Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using proof-theoretic methods for obtaining certified algorithms in exact real arithmetic. 1

AVOCS, the workshop on Automated Verification of Critical Systems, is an annual meeting that brings together researchers and practitioners to exchange new results on tools and techniques for the verification of critical systems. Topics of interest cover all aspects of automated verification, including model checking, theorem proving, abstract interpretation, and refinement pertaining to various types of critical systems (safety-critical, security-critical, business-critical, performance-critical, etc.). Contributions that describe different techniques, or industrial case studies are encouraged.

Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is not a single simulation environment that comes with a guarantee that the results of the simulation are determined purely by a real-valued model and not by artifacts of the digitized implementation. As such, simulation with guaranteed fidelity does not yet exist. Towards addressing this problem, we offer an expository account of what is known about exact real arithmetic. We argue that this technology, which has roots that are over 200 years old, bears significant promise as offering exactly the right technology to build simulation environments with guaranteed fidelity. And while it has only been sparsely studied in this large span of time, there are reasons to believe that the time is right to accelerate research in this direction. ∗ This research was sponsored by the NSF under Award 0439017,

Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation. 1