On the Tractable Acquisition of Heuristics for Software Synthesis Demonstrating that P~NP

Abstract

The Traveling Salesman Problem (TSP) was first formulated in 1930 and is one of the most studied problems in optimization. If the optimal solution to the TSP can be found in polynomial time, it would then follow that every NP-hard problem could be solved in polynomial time, proving P = NP. It will be shown that our algorithm finds P~NP with scale. Using a \( \delta - \varepsilon \) proof, it is straightforward to show that as the number of cities goes to infinity, P goes to NP (i.e., \( \delta > 0 \)). This was demonstrated using a quadratic number of parallel processors because that speedup, by definition, is polynomial. A fastest parallel algorithm is defined. Six distinct 3-D charts of empirical results are supplied. For example, using an arbitrary run of 5,000 cities, we obtained a tour within 0.00001063 percent of the optimal using 4,166,667 virtual processors (Intel Xenon E5-1603 @ 2.8 GHz). To save the calculated extra 209 miles would take a quantum computer, the fastest possible computer, over (5,000!/(2**4,978 * 22!)) * 267,782 centuries. Clearly, the automated acquisition of heuristics and the associated P~NP solutions are an important problem warranting attention. Machine learning through self-randomization is demonstrated in the solution of the TSP. It is also shown, in the small using property lists, for an inductive logic of abduction. Finally, it is argued that self-randomizing knowledge bases will lead to the creation of a synthetic intelligence, which enables cyber-secure software automation.