Classical proofs rely on effective numberings of partial recursive functions, which obscure key structural details behind arbitrary numerical reductions and hand-wavy treatments. Setting the identity function within the traditional recursion theorem technically generates an index that outputs itself, but lambda calculus offers a clearer, proof-theoretic alternative. This refined technique draws on functional programming principles by conceptualizing a self-reflective language that quotes terms into abstract syntax trees. Incorporating typed perspectives further illuminates hidden conceptual connections, even while conducting the underlying proof in an untyped environment.
Key points
Untyped lambda calculus serves as an alternative model that prioritizes underlying system…: Untyped lambda calculus serves as an alternative model that prioritizes underlying system structure over low-level numerical encodings.
The updated construction uses functional programming concepts to model self-reflection by…: The updated construction uses functional programming concepts to model self-reflection by quoting terms into abstract syntax trees.
Informal perspectives from type systems help expose valuable theoretical connections thro…: Informal perspectives from type systems help expose valuable theoretical connections throughout the untyped formalization process.
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Table of contents
- Introduction
- Kleene, Classical Perspective
- Models of Computation
- Kleene, Functional Programming Perspective
- Building a Quine in Untyped Lambda Calculus
- Putting It All Together
- References
Introduction
It is a well-known fact that the existence of Quines are a consequence of Kleene’s 2nd recursion theorem. I was curious about how easy or difficult it was to use this construction in practice, so I decided to try to formalize it.
One of the virtues of formalization is that it naturally motivates the mathematician to structure their definitions and proofs in a clearer and more concise fashion. The pain of a clunky definition or ill-structured argument is actually felt, and so the goals of expedient proofs and well-structured proofs often align. Far from being a mechanical chore, the act of formalization can actually be a source of insight and creativity (incidentally, this is why I think the popular view of formal verification as a mere exercise in transcription misses the mark).
With this in mind, I hoped that I could find or develop a clean…
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