Complete Completion using Types and Weights

Developing modern software typically involves composing functionality from existing libraries. This task is difficult because libraries may expose many methods to the developer. To help developers in such scenarios, we present a technique that synthesizes and suggests valid expressions of a given type at a given program point. As the basis of our technique we use type inhabitation for lambda calculus terms in long normal form. We introduce a succinct representation for type judgements that merges types into equivalence classes to reduce the search space, then reconstructs any desired number of solutions on demand. Furthermore, we introduce a method to rank solutions based on weights derived from a corpus of code. We implemented the algorithm and deployed it as a plugin for the Eclipse IDE for Scala. We show that the techniques we incorporated greatly increase the effectiveness of the approach. Our evaluation benchmarks are code examples from programming practice; we make them available for future comparisons

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.Comment: 53 pages. To appear in Geometry & Topology. Changes in response to helpful referee: replace the erroneous Corollary 6.2 with a new version, specify that we consider 2x3 Cohen-Macaulay singularities, calculate more entries of Table 5, improve wording, format for publicatio

Technical report and user guide: the 2010 EU kids online survey

This technical report describes the design and implementation of the EU Kids Online survey of 9-16 year old internet using children and their parents in 25 countries European countries