Separation of Complexity Classes in Koiran's Weak Model

Very recently Pascal Koiran introduced in [10] a model of computation that comes from a modification of the cost notion of the real Turing machine of [2]. This new model —that following Koiran will be called weak— drops the unit cost assumption for the arithmetical operations and only allows a “moderate use of multiplication” ([11]). The main result of [10] states that when restricted to Boolean inputs the class of sets decided by these machines in polynomial time coincides with P/poly. As a consequence, if P = NP in the weak model then the Boolean polynomial hierarchy collapses at the second level. In the present paper we continue the study of the computational power of the weak model. In particular, several separations between complexity classes for that model are proved, the most important one being P 6= NP. In fact, it is shown that NPW (the subscript stands for “weak”) strictly contains its subclass NPWD consisting of those sets that can be decided using binary guesses. Note that since PW ⊂ NPW the above mentioned separation holds. The problem of whether PW= NPWD remains open and we provide two kinds of partial answers to it. On the one hand, in section 3 and following the line of ideas of [10] we show that the above equality would imply the collapse of the polynomial hierarchy at its second level, a fact seen as unlikely in complexity theory. On the other hand, we prove in section 5 that if we restrict our attention to machines that branch only on equality tests, we can prove that the forementioned equality does not hold. This is done by showing that a well known problem (the Knapsack problem) belongs to NPWD and can not be solved in deterministic polynomial time. Finally, in section 4, we consider the alternating variation of the weak model, and we give a doubly exponential lower bound for the parallel time needed to decide problems solvable in polynomial alternating time.