We show that the satisfiability problem for quantifier-free theory of product structures with equicardinality relation is in NP. As an application, we extend combinatory array logic fragment to handle cardinality constraints. The resulting independent base element and index set theories.

Let M be a real r × c matrix, and let k positive integer. In the column subset selection problem (CSSP), we need to minimize quantity ‖ − S A , where can an arbitrary runs over all submatrices of . This its applications in numerical linear algebra are being discussed for several decades, but algorithmic complexity remained open issue. We show that CSSP is NP-complete.

We prove that the class NP has Co-NP-immune sets and the class Co-NP has NP-immune sets relative to a random oracle.

We prove that the class NP has Co-NP-immune sets relative to a random oracle. Moreover, we prove that, relative to a random oracle, there are L 2 P and NP-set L 1 L such that L 1 is Co-NP-immune and L n L 1 is NP-immune. The second theorem implies the theorem in 4] that Co-NP has NP-immune sets relative to a random oracle.

The P=?NP problem is philosophically solved by showing P is equal to NP in the random access with unit multiply (MRAM) model. It is shown that the MRAM model empirically best models computation hardness. The P=?NP problem is shown to be a scientific rather than a mathematical problem. The assumptions involved in the current definition of the P?=NP problem as a problem involving non deterministi...