Boolean Threshold Research Articles

Computational limitations on learning from examples

The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distribution-free sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) Boolean threshold functions, and (c) Boolean formulas in which each variable occurs at most once. Relationships between learning of heuristics and finding approximate solutions to NP-hard optimization problems are given.

An Algebraic Model for Bounding Threshold Circuit Depth

We define a new structured and general model of computation: circuits using arbitrary fan- in arithmetic gates over the characteristic two finite fields (<strong>F</strong>_2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold circuits that allow arbitrary integer weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model only differs by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Sum and ½ gates; there is no bound on the number of ''wires'' . We show that these results are provable for any ''reasonable'' correspondance between bit strings of n-bits and elements of <strong>F</strong>_ 2n. And, we find two distinct characterizations of ''reasonable''. Thus, we have shown that arbitrary fan-in arithmetic computations over <strong>F</strong>_ 2n constitute a precise abstraction of boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

Iterative behaviour of generalized majority functions

Let us consider a society of n persons {P1,…,Pn} with interaction coefficient aij between Pi and Pj. Let {θ1,…,θp} be the set of opinions which may be assumed by any person. We study the dynamical behaviour of such a society under local majority rules. The results presented in this paper generalize many partial results previously obtained by several authors (Tchuente, 1977; Goles and Olivos, 1980; Poljak and Sura, 1982). Furthermore, by introducing algebraic invariants, analogous to those exhibited in Goles (1980) for the study of Boolean threshold functions, we provide a tool for a general approach of such dynamical models.

Iterative synthesis of threshold functions

Iterative procedure for determining linear separability of Boolean function and threshold functions for logical switching circuit design