George Epstein

Theory and uses of Post algebras of order ω+ω*. II

A monotonic system of Post algebras of the order omega + omega *, whose chain of Post constants is isomorphic with 0<or=1<or=2<or= . . . <or=-3<or=-2<or=-1, is presented. Besides monotonic operations, other unary operations are considered, namely disjoin operations, the quasi-complement, successor and predecessor operations. The successor and predecessor operations are basic for number theory. Among other uses of Post algebras of order omega + omega * are set-theoretic uses and approximation reasoning.<<ETX>>

Three-valued Karnaugh maps

A method is presented for the disjunctive simplification of three-valued logic functions using three-valued Karnaugh maps. The number of variables that can be depicted is at most four. While a particular set of primitive operations is used with the simplification method, the method is not restricted to any particular operation set, provided only that the operations of disjunction and conjunction are included.<<ETX>>

A partially ordered extention of the integers

This paper presents a monotonic system of Post algebras of order ω+ω* whose chain of Post constans is isomorphic with 0≤1≤2≤ ... ≤-3≤-2≤-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.

A weak propositional calculus for signal processing with thresholds

A weak propositional calculus is presented for signal processing with lower threshold z and upper threshold u. For this result all signals are scaled to lie within the linearly ordered real interval [0,1], with focus on the case 0<z<u<1. A still weaker propositional calculus is given where this linear ordering is related to partial ordering in bounded distributive lattices.<<ETX>>

Theory and uses of Post algebras of order omega + omega *. II

For pt.I see Proc. 20/sup th/ International Symposium on Multiple- Valued Logic ISMVL 1990 p42-47. The paper is a continuation of the authors previous work that presents a stronger version of post algebras of order omega + omega whose chain of constants is isomorphic with (0<or=1<or=2<or=. . .<or=-3<or=-2<or=-1). The algebras are a generalization of Post algebras of finite order, and preserve more of their properties than any other generalizations of infinite orders.<<ETX>>

Almost orthogonal functions

Almost-orthogonal functions, which are obtained by perturbing orthogonal functions, do not exactly satisfy the orthogonality conditions. The notion of almost-orthogonal functions is explained with examples. An error analysis of expansion coefficients based on almost-orthogonal functions is given. Using examples derived from sinusoids, it is shown how computational advantage can be obtained by using almost-orthogonal functions in signal processing.<<ETX>>

Memory length as a feedback parameter in learning systems

In a classic learning experiment, a higher vertebrate is presented with two levers. To start, a reward is given if the left lever is pushed, no reward is given if the right lever is pushed. After a certain period of time, T, the meaning of the two levers is interchanged. Now a reward is given if the right lever is pushed and no reward is given if the left lever is pushed. This happens for the same period of time T. Once again there is an interchange in the meaning of the two levers for the period of time T. This alternation is repeated a number of times. Higher vertebrates exhibit learning in this situation. With each repeated period of time T, they gradually adapt their response to the correct lever. In other words, at first they are slow to change from one lever to another, but they gradually learn that the reward at each lever is being changed with period T. The above experiment strongly suggests that memory retention is being adjusted to the length of time that a reward is given at each lever. While it is difficult to determine the exact mechanism by which this is done, there is an easy feedback control system which models this behavior. This is shown below. In the system of Figure 1, if the error signal is too large, say exceeds a specified critical error, there is a decrease, s, in the memory length. If the error signal is low, say falls below the critical error, there is an increase, p, in the memory length. As described, this is a binary alternative in the change of memory length, as shown in Figure 2. It is of course possible to have more than two alternatives in the changes of memory length. In [1] the above system was applied to the tracking of a maneuvering aircraft, where it is assumed that the aircraft follows a periodic linear spline function. That is, the aircraft in the absence of noise follows the path shown in Figure 3. There are a certain number of samples of the aircraft position during each time interval T. The noise disturbances at each sample are chosen to be independent samples of normally distributed noise of mean 0 and constant standard of deviation &sgr;. To simplify, let the filter model be a straight line, least squares model. Other models are possible, for example a straight line, weighted least squares model. Let E be the error signal. In [1], p = 0 when |E| ≤ &Kgr;&sgr;, s = 1 when |E| > &Kgr;&sgr;, where &Kgr; is a constant and &Kgr;&sgr; is the critical error. While recursive filter models are given in [1], the operation may be viewed here as a sequence of nonrecursive linear regressions over f sample points, where f is the adjusting memory length. In Figure 4, if the error signal is less than &Kgr;&sgr;, the memory length increases from 7 to 8, fitting points 11 to 18. This corresponds to p = 0 because there is no penetration into past history prior to point 11. In Figure 4, if the error signal is greater than &Kgr;&sgr;, the memory length decreases from 7 to 6, fitting points 13 to 18. This corresponds to s = 1 because there is a shrinkage of memory length by 1. Other values for p and s are possible. In the above we have seen a binary alternative where either f increases by 1 (p = 0) or f decreases by 1 (s = 1). Many questions remain on the choice of alternatives. In what follows, we answer the questions below. Is there any improvement in going from 2 alternatives, where f increases or decreases by 1, to 3 alternatives, where f increases by 1, decreases by 1, or remains the same? Is there any improvement in using 2 alternatives corresponding to p = 1 (f increases by 2) and s = 2 (f decreases by 2)? Is there any improvement in using 2 alternatives corresponding to high values of s and p? First, is there any improvement in allowing f to remain unchanged, as well as increasing or decreasing by one? We have made a number of simulations to test this, and find no discernible improvement over the binary alternative (just increasing or decreasing by one). From the standpoint of models for learning systems, this result is not surprising. Second, the system in which p = 1 and s = 2 shows improvement over the system in which p = 0 and s = 1. The latter may be found in [1]. The improved performance of the former is shown in Figure 5. Apparently in Figure 5, the improvement stems from the fact that f increases or decreases by a greater amount. For p = 1, the model penetrates into past history by 1, increasing f by 2 rather than 1. For s = 2, the model shrinks f by 2, rather than 1. Note that in Figure 5 the learning is rapid, in that the overshoot at each knot of the spline function rapidly decreases, through the sequence at knots B, C, D, E, F. As can be seen in Figure 5, the period T is rapidly learned by the filter output. This is apparent by viewing the filter output from knot E to knot F. In fact, the filter value of f in steady state has a root-mean-square error of only 3.5 sample points with essentially no bias error. Third, could high increases or decreases in f lead to improved performance? The answer is a definite no. The performance in such cases is very poor. Apparently the best performance may be found around p = 0 and s = 1, or p = 1 and s = 2, as noted above.