Stephen A. Bloch, Jonathan F. Buss, Judy Goldsmith

We define the sharply bounded hierarchy, SBHQL, a hierarchy of

classes within P, using quasilinear-time computation and

quantification over values of length log n. It generalizes the

limited nondeterminism hierarchy introduced by Buss and Goldsmith,

while retaining the invariance properties. The new hierarchy has

several alternative characterizations.

We define ... more >>>

Ke Yang

In this paper, we address the problem of evaluating the

Integer Circuit (IC), or the $\{\cup, \times, +\}$-circuit over

the set of natural numbers. The problem is a natural extension

to the integer expression by Stockmeyer and Mayer, and is also studied

by Mckenzie, Vollmer and Wagner in ...
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Olivier Powell

An almost complete set A for a complexity class C is a language of C which is not complete, but that has the property that ``many'' languages of C reduce to A, where the term ``many'' is used in reference to Lutz's resource bounded measure (rbm). The question of the ... more >>>

Scott Aaronson, John Watrous

While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the ... more >>>

Or Meir

The IP theorem, which asserts that IP = PSPACE (Lund et. al., and Shamir, in J. ACM 39(4)), is one of the major achievements of complexity theory. The known proofs of the theorem are based on the arithmetization technique, which transforms a quantified Boolean formula into a related polynomial. The ... more >>>

Eric Allender, Luke Friedman, William Gasarch

Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R_C and R_K denote the sets of strings that are ``random'' according to these measures; both R_K and R_C are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both R_K ... more >>>

Scott Aaronson, Andrew Drucker

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states ... more >>>

Eric Allender, Harry Buhrman, Luke Friedman, Bruno Loff

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out by [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it ... more >>>

Chiranjit Chakraborty, Rahul Santhanam

We define instance compressibility for parametric problems in PH and PSPACE. We observe that

the problem \Sigma_{i}CircuitSAT of deciding satisfiability of a quantified Boolean circuit with i-1 alternations of quantifiers starting with an existential uantifier is complete for parametric problems in \Sigma_{i}^{p} with respect to W-reductions, and that analogously ... more >>>

Christian Glaßer, Maximilian Witek

We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain:

- For NP and all other classes of the PH: each logspace many-one-complete set is logspace Turing-autoreducible.

- For P, the delta-levels of ...
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Eric Allender, Dhiraj Holden, Valentine Kabanets

We consider variants of the Minimum Circuit Size Problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MCSP$^{QBF}$ is known to be complete for PSPACE under ZPP reductions. We show that it is not ... more >>>

Michael Forbes, Amir Shpilka

In this paper we study the complexity of constructing a hitting set for $\overline{VP}$, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given ... more >>>

Zeyu Guo, Nitin Saxena, Amit Sinhababu

Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). ... more >>>

Joshua Cook

Let $\mathbf{TISP}[T, S]$, $\mathbf{BPTISP}[T, S]$, $\mathbf{NTISP}[T, S]$, and $\mathbf{CoNTISP}[T, S]$ be the set of languages recognized by deterministic, randomized, nondeterminsitic, and co-nondeterministic algorithms, respectively, running in time $T$ and space $S$. Let $\mathbf{ITIME}[T_V]$ be the set of languages recognized by an interactive protocol where the verifier runs in time $T_V$. ... more >>>