Lambda the Ultimate

I mentioned this book in a recent discussion, but I think it might interest members not following that discussion.

Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises.

The book first develops the necessary logical machinery emphasizing novel forms of Gödel's famous functional ('Dialectica') interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.

The site includes some sample pages for your reading pleasure. Not ten lines into the preface does Dana Scott appear, and he is clearly one of us...

Read the preface and share your thoughts!

This probably isn't a critical bug, but just for the record, I stumbled across this behavior while writing a post. Fortunately I caught my mistake and fixed it quickly. See my comments for details.

Map-reduce-merge: simplified relational data processing on large clusters (freely-accessible slides). Hung-chih Yang, Ali Dasdan, Ruey-Lung Hsiao, D. Stott Parker. 2007 ACM SIGMOD conference.

Map-Reduce is a programming model that enables easy development of scalable parallel applications to process a vast amount of data on large clusters of commodity machines. Through a simple interface with two functions, map and reduce, this model facilitates parallel implementation of many real-world tasks such as data processing jobs for search engines and machine learning.

However,this model does not directly support processing multiple related heterogeneous datasets. While processing relational data is a common need, this limitation causes difficulties and/or inefficiency when Map-Reduce is applied on relational operations like joins.

We improve Map-Reduce into a new model called Map-Reduce-Merge. It adds to Map-Reduce a Merge phase that can efficiently merge data already partitioned and sorted (or hashed) by map and reduce modules. We also demonstrate that this new model can express relational algebra operators as well as implement several join algorithms.

They seem to add a third phase – merge: ((k1, [v1]), (k2, [v2])) → (k3, [v3]) – which combines the outputs of two separate, parallel MapReduce tasks.

This makes it possible to do things like joins and build cartesian products.