In this paper, we give a deterministic pushdown transducer and a normal sequence of digits compressed by it. This solves positively a question left open in a previous paper by V. Becher, P. A. Heiber and the first author.

We consider the cyclotomic identity testing (CIT) problem: given a polynomial f(x1,…,xk), decide whether f(ζne1, …,ζnek) is zero, where ζn = e2π i/n is a primitive complex n-th root of unity and e1,…,ek are integers, represented in binary. When f is given by an algebraic circuit, we give a randomized polynomial-time algorithm for CIT assuming the generalised Riemann hypothesis (GRH), and show that the problem is in NP unconditionally. When f is given by a circuit of polynomially bounded degree, we give a randomized NC algorithm. In case f is a linear form we show that the problem lies in NC. Towards understanding when CIT can be solved in deterministic polynomial-time, we consider so-called diagonal depth-3 circuits, i.e., polynomials f ∑mi=1 g+idi, where gi is a linear form and di a positive integer given in unary. We observe that a polynomial-time algorithm for CIT on this class would yield a sub-exponential-time algorithm for polynomial identity testing. However, assuming GRH, we show that if the linear forms gi are all identical then CIT can be solved in polynomial time. Finally, we use our results to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in randomized NC.

The so-called “one-bit catastrophe” for the compression algorithm LZ’78 asks whether the compression ratio of an infinite word can change when a single bit is added in front of it. We answer positively this open question raised by Lutz and others: we show that there exists an infinite word w such that ρsup(w) = 0 but ρinf (0w) > 0, where ρsup and ρinf are respectively the lim sup and the lim inf of the compression ratios ρ of the prefixes (Theorem 2.1). To that purpose we explore the behaviour of LZ’78 on finite words and show the following results: • There is a constant C > 0 such that, for any finite word w and any letter . Thus, sufficiently compressible words (ρ(w) = o(1/ log |w|)) remain compressible with a letter in front (Theorem 2.2); • The previous result is tight up to a multiplicative constant for any compression ratio ρ(w) = O(1/ log |w|) (Theorem 2.4). In particular, there are infinitely many words w satisfying ρ(w) = O(1/log |w|) but ρ(0w) = Ω(1).

This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of multilinear ABPs to that of multilinear arithmetic formulas, and prove a tight super-polynomial separation between the two models. Specifically, we describe an explicit n-variate polynomial F that is computed by a linear-size multilinear ABP but every multilinear formula computing F must be of size nΩ(log n).

We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.

The pressing need for eficient 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.