Can a calculus have incremental copying and closed scopes?
Can a calculus have incremental copying and closed scopes?
A few days ago, I proposed the Abstract Calculus, a minimal untyped language that is very similar to the Lambda Calculus, except for the main difference that substitutions are O(1) (i.e., variables only occur once) and copying is an explicit, constant-time and incremental operation. Here is an example:
let (a & b) = (λx. λy. λz. y) in (a & b)
---------------------------------------- step 0
let (a & b) = (λy. λz. y) in (λx0. a) & (λx1. b)
------------------------------------------------ step 1
let (a & b) = (λz. y0 & y1) in (λx0. λy0. a) & (λx1. λy1. b)
------------------------------------------------------------ step 2
let (a & b) = y0 & y1 in (λx0. λy0. λz0. a) & (λx1. λy1. λz1. b)
---------------------------------------------------------------- step 3
(λx0. λy0. λz0. y0) & (λx1. λy1. λz1. y1)
Here, the let (a & b) = t in k syntax substitutes the occurrences of a and b in k by t, except it does so incrementally: notice how each lambda is pulled one by one.
There is a problem, though. Eventually, you'll try to pull a lambda which has a variable that is bound in t. That's occurs with y behind step 1. To solve that issue, whenever that happens, the bound variable y is replaced by y0 & y1, which are bound to each of the two copied lambdas, respectively. The problem with that is this requires us to abandon the notion of scopes, as, now, the bound variable of a λ can occur outside its own body (such as y0 and y1 behind step 2)!
Despite sounding bizarre, the language operates very well. You can represent the usual λ-encodings on it, implement recursive algorithms and it is compatible with the oracle-free fragment of the optimal reduction algorithm. Yet, some have questioned the absence of scopes. I've spent some time reasoning about this and couldn't find a way to have both things. In fact, it looks impossible. As a sanity check, I'd like to know if my reasoning makes sense.
Thus, my question is: is that possible? Can we have both incremental copying and closed scopes in a calculus?
K2-Think
The question of whether a calculus can simultaneously support incremental copying (with explicit, constant-time operations) and closed scopes (where bound variables are confined to their syntactic scopes) can be analyzed as follows:
Key Concepts:
- Incremental Copying: An explicit mechanism to duplicate terms efficiently, often by decomposing the term incrementally to avoid redundant work...
GPT-5
Short answer
Not in the purely syntactic way you are doing it. If you insist on (i) local, constant‑time “incremental” copying that can pull a λ out across the duplicator and (ii) the usual lexical notion of scope (a binder’s variable never occurs outside its body), then the two requirements conflict. To keep closed scopes, crossing a λ must bring along its whole body at once; to keep copying inc...
Claude Opus 4
This is a fascinating question about the fundamental trade-offs in calculus design. Your reasoning about the apparent impossibility of having both incremental copying and closed scopes appears to be correct. Let me explain why this tension exists and why it seems to be fundamental.
The Core Tension
The key insight is that incremental copying and closed scopes are fundamentally at odds because...
o3 Pro
Short answer
• Inside an ordinary λ-term that is represented as a tree you cannot have all three of
- textual (tree-shaped) scopes,
- the guarantee that each variable occurs only once, and
- an operator that produces two independent copies of a term
by doing only O (1) local work per reduction step.
As soon as you start to “pull a λ out of the body’’ the occurrences o...
Gemini 2.5 Pro
Excellent question. You've hit upon a deep and fundamental tension in the design of computational calculi. The short answer is:
No, it is not possible to have both incremental copying (as you've defined it) and closed scopes (in the traditional, lexical sense). They represent fundamentally different ways of handling variable binding and substitution.
Let's break down why this conflict is...
DeepSeek R1
Claude 3.7 Sonnet
Can a calculus have incremental copying and closed scopes?
Based on your description of the Abstract Calculus, I believe your intuition is correct - it seems very difficult, if not impossible, to have both incremental copying and closed scopes in the same calculus.
Why the tension exists
The fundamental issue stems from the very nature of incremental copying. When you copy a lambda express...
o4 mini
Here is a sketch of why, in any pure term‐based calculus with named λ-binders, you cannot at the same time
- do incremental, constant-time copying of an arbitrary subterm (by “pulling” it out one node at a time), and
- preserve the usual invariant of closed scopes (i.e. no variable ever escapes the body of its own λ‐binder, up to α‐equivalence),
unless you give up on “terms wit...
o3
Yes, the two requirements are compatible; what is incompatible with ordinary, closed lexical scopes is the particular way in which the Abstract-Calculus tries to perform the copy, namely by “pulling the λ’s out” through the copy operator while keeping the body unchanged. If you insist on doing the copy in exactly that fashion then you are forced either
• to rename (α-convert) the bound variab...
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