DeepAI

# Combinatorics of explicit substitutions

λυ is an extension of the λ-calculus which internalises the calculus of substitutions. In the current paper, we investigate the combinatorial properties of λυ focusing on the quantitative aspects of substitution resolution. We exhibit an unexpected correspondence between the counting sequence for λυ-terms and famous Catalan numbers. As a by-product, we establish effective sampling schemes for random λυ-terms. We show that typical λυ-terms represent, in a strong sense, non-strict computations in the classic λ-calculus. Moreover, typically almost all substitutions are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue that λυ is an intrinsically non-strict calculus of explicit substitutions. Finally, we investigate the distribution of various redexes governing the substitution resolution in λυ and investigate the quantitative contribution of various substitution primitives.

• 8 publications
• 6 publications
02/02/2018

### Counting Environments and Closures

Environments and closures are two of the main ingredients of evaluation ...
12/11/2018

### Towards the average-case analysis of substitution resolution in λ-calculus

Substitution resolution supports the computational character of β-reduct...
04/27/2021

### The ksmt calculus is a δ-complete decision procedure for non-linear constraints

ksmt is a CDCL-style calculus for solving non-linear constraints over re...
01/11/2022

### A Faithful and Quantitative Notion of Distant Reduction for Generalized Applications (Long Version)

We introduce a call-by-name lambda-calculus λ J with generalized applica...
11/02/2021

### A strong call-by-need calculus

We present a call-by-need λ-calculus that enables strong reduction (that...
05/17/2018

### Strict Ideal Completions of the Lambda Calculus

The infinitary lambda calculi pioneered by Kennaway et al. extend the ba...
08/20/2018

### Lambda Calculus with Explicit Read-back

This paper introduces a new term rewriting system that is similar to the...