Information Theory Defines “Mathematically Conceivable Communication System”

Ray Dougherty

New York University

Abstract

Information Theory (IT), introduced by Shannon (The Mathematical Theory of Communication, 1949) offers a purely mathematical theory of signals traveling over a channel from a transmitter to a receiver. Transmitter, receiver, signal, channel, and so on must be abstracted, idealized, and expressed as mathematical functions. We focus only on communication systems in which the signal structures are complex sinusoidal functions (sound waves). We precisely follow the formalisms and the mathematics of Poincaré on coupled oscillators presented in Nicolis and Prigogine (Exploring Complexity, 1989).

The concept of a mathematically conceivable communication system (MCCS) exists as pure abstraction, even before matter or biology existed to implement any system. We “name” (or “label”) each MCCS by a mathematical function that defines the signal set complexity following Poincaré’s system for defining complexity of coupled nonharmonic oscillators:

1. Plot[A*Sin[a*x],{x,0,2*Pi}]

1 Sine Wave

2. Plot[A*Sin[a*x] + B*Sin[b*x], {x,0,2*Pi}]

2 Sine Waves

3. Plot[A*Sin[a*x] * B*Sin[b*x],{x,0,2*Pi}]

j. Plot[A*Sin[(a + B*Sin[b*x])*x + C*Sin[c*x]],{x,0,2Pi}]

3 Sine Waves

i. Plot[A*Sin[a*x +B*Sin[b*x]] + C*Sin[c*x + D*Sin[d*x]], {x, 0, 2*Pi}]

4 Sine Waves

...and on to 5, 6 (the whale and porpoise)... and N Sine
waves.

(two formant system)

We present a Chomsky-type generative grammar with an alphabet of these symbols +, *, [, ], (, ), {, }, Sine, x; A, B,…, a, b… that defines a grammatical sentence in webMathematica such that each sentence names a mathematical function (that runs in Mathematica) defining the signal set complexity of a mathematically conceivable communication system. With Mathematica programming, the “Name” of any mathematically conceivable communication system can be plotted with ContourPlot, DensityPlot, and three-dimensional Plots, and also Played over loudspeakers, or fed to a Fourier spectrogram program.

We present an IT perspective and a Reverse Engineering (RE) methodology to show how our simple illustrations can be fleshed out by more Mathematica functions to generate precise testable mathematical models for the sinusoidal complexity of any animal communication system. How can our project to generate mathematical models of any and all animal communication systems be tested? (a) The predicted set of signal complexities can be compared with data such as that presented by Hauser, Konishi, and others on the spectral complexity of insect, bird, frog, bat, and animal signals. (b) They must match the transmitter and the receiver mechanisms of creatures. (c) They can be played and spectrally analyzed.

The IT Darwinian comparative method does not compare any two animals. Rather we compare each animal against the set of mathematically conceivable communication systems. We do not ask: what are the properties of an animal CS? Rather we ask: which (or which logical combination) of the mathematically conceivable systems is this animal using?