Asymptotic Properties of Combinatory Logic

Rewriting Properties of Combinators for Intuitionistic Linear Logic

3 1 M ar 2 00 9 Some properties of random λ - terms

We show various (syntactic and behavioral) properties of random λ-terms. Our main results are that at least 3/4 of the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the λ-calculus into combinators) the result is different. We show that almost all terms are not strongly normalizing, beca...

On the likelihood of normalisation in combinatory logic

We present a quantitative basis-independent analysis of combinatory logic. Using a general argument regarding plane binary trees with labelled leaves, we generalise the results of David et al. (see [9]) and Bendkowski et al. (see [6]) to all Turing-complete combinator bases proving, inter alia, that asymptotically almost no combinator is strongly normalising nor typeable. We exploit the structu...

Some properties of random λ-terms∗

We present quantitative analysis of various (syntactic and behavioral) properties of random λ-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the λ-calculus into combinators) the result is exactly opposite. We show that almost all te...

Towards Combinatory Logic Synthesis

A theory of composition synthesis based on inhabitation in combinatory logic is described and illustrated with examples. Composition synthesis automatically generates applicative combinations of typed combinators inhabiting a goal type. In the intended applications, combinatory type environments model repositories of components, and types are used to specify their interfaces enriched with seman...