166 research outputs found
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
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
LWPP and WPP are not uniformly gap-definable
AbstractResolving an issue open since Fenner, Fortnow, and Kurtz raised it in [S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1) (1994) 116–148], we prove that LWPP is not uniformly gap-definable and that WPP is not uniformly gap-definable. We do so in the context of a broader investigation, via the polynomial degree bound technique, of the lowness, Turing hardness, and inclusion relationships of counting and other central complexity classes
Efficient holographic proofs
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 57-63).by Alexander Craig Russell.Ph.D
A note on a problem in communication complexity
In this note, we prove a version of Tarui's Theorem in communication
complexity, namely . Consequently, every
measure for leads to a measure for , subsuming a result of
Linial and Shraibman that problems with high mc-rigidity lie outside the
polynomial hierarchy. By slightly changing the definition of mc-rigidity
(arbitrary instead of uniform distribution), it is then evident that the class
of problems with low mc-rigidity equals . As , this rules out the possibility, that had been
left open, that even polynomial space is contained in
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