# Reductions in computability theory from a constructive point of view

Here are the slides from my Logic Coloquium 2014 talk in Vienna. This is joint work with Kazuto Yoshimura from Japan Advanced Institute for Science and Technology.

Abstract: In constructive mathematics we often consider implications between non-constructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility, which turns out to have a very rich structure. Even better, under Kleene’s function realizability interpretation instance reducibility corresponds to Weihrauch reducibility, while Kleene’s number realizability relates it to truth-table reducibility. We may also ask about a constructive treatment of other reducibilities in computability theory. I shall discuss how one can tackle Turing reducibility constructively via Kleene’s number realizability.

Slides with talk notes:  lc2014-slides-notes.pdf

# Univalent foundations subsume classical mathematics

A discussion on the homotopytypetheory mailing list prompted me to write this short note. Apparently a mistaken belief has gone viral among certain mathematicians that Univalent foundations is somehow limited to constructive mathematics. This is false. Let me be perfectly clear:

Univalent foundations subsume classical mathematics!

# Free variables are not “implicitly universally quantified”!

Mathematicians are often confused about the meaning of variables. I hear them say “a free variable is implicitly universally quantified”, by which they mean that it is ok to equate a formula $\phi$ with a free variable $x$ with its universal closure $\forall x \,.\, \phi$. I am addressing this post to those who share this opinion.

# Substitution is pullback

I am sitting on a tutorial on categorical semantics of dependent type theory given by Peter Lumsdaine. He is talking about categories with attributes and other variants of categories that come up in the semantics of dependent type theory. He is amazingly good at fielding questions about definitional equality from the type theorists. And it looks like some people are puzzling over pullbacks showing up, which Peter is about to explain using syntactic categories. Here is a pedestrian explanation of a very important fact:

Substitution is pullback.

# On the Bourbaki-Witt Principle in Toposes

Abstract: The Bourbaki-Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.