Relativizable and Nonrelativizable Theorems in the Polynomial Theory of Algorithms

Abstract

Starting with the paper of Baker, Gill and Solovay BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision trees of a certain type or for machines with polylogarithmic restrictions on time. The following question arises: Are these methods of proving \relativized" results universal? In the rst part of the present paper we propose a general framework in which assertions of universality of this kind may be formulated and proved as convenient criteria. Using these criteria we obtain, as easy consequences of the known results on boolean decision trees, some new \relativized" results and new proofs of some known results. In the second part of the present paper we apply these general criteria to many particular cases. For example, for many of the complexity classes studied in the literature all relativizable inclusions between the classes are found.

Reducing the Number of Solutions of NP Functions

“…There, notions of ''acceptance types'' and ''promises about numbers of accepting paths'' are natural. In fact, a language notion ''NP A '' can be found in the literature [6], and unifies many notions of counting-based acceptance (and see more generally the notion of leaf languages [3,27]). In our function case, we view the A of NP A V as a cardinality type since it specifies the allowed nonzero numbers of solutions.…”

Restrictive Acceptance Suffices for Equivalence Problems

“…For example, Cai and Hemachandra, after showing that FewP is in ⊕P, then easily applied their technique to show that even Few is in ⊕P [15]. Similarly, it is immediately clear that FewP has Turing-complete sets if and only if Few has Turing-complete sets, and so the proof that there is a relativized world in which FewP lacks Turing-complete sets [28] implicitly proves that there is a world in which Few lacks Turing-complete sets (see also [38]). However, in the case of Corollary 3.5, it is unlikely that by modifying the technique in a way similar to that done by Cai and Hemachandra one could hope to establish the slightly stronger result that EP even contains Few.…”

Generic separations and leaf languages

“…This computation model was introduced by Papadimitriou and Sipser around 1979, and published for the first time by Bovet, Crescenzi and Silvestri [5], and independently by Vereshchagin [18] (see also the textbook [16, pp. 504f]).…”

“…Important for us will be reductions among leaf languages, in particular: ≤ plt m -reductions as introduced in [5,18]. First, we define the class of those functions that will constitute our reductions:…”

“…Bovet, Crescenzi, and Silvestri [5] and Vereshchagin [18] showed how the question of the separability of complexity classes by an oracle can be expressed equivalently by a relation among the characterizing leaf languages. In this way, the task to construct an oracle separation often reduces to a combinatorial problem, without the need to perform an actual stage construction.…”