### Multi-way Joins in Links

Multi-way joins are now working (transformed from simple list comprehensions, that is) in the Links interpreter. There were a number of subtleties in the rewriting code that prevented this from working.

Currently, my cohort Jeremy and I are not sure whether it's better to use a deep-searching algorithm, like the one we have now, or rather to use simple, local rewrite rules with a fixed depth. The advantage of simple rules would be the ease of showing correctness. But it might be hard to show that such rules must produce more optimal behavior—there is the possibility of hill-climbing, reaching a local optimum and thereby missing a global optimum. The deep-searching rewriter we have now is rather hard to keep track of, as it relyies on simultaneous knowledge of several different kinds of AST nodes, whereas we suspect that a shallow-rule system could be built with only about one rule for each way of combining two node constructors.

To decide this question, I think we need to flesh out the shallow-rule system, and in parallel we need to pursue some kind of assertions about the deep-search system. I still haven't been able t find any tidy results about this in the literature. There are scads of papers by Libkin & Wong that sound related, but I haven't yet done a thorough search to see what's there.

Some things I discovered: Codd, 1970, A relational model for large shared data banks is the original paper for (flat) 'relational algebra' which is the formalism ours needs to compile into. Libkin & Wong's 1994 paper New Techniques for Studying Set Languages, Bag Languages and Aggregate Functions argues about the comparative power of various nested calculi, including those called *BQL* and *NRC*^{aggr}.