We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let

this paper, if T is time-constructible, then

wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constant-factor overhead ; if g = O(t log t) it has a factor-of-O(log t) overhead , and so on. The simulation is on-line if each step of M 1 i

We continue an investigation into resource-bounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resource-bounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABK + 06]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resource-bounded Kolmogorov complexity are two-fold: • to demonstrate that other complexity measures such as branching-program size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL02] also fit well into this framework. The main theorems that we provide using this new approach to resource-bounded Kolmogorov complexity are: • A complete set (RKNt) for NEXP/poly defined in terms of strings of high Kolmogorov complexity.

Abstract. A realistic model of computation called the Block Move (BM) model is developed. The BM regards computation as a sequence of finite transductions in memory, and operations are timed according to a memory cost parameter µ. Unlike previous memory-cost models, the BM provides a rich theory of linear time, and in contrast to what is known for Turing machines, the BM is proved to be highly robust for linear time. Under a wide range of µ parameters, many forms of the BM model, ranging from a fixed-wordsize RAM down to a single finite automaton iterating itself on a single tape, are shown to simulate each other up to constant factors in running time. The BM is proved to enjoy efficient universal simulation, and to have a tight deterministic time hierarchy. Relationships among BM and TM time complexity classes are studied. Key words. Computational complexity, theory of computation, machine models, Turing machines, random-access machines, simulation, memory hierarchies, finite automata, linear time, caching. AMS/MOS classification: 68Q05,68Q10,68Q15,68Q68.

. For any fixed k, a remarkably simple single-tape Turing machine can simulate k independent counters in real time. Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of Computation---relations among models; bounded-action devices; E.2 [Data]: Data Storage Representations---contiguous representations; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems---sequencing and scheduling; G.2.1 [Discrete Mathematics]: Combinatorics ---combinatorial algorithms; F.2.2 [Analysis of Algorithms and Problem Complexity ]: Tradeoffs among Complexity Measures General Terms: Theory, Algorithms, Design, Verification Additional Key Words and Phrases: Counter, abstract storage unit, counter machine, multicounter machine, one-tape Turing machine, simulation between models, real-time simulation, on-line simulation, oblivious simulation, redundant number representation, signed-digit number representation, recursion elimination 1. Int...