Abstract. We extend a result of Long and give a simple proof of the extension: For k>0, if S is a sparse set in ∑Pk, then ∑Pk(S)⊆ ΔPk+1 so that ΔPk+1(S)=ΔPk+1.

Long and Selman first proved that the polynomial-time hierarchy collapses if and only if for every sparse set S, the hierarchy relative to S collapses. This ...

On Certain Polynomial-Time Truth-Table Reducibilities of Complete Sets to Sparse Sets. Let $\Sigma $ be a finite alphabet. A set $S \subset \Sigma ^ * $ is ...

We study sets that are truth-table reducible to sparse sets in polynomial time. The principal results are as follows: (1) For every integer $k > 0$, there is a ...

Bibliographic details on A Note on Sparse Sets and the Polynomial-Time Hierarchy.

The main problem is whether the collapse EXPTIME= NEXPTIME which forces all sparse sets from NP into P also forces all sparse sets from the polynomial-time ...

Mar 7, 1993 · It is the purpose of this paper to give simple proofs, in a uniform format, of the major known (pre-1992) results relating how polynomial-time ...

Sep 18, 2003 · Definition 3.1 A set S ⊆ Σ∗ is called a sparse set if |S=n| = nO(1), i.e. if the number of strings at length n in S is at most polynomial in n.

How reductions to sparse sets collapse the polynomial-time hierarchy: a primer: Part II restricted polynomial-time reductions · Paul Young. Computer Science ...

The hierarchy in polynomial-size can be extended to only distinct finitely level if a sparse set in NP. Hence, it can be said that if PH=PSPACE, then the ...