Abstract. An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if K(X n) ≤ + K(0 n ) for all n, where K is the prefix-free complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the information in a sequence of 0s of the same length. We study the gap between the trivial complexity K(0 n ) and the complexity of a non-trivial sequence, i.e. the functionsfor a non-trivial (in terms of initial segment complexity) sequence X. We show that given any ∆ 0 2 unbounded non-decreasing function f there exist uncountably many sequences X which satisfy ( ). On the other hand there exists a ∆ 0 3 unbounded non-decreasing function f which does not satisfy ( ) for any X with non-trivial initial segment complexity. This improves the bound ∆ 0 4 that was known from [CM06]. Finally we give some applications of these results.

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In this paper we show that the PCP theorem holds as well in the real number computational model introduced by Blum, Shub, and Smale. More precisely, the real number counterpart NP R of the classical Turing model class NP can be characterized as NP R = PCP R (O(log n), O(1)). Our proof structurally follows the one by Dinur for classical NP. However, a lot of minor and major changes are necessary due to the real numbers as underlying computational structure. The analogue result holds for the complex numbers and NP C . ACM Subject Classification F.2.2 Complexity of Proof Procedures Keywords and phrases IntroductionOne of the major achievements in theoretical computer science within the last two decades certainly was the PCP theorem. It gave a new characterization of the complexity class NP via so called probabilistically checkable proofs and had tremendous impact on the field of approximation algorithms in combinatorial optimization. Starting point for the present paper is the real number model of computation and its related complexity theory as introduced by Blum, Shub, and Smale, henceforth called BSSmodel for short, see [6,5]. The model focusses on algebraic aspects of computation over arbitrary structures, and here in particular the real numbers. As with the Turing model the major open question in this real number framework is whether the real complexity classes P R and NP R of problems decidable and verifiable, respectively, in polynomial time are different.The definition of probabilistically checkable proofs makes sense as well in the real number model. Given the importance of the classical PCP theorem it is only natural to ask whether the corresponding characterization holds as well in the BSS framework for NP R . The goal of this paper is to prove the PCP theorem in this setting.

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