Maximal sharing in the Lambda calculus with letrec

Abstract

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of 'maximal compactness' for λ letrec -terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. λ letrec -terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation.We describe practical and efficient methods for the following two problems: transforming a λ letrec -term into a maximally compact form; and deciding whether two λ letrec -terms are unfolding-equivalent. The transformation of a λ letrec -term L into maximally compact form L0 proceeds in three steps: (i) translate L into its term graph G = L ; (ii) compute the maximally shared form of G as its bisimulation collapse G0 ; (iii) read back a λ letrec -term L0 from the term graph G0 with the property L0 = G0. This guarantees that L0 and L have the same unfolding, and that L0 exhibits maximal sharing.The procedure for deciding whether two given λ letrec -terms L1 and L2 are unfolding-equivalent computes their term graph interpretations L1 and L2 , and checks whether these term graphs are bisimilar.For illustration, we also provide a readily usable implementation.

Maximal sharing in the Lambda calculus with letrec

“…The implementation is available at http://hackage.haskell.org/package/maxsharing/. Output produced for three examples from this paper, and explanations for it, can be found in Appendix B; for all examples in [16].…”

“…In order to formally define the infinite unfolding of λ letrec -terms we utilise a CRS whose rewrite rules formalise unfolding steps [11]. Every λ letrec -term L that represents an infinite λ-term M can be rewritten by a typically infinite rewrite sequence that converges to M in the limit.…”

“…We focused on eager-scope translations, because they facilitate maximal sharing, and guarantee that interpretations of unfolding-equivalent λ letrec -terms are bisimilar. Yet every scope-closure strategy [11] induces a translation and its own notion of maximal sharing. For adapting our maximal sharing method it is necessary to modify the translation into first-order term graphs in such a way that the image class obtained is closed under homomorphism (T is not closed under →, unlike its subclass Teag).…”

Term Graph Representations for Cyclic Lambda-Terms

“…There are two variants, one with and one without an edge (a back-link) from each variable occurrence to its corresponding abstraction vertex. The class with back-links is related to higher-order term graphs as defined by Blom in [4], and in fact is an adaptation of that concept for the purpose of representing λ -terms.abstraction-prefix based λ -higher-order-term-graphs (Section 4) do not have a scope function but assign, to each vertex w, an abstraction prefix consisting of a word of abstraction vertices that includes those abstractions for which w is in their scope (it actually lists all abstractions for which w is in their 'extended scope' [7]). Abstraction prefixes are aggregations of scope information that is relevant for and locally available at individual vertices.λ -term-graphs with scope delimiters (Section 5) are plain first-order term graphs intended to represent higher-order term graphs of the two sorts above, and in this way stand for λ -terms.…”

“…abstraction-prefix based λ -higher-order-term-graphs (Section 4) do not have a scope function but assign, to each vertex w, an abstraction prefix consisting of a word of abstraction vertices that includes those abstractions for which w is in their scope (it actually lists all abstractions for which w is in their 'extended scope' [7]). Abstraction prefixes are aggregations of scope information that is relevant for and locally available at individual vertices.…”