Bohm Trees Research Articles

Revisiting Call-by-value B\"ohm trees in light of their Taylor expansion

The call-by-value lambda calculus can be endowed with permutation rules, arising from linear logic proof-nets, having the advantage of unblocking some redexes that otherwise get stuck during the reduction. We show that such an extension allows to define a satisfying notion of Bohm(-like) tree and a theory of program approximation in the call-by-value setting. We prove that all lambda terms having the same Bohm tree are observationally equivalent, and characterize those Bohm-like trees arising as actual Bohm trees of lambda terms. We also compare this approach with Ehrhard's theory of program approximation based on the Taylor expansion of lambda terms, translating each lambda term into a possibly infinite set of so-called resource terms. We provide sufficient and necessary conditions for a set of resource terms in order to be the Taylor expansion of a lambda term. Finally, we show that the normal form of the Taylor expansion of a lambda term can be computed by performing a normalized Taylor expansion of its Bohm tree. From this it follows that two lambda terms have the same Bohm tree if and only if the normal forms of their Taylor expansions coincide.

Redexes are stable in the λ-calculus

In the theory of sequential programming languages, Berry's stability property is an important step claiming that values have unique origins in their calculations. It has been shown that Bohm trees are stable in the λ-calculus, meaning that there is a unique minimum prefix of any lambda term which computes its Bohm tree. Moreover this property is also true for finite prefixes of Bohm trees. The proof relies on Curry's standardization theorem as initially pointed out by Plotkin in his famous articleLCF considered as a programming language. In this paper we will show that the stability property also holds for redexes. Namely for any redex family, there is a unique minimum calculation and a unique redex which computes this family. This property was already known in the study of optimal reductions, but we stress here on stability and want to show that stability is inside the basic objects of calculations. The proof will be based on nice commutations between residuals and creations of new redexes. Our tool for proving this property will be the labelled lambda calculus used in the study of optimal reductions.

Relating graph and term rewriting via Böhm models

Dealing properly with sharing is important for expressing some of the common compiler optimizations as source-to-source transformations, such as common subexpressions elimination, lifting of free expressions and removal of invariants from a loop. Term graph rewriting is a computational model to accommodate these concerns. In this paper we are interested in defining a term model for term graph rewriting systems, which allows us to prove total correctness of those optimizations. We introduce the notion of Bohm tree, and show that for orthogonal term graph rewriting systems, Bohm tree equivalence defines a congruence. Total correctness then follows in a straightforward way from showing that if a programM contains less sharing than a programN, then bothM andN have the same Bohm tree. Using Bohm trees we also show that orthogonal term graph rewriting systems are a correct implementation of orthogonal term rewriting systems. This boils down to showing that the behavior of a term graph can be deduced from its finite approximations, that is, graph rewriting is a continuous operation. Our approach differs from that of other researchers which is based on infinite rewriting.