We study the relative power of determinism, randomness and nondeterminism for search problems in the Boolean decision tree model. We show that the gaps between the nondeterministic, the randomized and the deterministic complexities can be arbitrary large for search problems. We also mention an interesting connection of this model to the complexity of resolution proofs. Boolean functions, decision trees 68R05, 94A15, 68Q10. 1 Introduction Ramsey's theorem asserts that every graph on n vertices has either a complete graph or an independent set of size 1 2 log n. A natural search problem associated with this theorem is to find such a subgraph. Many other problems, like the ones below, have a similar flavor: Given an assignment of n pigeons into n \Gamma 1 pigeonholes, find two pigeons assigned to the same hole. Given a k-chromatic graph and a coloring of its nodes with fewer than k colors, find two neighbors which have the same color. Given an unsatisfiable 3-CNF formula and an ...

We present a method to prove oracle theorems of the following type.

We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type-2 classes are distinct.

this paper appeared as #AG91a#

It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM Journal on Computing, 20(6):999--1007, 1991). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur-Merlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is bounded and the second if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one round Merlin-Arthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a one-round Arthur-Merlin protocol. Moreover, we show that sometimes a Merlin-Arthur protocol can be more efficient than an Arthur-Merlin protocol, and than a Me...

In this paper we give arguments both for and against the existence of an oracle A, relative to which BPP equals probabilistic linear time. First, we prove a structure theorem for probabilistic oracle machines, which says that either we can fix the output of the machine by setting the answer to only polynomially many oracle strings, or else we can set part of the oracle such that the machine becomes improper. This theorem could help complete the construction of the oracle A, which was proposed by Fortnow and Sipser in [2]. Second, we show that there are previously unknown problems with this construction. Thus the question whether probabilistic polynomial time has a hierarchy relative to all oracles remains completely open.

We investigate the issues and obstacles involved in extending Lutz's notion of measure to provide a measure for P. We provide one natural definition that, under a plausible but unproven assumption, provides a reasonable notion of measure for P. We also provide a more complicated definition that does provide a measure for P (with no unproven assumption). 1 Introduction Lutz has defined a notion of measure on complexity classes E = DTIME(2 O(n) ) and higher, where languages are the analogs of the classical points, and complexity subclasses of E are the sets to be measured [L92]. This notion of measure has been investigated in a series of related papers, and many of its properties are now understood. Unfortunately, Lutz's definitions are meaningful only for "large" complexity classes; they provide no notion of measure for P or PSPACE, etc. More recently, Mayordomo defined an extension of Lutz's notions, and provided a notion of measure for PSPACE [M]. Many of the most interesting que...

There is a family of questions in relativized complexity theory---weak analogs of the Friedberg Jump-Inversion Theorem---that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets, i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2-generic set G such that NP G " co-NP G 6` P G\PhiX . (On the other hand, if G is 1-generic, then NP G " co-NP G ` P G\PhiSAT , where SAT is the NP- complete Satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above 0 0 contains an aw2-generic set; no aw2-generic set exists below any incomplete r.e. set; there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0 0 is the jump of an aw2-generic degree. 1

Mundici considered the question of whether the interpolant of two propositional formulas of the form F → G can always have a short circuit description, and showed that if this is the case then every problem in NP ∩ co-NP would have polynomial size circuits. In this note we observe further consequences of the interpolant having short circuit descriptions, namely that UP ⊆ P/poly, and that every single valued NP function has a total extension in FP/poly. We also relate this question with other Complexity Theory assumptions. 1

A well-known theorem of type-two recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduction Type-two computability theory deals with the computability of functionals, which take functions and numbers as input, and produce numbers as output. A surprising and pleasing aspect of type-two computability is its close connections with topology on Baire space. Notions of relative typetwo computability (that is, computability with respect to some oracle,) can be characterized using purely topological notions. In particular, a type-two functional is computable relative to an oracle if and only if it is continuous. While the theory of type-two computability has been widely successful, relatively little work has been done on the development of a complexity theory for type-two functionals...