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Deriving Efficient Sequential and Parallel Generators for Closed Simply-Typed Lambda Terms and Normal Forms

Tarau Paul

Fundamenta Informaticae

2020

Vol. 177, nr 3/4

385--415

Contrary to several other families of lambda terms, no closed formula or generating function is known and none of the sophisticated techniques devised in analytic combinatorics can currently help with counting or generating the set of simply-typed closed lambda terms of a given size. Moreover, their asymptotic scarcity among the set of closed lambda terms makes counting them via brute force generation and type inference quickly intractable, with previous published work showing counts for them only up to size 10. By taking advantage of the synergy between logic variables, unification with occurs check and efficient backtracking in today’s Prolog systems, we climb 4 orders of magnitude above previously known counts by deriving progressively faster sequential Prolog programs that generate and/or count the set of closed simply-typed lambda terms of sizes up to 14. Similar counts for closed simply-typed normal forms are also derived up to size 14. Finally, we devise several parallel execution algorithms, based on generating code to be uniformly distributed among the available cores, that push the counts for simply typed terms up to size 15 and simply typed normal forms up to size 16. As a remarkable feature, our parallel algorithms are linearly scalable with the number of available cores.

Rank 2 Intersection for Recursive Definitions

Damiani F.

Fundamenta Informaticae

2007

Vol. 77, nr 4

451-488

Let |- be an intersection type system. We say that a term is |--simple (or just simple when the system |- is clear from the context) if system |- can prove that it has a simple type. In this paper we propose new typing rules and algorithms that are able to type (with rank 2 intersection types) recursive definitions that are not simple. Typing rules for assigning intersection types to (non-simple) recursive definitions have been already proposed in the literature. However, at the best of our knowledge, previous algorithms for typing recursive definitions in the presence of intersection types allow only simple recursive definitions to be typed. The rules and algorithms proposed in this paper are also able to type interesting examples of polymorphic recursion (i.e., recursive definitions rec {x=e} where different occurrences of x in e are used with different types). Moreover, the underlying techniques do not depend on particulars of rank 2 intersection, so they can be applied to other type systems.