Dataflow programming and linear models of computation

More on linear models of computation:

https://github.com/anhinga/fluid contains open source code, demo videos, and links to the reference paper and other preprints written by our group on this topic.

We finally have a model of computation which allows us to continuously deform programs and, moreover, to represent large classes of programs by matrices of real numbers.

We consider classes of computations which admit taking linear combinations of execution runs. Two well known large classes of computations with this property are probabilistic sampling and generalized animation. Both allow for enough higher-order constructions to make us believe that limiting consideration to either of these two classes is not a restriction of generality compared to general-purpose computations.

Since both probabilistic sampling and generalized animation represent stream-based computing architecture, it is natural to consider dataflow programming in this context. We experimented with several different dataflow architectures (the animation above demonstrates the first of the architectures we considered).

We eventually obtained the following solution. If one takes a countable number of built-in stream transformers of one's choice (usually, one takes a finite number of those, and then considers a countable number of copies of each), one can then take each input to be a countable sum of all outputs.

The way to think about this is to say that the set of built-in transformers defines a language, and the coefficients in the countable sums define a program in that language. So a program is defined by a countable-sized matrix; usually one requires that only a finite number of those coefficients are non-zero in order to fit all this into a finite machine.

Then one can change the program in a continuous fashion by continuously changing the matrix coefficients. Because a countable-size matrix has a countable number of matrix elements, one can compute the matrix elements themselves via the streams of real numbers computed by the program itself, thus enabling a variety of higher-order mechanisms (the continuous version of self-modifying code).

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