We study constructive and resource-bounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/poly-Turing reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rd-order scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turing-hard sets for ESPACE. 1

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Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that

Abstract Resource-bounded dimension is a notion of computational information density of in-finite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension.The pushdown dimension of any sequence is trivially bounded above by its finite-state dimension, since a pushdown gambler can simulate any finite-state gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finite-state dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewell-known fact that pushdown automata decide strictly more languages than finite-state

E-mail address: asylvain.perifel(at)ens-lyon.fr

Abstract This paper develops new relationships between resource-bounded dimension, entropy rates,and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes.Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in \Delta E3, the third level of the exponential-time hierarchy. This generalresult is applied to simultaneously improve the results of Mayordomo (1994) on the measure on P/poly in \Delta E3 and of Lutz (2000) on the dimension of exponential-size circuit complexity classesin ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied inconjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz (1992) aboutpolynomial-size circuit complexity classes from resource-bounded measure to dimension.

Abstract. The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. The celebrated Lempel-Ziv algorithm LZ78 [10] was introduced as a general purpose compression algorithm that outperforms finite-state compressors on all sequences. We compare the performance of the Lempel-Ziv algorithm with that of the pushdown compressors, or compression algorithms that can be implemented with a pushdown transducer. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that Lempel-Ziv is incomparable with pushdown compressors. 1.

space compression is incomparable with

The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [11, 17], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose Lempel-Ziv algorithm. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.