1,269 research outputs found
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
Relativized Propositional Calculus
Proof systems for the Relativized Propositional Calculus are defined and
compared.Comment: 8 page
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
On the Impossibility of Probabilistic Proofs in Relativized Worlds
We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle.
This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it.
We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent
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