Published: 3rd June 2010 DOI: 10.4204/EPTCS.24 ISSN: 20752180 
This volume of the Electronic Proceedings in Theoretical Computer Science (EPTCS) contains abstracts/extended abstracts of talks to be presented at the Seventh International Conference on Computability and Complexity in Analysis (CCA) that will take place in Zhenjiang, China, June 2125, 2010. This conference is the seventeenth event in the series of CCA annual meetings. The CCA conferences are aimed at promoting the study and advancement of the theory of computability and complexity over realvalued data and its application.
We thank authors for their contributions to this EPTCS volume and members of the CCA2010 Program Committee (see below) and external reviewers for their help. We also thank Professor Rob van Glabbeek, the editorinchief of EPTCS, for his support and the local organizers (see below) for all their help with the organization of CCA2010. The CCA 2010 is cosponsored by Arcadia University, Guizhou University, Jiangsu University, Nanjing University, Peking University, and Sun YatSen University.CCA 2010 Program Committee:
CCA2010 Local Organizing Committee:
Xizhong Zheng and Ning Zhong
Program Committee CoChairs
Glenside and Cincinnati, May 2010
Following Lutz (1995) we consider a problem to be weakly hard for a complexity class C if a nonnegligible part of the problems in C can be reduced to C. For the exponentialtime class E (the union of DTIME(2^{kn}) for all k) Lutz formalized this concept by introducing a resourcebounded measure on this class and by saying that a subclass of E is negligible if it has measure 0 in E. In our talk we introduce and discuss some new weak hardness notions for E, called Enontriviality and strong Enontriviality, which generalize Lutz's weak hardness notion for E and which are conceptually much simpler than Lutz's concept. Here a set A is Enontrivial if the problems in E which can be reduced to A are not contained in a single level DTIME(2^{kn}) of the hierarchy E, and A is strongly Enontrivial if, for any number k there is a problem in E which can be reduced to A and which is almosteverywhere 2^{kn}complex (i.e., biimmune for the kth level DTIME(2^{kn}) of the hierarchy E). This work is joint work with Timur Bakibayev.
This talk will present a brief survey of the complexity of real functions, including: computational models for realvalued computation, complexity hierarchy of basic numerical operations, computational complexity of differential and integral equations, and applications to computational geometry and complex analysis.
Whereas computational complexity is well established on discrete structures, it is only very rudimentary treated in Type2theory of effectivity. The aim of this talk is to shed some (subjectiv) light on a few aspects of computational complexity and some recent developments in this field. Problems we will consider are among others:
Measure theory is a fundament of modern analysis. Computable measure theory studies those functions from measure theory which are computable. Compared to the immense number of publications in measure theory and the increasing number of publications in computable analysis there are still very few articles on computable measure theory. In the talk we introduce and discuss some basic concepts of computable measure theory. Furthermore, we present some of the results obtained so far. As in computable analysis numerous new results are waiting for becoming detected. The talk is considered as an invitation to study computable measure theory.
In classical reverse mathematics, the monotone completeness theorem:


In this talk, we investigate the monotone completeness theorem
(MCT) within a framework of constructive reverse mathematics aiming
at classifying various theorems in intuitionistic, constructive
recursive and classical mathematics by logical principles, function
existence axioms and their combinations [5]; see also [6,10].
We distinguish a Cauchy sequence with modulus from a Cauchy sequence
without modulus, and we say that a sequence (x_{n})_{n} of real numbers
is a Cauchy sequence if there exists γ such that


We give a logical principle Σ^{0}_{1}PEM^{+} which is stronger than Σ^{0}_{1}PEM, and show that Σ^{0}_{1}PEM^{+} is equivalent to the statement:
[1]  Errett Bishop, Foundations of Constructive Mathematics, McGrawHill, New York, 1967. 
[2]  Errett Bishop and Douglas Bridges, Constructive Analysis, Springer, Berlin, 1985. 
[3]  Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, London Math. Soc. Lecture Notes 97, Cambridge Univ. Press, London, 1987. 
[4]  Douglas Bridges and Luminiţa Vîţa, Techniques of Constructive Analysis, Springer, New York, 2006. 
[5]  Hajime Ishihara, Constructive reverse mathematics: compactness properties, In: L. Crosilla and P. Schuster eds., From Sets and Types to Analysis and Topology, Oxford Logic Guides 48, Oxford Univ. Press, 2005, 245267. 
[6]  Iris Loeb, Equivalents of the (weak) fan theorem, Ann. Pure Appl. Logic 132 (2005), 5166. 
[7]  Mark Mandelkern, Limited omniscience and the BolzanoWeierstrass principle, Bull. London Math. Soc. 20 (1988), 319320. 
[8]  Stephen G. Simpson, Subsystems of Second Order Arithmetic, Springer, Berlin, 1999. 
[9]  Anne S. Troelstra and Dirk van Dalen, Constructivism in Mathematics, Vol.I and II, NorthHolland, Amsterdam, 1988. 
[10]  Wim Veldman, Brouwer's fan theorem as an axiom and as a contrast to Kleene's alternative, preprint, Radboud University, Nijmegen, 2005. 