Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.Comment: 40 page

Precision Higgs physics at a γγ\gamma \gamma collider

The loop induced coupling of an intermediate mass Higgs boson to two photons is a sensitive and unique measure for precision tests of physics beyond the Standard Model. In this work we summarize recent results on the expected precision of the partial $\Gamma (H \to \gamma \gamma)$ width at the $\gamma \gamma$ option of a future linear collider. Heavy particles do not decouple in general and differences between the SM and MSSM predictions or 2HD-models can differ in the percentile regime. Large non-Sudakov DL corrections need to be resummed and consistency requirements demand the use of the Sterman-Weinberg jet definition in order to avoid additional DL terms from three jet final states. We find that the well understood background process $\gamma \gamma \to q \bar{q}$ allows for a ${\mathcal O}$(2%) determination of $\Gamma (H \to \gamma \gamma)$ using conservative collider parameters. Recent improvements in the expected $\gamma \gamma$ luminosity suggest that the precision for the diphoton partial Higgs width can be further improved and is dominated by the error in BR($H \to b \bar{b}$) from the $e^\pm$ mode, which is presently estimated to be in the one percent regime.Comment: 6 pages, 3 epsfigures, Latex2e plus style files. Contribution to the International Workshop on High Energy Photon Photon Colliders at DESY, June 200

Turing Impossibility Properties for Stack Machine Programming

The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts concerning Turing impossibility for stack machine programming are trivially adapted from previous work. Several intriguing questions are raised about the Turing impossibility properties concerning different method interfaces for stack machine programming.Comment: arXiv admin note: substantial text overlap with arXiv:0910.556

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis