55 research outputs found
Different Approaches to Proof Systems
The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper
P-Selectivity, Immunity, and the Power of One Bit
We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is
not EXP/1-immune. That is, we prove that some infinite P-selective set has no
infinite EXP-time subset, but we also prove that every infinite P-selective set
has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so
fragile that it is pierced by a single bit of information.
The above claims follow from broader results that we obtain about the
immunity of the P-selective sets. In particular, we prove that for every
recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is
not \Pi_2^p/1-immune
Boolean Operations, Joins, and the Extended Low Hierarchy
We prove that the join of two sets may actually fall into a lower level of
the extended low hierarchy than either of the sets. In particular, there exist
sets that are not in the second level of the extended low hierarchy, EL_2, yet
their join is in EL_2. That is, in terms of extended lowness, the join operator
can lower complexity. Since in a strong intuitive sense the join does not lower
complexity, our result suggests that the extended low hierarchy is unnatural as
a complexity measure. We also study the closure properties of EL_ and prove
that EL_2 is not closed under certain Boolean operations. To this end, we
establish the first known (and optimal) EL_2 lower bounds for certain notions
generalizing Selman's P-selectivity, which may be regarded as an interesting
result in its own right.Comment: 12 page
Nondeterministic Instance Complexity and Proof Systems with Advice
Motivated by strong Karp-Lipton collapse results in bounded arithmetic, Cook and Krajíček [1] have recently introduced the notion of propositional proof systems with advice. In this paper we investigate the following question: Given a language L , do there exist polynomially bounded proof systems with advice for L ? Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that the above question is tightly linked with the question whether L has small nondeterministic instance complexity
Self-Specifying Machines
We study the computational power of machines that specify their own
acceptance types, and show that they accept exactly the languages that
\manyonesharp-reduce to NP sets. A natural variant accepts exactly the
languages that \manyonesharp-reduce to P sets. We show that these two classes
coincide if and only if \psone = \psnnoplusbigohone, where the latter class
denotes the sets acceptable via at most one question to \sharpp followed by
at most a constant number of questions to \np.Comment: 15 pages, to appear in IJFC
A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Cook and Krajíček [9] have obtained the following Karp-Lipton result in bounded arithmetic: if the theory proves , then collapses to , and this collapse is provable in . Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even p-optimal proof systems using advice. We further show that these p-optimal systems are equivalent to natural extensions of Frege systems
Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard
Maximum-likelihood decoding is one of the central algorithmic problems in
coding theory. It has been known for over 25 years that maximum-likelihood
decoding of general linear codes is NP-hard. Nevertheless, it was so far
unknown whether maximum- likelihood decoding remains hard for any specific
family of codes with nontrivial algebraic structure. In this paper, we prove
that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon
codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes
remains hard even with unlimited preprocessing, thereby strengthening a result
of Bruck and Naor.Comment: 16 pages, no figure
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