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

Abstract

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.