Enumeration conservation law and natural parallelism of the $D$-algorithms for test generation and modeling in engineering diagnostics

In Part I, typical and minimum diagnostic labyrinths that graphically model generation and modeling of tests were introduced using the $D$-algorithms. Notions of the value of diagnostic enumeration and the minimum value $\mathrm{Pr}_{\min}$ of this enumeration were introduced. A diagnostic unit was introduced for measuring this enumeration. A counterpart of the well-known “physical” Noether theorem was proved relying on the symmetry of the minimum diagnostic labyrinths. The theorem gives rise to the enumeration conservation law $\mathrm{Pr}_{\min}=2N-n-m=\mathrm{const}$ for sequential $D$-algorithms, where $N$ is the number of arcs in the logical network of a combinational device, and $n$ and $m$ are, respectively, the numbers of its external inputs and outputs. Efficiency of the $D$-algorithm was defined, and a formula to calculate it was presented. For the ideal test generator which is a theoretical prototype of the physical generators with scaled, potentially unlimited, throughput, the mechanism of natural parallelism was substantiated. In Part II, described was a mechanism of natural (hypermass) parallelism of the ideal test generator, which uses new parallel-sequential $D$-algorithms to design discrete devices and, thus, appreciably enhance test generation and diagnostic modeling. The mechanism was shown to overcome the lower constraint on the time of solution of the direct and inverse problems which is imposed by the enumeration conservation law for the sequential $D$-algorithms. The spectrum of the parallel-sequential $D$-algorithms was described in general terms. A diagnostic counterpart of the Noether theorem was proved, and the corresponding numerical forms of the enumeration conservation law were deduced for them, as well as for the sequential algorithms. Conclusions substantiated a new direction of the research program for the diagnostic theory of tolerance and similitude – the possibility of using the mechanism of natural parallelism to solve systems of logical, algebraic, and differential equations.

Presented by the member of Editorial Board: P. P. Parkhomenko