Abstract
A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.
Highlights
Network C consists of two subnetworks A and B, where subnetwork A has the single input X and the output Z(X), while network B has the two vectorial inputs Z(X) and Y and the output s(Z, Y), which is exactly the same as the output t(X, Y) of network C
The Type-2 Problem is, the inverse problem of logic. This problem was treated extensively by nineteen-century logicians such as Jevons (1872, 1874), Venn (1894) and Poretsky (1898). Though interest in this problem faded away for more than half a century, it witnessed a revival at the hands of pioneers of modern digital design including Ledley (1960), Bell (1968), Cerny and Marin (1974, 1977), Cerny (1976) and Brown (1974, 1975a, 1975b), and it remains an essential element in contemporary digital design practice (Brown, 1990, 2003; Rushdi and Ba-Rukab, 2003; Steinbach and Posthoff, 2003; Baneres et al, 2009; Brown and Vranesic, 2014; Knodel et al, 2014; Rushdi, 2018b; Rushdi and Ahmad, 2018)
The expressions (18) will not be as compact as they are in the conventional case, but the independent parameters pi belong to the two-valued Boolean algebra B2 (Brown, 2003; Rushdi and Amashah, 2011), a fact that facilitates the generation of all particular solutions as will be seen shortly in the subsection
Summary
Done more than half a century ago, Ledley (1959, 1960) posed three ‘elementary problems of digital circuit design’ inspired by the arrangement, which entails five quantities X, Y, Z, s and t that belong to B2k, B2m, B2n, B2l , and B2l , respectively. This arrangement includes a ‘parent’ combinational network C of two (vectorial) inputs X and Y and a vectorial output t(X, Y).
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