2018-05-04: Fredrik Nordvall Forsberg (MSP)

Sat, 21 Oct 2023 21:24:08 +0100

Fredrik Nordvall Forsberg (MSP)

MSP 101: The Dialectica Interpretation

Abstract

A proof interpretation translates proofs of one logical system into proofs of another (example: the double-negation interpretation of classical logic into constructive logic). This often reveals some information about the original system (e.g. classical logic is equiconsistent with constructive logic). Gödel's Dialectica interpretation (named after the journal it was published in) translates Heyting arithmetic (the constructive theory of the natural numbers, including induction) into System T (the quantifier-free theory of the simply typed lambda calculus with natural numbers) — quantifier complexity is traded for higher type complexity. Combining this translation with (a refined) double negation translation, one can extract System T programs from a proof of a forall-exists statement, even if this proof is using non-constructive priciples such as Markov's Principle, Excluded Middle, or the Quantifier-Free Axiom of Choice. I've always found the Dialectica translation mystifying, so I'll try to explain the intuition behind it.

2018-05-04 11:00:00 UTC
LT1310