PROSE

PROSE was the mathematical 4GL virtual machine which established the holistic modeling paradigm known as Synthetic Calculus (AKA MetaCalculus). A successor to the SLANG/CUE simulation and optimization language developed at TRW Systems, it was introduced in 1974 on Control Data supercomputers. It was the first commercial language to employ automatic differentiation (AD), which was optimized to loop in the instruction-stack of the CDC 6600 CPU.

Although PROSE was a rich block-structured procedural language, its focus was the blending of simultaneous-variable mathematical systems such as:

• implicit non-linear equations systems,
• ordinary differential-equations systems, and
• multidimensional optimization.

Each of these kinds of system models were distinct and had operator templates to automate and solve them, added to the procedural syntax. These automated system problems were considered "holistic" because their unknowns were simultaneous, and they could not be reduced in formulation to solve piecewise, or by algebra manipulation (e.g. substitution), but had to be solved as wholes. And wholeness also pertained to algorithmic determinacy or mathematical "closure", which made solution convergence possible and certain in principle, if not corrupted by numerical instability.

Holarchies of Differential Propagation

Since these holistic problem models could be independently automated and solved due to this closure, they could be blended into higher wholes by nesting one inside of another, in the manner of subroutines. And users could regard them as if they were ordinary subroutines.

Yet semantically, this mathematical blending was considerably more complex than the mechanics of subroutines, because an iterative solution engine was attached to each problem model by its calling operator template above it in the program hierarchy. In its numerical solution process, this engine would take control and would call the problem model subroutine iteratively, not returning to the calling template until its system problem was solved. During some or maybe all of the iterative model-subroutine calls the engine would invoke automatic differentiation of the formulas in the model holarchy with respect to the model's input-unknowns (arguments) defined in the calling template. Additional mechanisms were performed in the semantics to accommodate ubiquitous nesting of these holistic models.

Differentiation of Prediction Processes

If the nested solution was a prediction (e.g. numerical integration), then its solution algorithm, in addition to the model formulas, would also be automatically differentiated. Since this differentiation propagated (via the chain rule) throughout the integration from initial conditions to boundary conditions, the differentiation of boundary conditions with respect to initial conditions (so called Frechet derivatives) would be performed. This enabled the routine solution of boundary-value problems by iterative "shooting" methods using Newton-method engines. Of course, at the same time this propagated differentiation could also be performed with respect to arbitrary parameters of the differential equations to further shape the integrated functions. And these parameters could be solved for as unknowns of any nest in the holarchy above the integration process, a significant convenience in overall problem formulation.

Differentiation of Search Processes

If the inner nested problem was a search, and the outer problem was also a search (e.g. optimization), then the partial derivatives produced with respect to the inner-search unknowns had to be converted into partial derivatives of the outer search via a differential-geometry coordinate transformation. This was also an iterative process involving higher order differentiation and sometimes different independent variables.

Yet these extended and iterative differential-arithmetic processes were totally hidden from the user, and were hardly more significant in his modeling task than if only ordinary subroutines and their calls were involved. The fact that they were iterative and the number and kind of iterations were indefinite, because a whole sub-problem was being solved which was also a part of a higher problem, it was natural to call each problem nest a "holon", as this dynamic entity perfectly fitted the theory of Arthur Koestler who coined that term. This was not done in the original PROSE documentation, because in those years, Koestler's theory was new, and somewhat controversial. This term was later used after Ken Wilber had ratified Koestler's holon concepts.