cyclic equivalence classes of length T. Then a method of generating a de Bruijn sequence of length 2n without using Boolean function is proposed which corresponds to the Hamilton directed loop in the graph Gn. The method consists of converting the initial n-th order Hamilton directed loop to another n-th-order Hamilton directed loop and requires less memory capacity than the traditional method that determines the next n-th-order de Bruijn sequence by utilizing all of the (n-1)-th order de Bruijn sequences. The method is effective since the conversion procedure is easy. Because of the greater number of sequences generated and the programmable property of the sequence conversion, the method can be expected to be applicable to spread spectrum communication.

Abstract: A synthesis procedure for the p-symmetric Boolean function is presented. The blocks of variables corresponding to the partition of the p-symmetry are first mapped into new sets of variables. The function itself is then synthesised from these new variables. The procedure is systematic and leads to circuit realisations with smaller depths. The procedure is illustrated by examples.

Abstract: This note consists of two independent parts. In the first part the concept of an (m,c)-system for a set of linear forms is introduced, and a lower bound is obtained for the algebraic complexity of the computation of (m,c)-systems on algebraic circuits of a special form. In the second part, the notion of an l-independent set of boolean functions is introduced and a lower bound is obtained for a certain complexity measure for circuits of boolean functions computing l-independent sets. As a corollary it is shown that the standard algorithm for multiplying matrices or polynomials may be realized by a circuit of boolean functions in a way that is optimal with respect to a selected complexity measure.