Strict Ideal Completions of the Lambda Calculus

The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of strictness. To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend $\beta$-reduction with infinitely many `$\bot$-rules', which contract meaningless terms directly to $\bot$. Three of the resulting B\"ohm reduction calculi have unique infinitary normal forms corresponding to B\"ohm-like trees. In this paper we develop a corresponding theory of infinitary lambda calculi based on ideal completion instead of metric completion. We show that each of our calculi conservatively extends the corresponding metric-based calculus. Three of our calculi are infinitarily normalising and confluent; their unique infinitary normal forms are exactly the B\"ohm-like trees of the corresponding metric-based calculi. Our calculi dispense with the infinitely many $\bot$-rules of the metric-based calculi. The fully non-strict calculus (called $111$) consists of only $\beta$-reduction, while the other two calculi (called $001$ and $101$) require two additional rules that precisely state their strictness properties: $\lambda x.\bot \to \bot$ (for $001$) and $\bot\,M \to \bot$ (for $001$ and $101$)

Confluence of nearly orthogonal infinitary term rewriting systems

We give a relatively simple coinductive proof of confluence, modulo equivalence of root-active terms, of nearly orthogonal infinitary term rewriting systems. Nearly orthogonal systems allow certain root overlaps, but no non-root overlaps. Using a slightly more complicated method we also show confluence modulo equivalence of hypercollapsing terms. The condition we impose on root overlaps is similar to the condition used by Toyama in the context of finitary rewriting

On Undefined and Meaningless in Lambda Definability

We distinguish between undefined terms as used in lambda definability of partial recursive functions and meaningless terms as used in infinite lambda calculus for the infinitary terms models that generalise the Bohm model. While there are uncountable many known sets of meaningless terms, there are four known sets of undefined terms. Two of these four are sets of meaningless terms. In this paper we first present set of sufficient conditions for a set of lambda terms to serve as set of undefined terms in lambda definability of partial functions. The four known sets of undefined terms satisfy these conditions. Next we locate the smallest set of meaningless terms satisfying these conditions. This set sits very low in the lattice of all sets of meaningless terms. Any larger set of meaningless terms than this smallest set is a set of undefined terms. Thus we find uncountably many new sets of undefined terms. As an unexpected bonus of our careful analysis of lambda definability we obtain a natural modification, strict lambda-definability, which allows for a Barendregt style of proof in which the representation of composition is truly the composition of representations

Glueability of Resource Proof-Structures: Inverting the Taylor Expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures