41 research outputs found
Robust Simulations and Significant Separations
We define and study a new notion of "robust simulations" between complexity
classes which is intermediate between the traditional notions of
infinitely-often and almost-everywhere, as well as a corresponding notion of
"significant separations". A language L has a robust simulation in a complexity
class C if there is a language in C which agrees with L on arbitrarily large
polynomial stretches of input lengths. There is a significant separation of L
from C if there is no robust simulation of L in C. The new notion of simulation
is a cleaner and more natural notion of simulation than the infinitely-often
notion. We show that various implications in complexity theory such as the
collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust
simulations. We then use these results to prove that most known separations in
complexity theory, such as hierarchy theorems, fixed polynomial circuit lower
bounds, time-space tradeoffs, and the theorems of Allender and Williams, can be
strengthened to significant separations, though in each case, an almost
everywhere separation is unknown.
Proving our results requires several new ideas, including a completely
different proof of the hierarchy theorem for non-deterministic polynomial time
than the ones previously known
Universality, Invariance, and the Foundations of Computational Complexity in the light of the Quantum Computer
Universality, Invariance, and the Foundations of Computational Complexity in the light of the Quantum Computer
Efficient and Elegant Subword-Tree Construction
A clean version of Weiner's linear-time compact-subword-tree construclion simultaneously constructs the smallest deterministic finite automaton recognizing the reverse subwords
Two heads are better than two tapes
We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set.</jats:p
An Information-Theoretic Approach toTime Bounds for On-Line Computation
Introduction
Static, descriptional complexity (program size) [16, 9] can be used to obtain lower bounds on dynamic,
computational complexity (such as running time). We describe and discuss this "information-theoretic approach" in the following section. Paul introduced it in [13], to obtain restricted lower bounds on the time complexity of sorting. We use the approach here to obtain lower time bounds for on-line simulation of one abstract storage unit by another. A major goal of our work is to promote the approach...
An Ω (n log n) Lower Bound for a Restricted Form of On-Line Labeling Algorithm
Under the Normalization Assumption on the algorithms, the labeling problem has a lower bound Ω(nlog n). This is proved by a mechanism dividing the label intervals into a list of collections whose corresponding costs are bounded from below by a suitably chosen function
Operations preserving recognizable languages
Given a subset S of N, filtering a word a0a1 ···an by S consists in deleting the letters ai such that i is not in S. By a natural generalization, denote by L[S], where L is a language, the set of all words of L filtered by S. The filtering problem is to characterize the filters S such that, for every recognizable language L, L[S] is recognizable. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. There are two main ingredients on our approach: the first one is the notion of residually ultimately periodic sequences, and the second one is the notion of representable transductions