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Extra info for Crisp and Soft Computing with Hypercubical Calculus: New Approaches to Modeling in Cognitive Science and Technology with Parity Logic, Fuzzy Logic, and Evolutionary Computing
Jx, that is, the generalized XO R-reduction operator. By the same token, if =f. is subjected to the cumulative scan operator\, we get the derived function z +--=f. \x, that is, the generalized XOR-scan operator. 3 below. Notice that each operator has its dual operator by virtue of DeMorgan's laws 1 . 1 An extensive treatment of monadic (or unary) operators is provided by Iverson ([IVE62], [IVE78]) and Lochner ([LOC89]). Their scope of applications in science is discussed by Langlet ([LAN92a], [LAN93], [LAN94]) and Zaus ([ZA94], [ZA95b], [ZA95cJ) .
What remains is system inscrutability and lack of explanatory power. 1 Standard Function Systems It also shows the aforementioned isomorphism between time functions x(t) and Boolean functions f(x(t)). 1 are part of a far more general and universal system of standard functions. 2 . 2 prepares the reader for the binary counterpart of Fourier analysis. We take the cue from linear systems analysis according to which the Gaussian normal distribution N(O, 1) is the invariant of the Fourier transform.
3. 4. elements X E sn as vectorial representations of binary signals, the eXclusive-OR operation X$ y, defined on elements x, y E sn' the XOR operator EB~=l Xi E X E sn the binary scalar integral, the XOR operator EBf=l Xi E X E sn' the binary vector integral, I are systematically applied to signals x( t) regarding representations by spectra vs. representations by sequences in minute detail. This includes the dual relationship between spectra and pulses, signal transforms, and the algorithmic compression of binary signal representations and their transformations.