The advent of increasingly dense and fast Very Large Scale Integrated (VLSI) circuits allows for the design of larger and more sophisticated digital logic circuits. Efficient logic representations are necessary for the synthesis, testing and verification of these circuits. This thesis introduces a new logic representation, called the Free Boolean Diagram (FBD). This representation can be manipu...

In this paper, we propose a recurrent neural network that can flexibly make inferences to satisfy given Boolean constraints. In our proposed network, each Boolean variable is represented in dual representation by a pair of neurons, which can handle four states of true, false, unknown, and contradiction. We successfully import Blake’s classical Boolean reasoning algorithm to recurrent neural net...

A pseudo-Boolean function is a real-valued function f(x) = f(x1, x2, . . . , xn) of n binary variables; that is, a mapping from {0, 1}n to R. For a pseudo-Boolean function f(x) on {0, 1}n, we say that g(x, y) is a quadratization of f if g(x, y) is a quadratic polynomial depending on x and on m auxiliary binary variables y1, y2, . . . , ym such that f(x) = min{g(x, y) : y ∈ {0, 1}m} for all x ∈ ...

Nečiporuk [Nec̆66] gave a method based on counting subfunctions to lower bound the formula size over an arbitrary binary basis and contact scheme (or Boolean branching program) size required to compute explicit n-input Boolean functions. Pudlak [Pud87] observed that Nečiporuk’s method also yields an Ω(n/ log n) lower bound on switching-and-rectifier network size as well as nondeterministic branc...

The relevance feedback process uses information obtained from a user about a set of initially retrieved documents to improve subsequent search formulations and retrieval performance. In extended Boolean models, the relevance feedback implies not only that new query terms must be identified and re-weighted, but also that the terms must be connected with Boolean And/Or operators properly. Salton ...

The maximum absolute value of integral weights sufficient to represent any linearly separable Boolean function is investigated. It is shown that upper bounds exhibited by Muroga (1971) for rational weights satisfying the normalized System of inequalities also hold for integral weights. Therewith, the previous best known upper bound for integers is improved by approximately a factor of 1/2.

The Rudin-Keisler ordering of ultrafilters is extended to complete Boolean algebras and characterised in terms of elementary embeddings of Boolean ultrapowers. The result is applied to show that the Rudin-Keisler poset of some atomless complete Boolean algebras is nontrivial.

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f . In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log 2 (n + 1)⌉ NO...