Viewing λ-terms through maps

Abstract

In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λ term. We use binary trees over 0 and 1 as maps, but some well-formedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λ term). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λ terms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively.The map representation has good properties. Substitution does not require adjustment of binding information: neither α conversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation.Using the notion of map we study conventional raw λ syntax. E.g. we give, and prove correct, a decision procedure for α equivalence of raw λ terms that does not require fresh names.We conclude with a definition of β reduction for map terms, some discussion on the limitations of our current work, and suggestions for future work.

Hashing modulo alpha-equivalence

“…That is, it is possible to implement rebuild :: ESummary -> Expression so that rebuild (summariseExpr e) is 𝛼-equivalent to e; in other words, an ESummary loses no necessary information. Indeed, others have suggested using a representation in which a lambda contains a list of the occurrences of its bound variable as the primary representation of lambda terms [1,12,17].…”

“…The idea of representing a lambda with a "map" of the occurrences of its bound variable, which we adopt for our e-summaries in Section 4, has been studied before [1,17]. Kennaway and Sleep describe another representation, director strings, in which information about occurrences is stored in the application nodes, rather than the lambdas [10].…”

Everybody's Got To Be Somewhere

“…My explorations follow Sato, Pollack, Schwichtenberg and Sakurai, whose λ -terms make binding sites carry maps of use sites [21]. E.g., the K and S combinators become (respectively) names λ c. λ e. c λ f .…”

“…My explorations follow Sato, Pollack, Schwichtenberg and Sakurai, whose λ -terms make binding sites carry maps of use sites [21]…”