Christian Lawson-Perfect @christianp

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**Andrej Bauer** @andrejbauer · Jan 13 *

I tried https://notebooklm.google on two papers of mine. It's advertised as "Your Personalized AI Research Assistant". The short summary is that the tool is exactly as good as an insolent incompetent science journalist. When confronted with its own factual mistakes, it tries to blame the paper instead. (1/3)

notebooklm.googleGoogle NotebookLM | Note Taking & Research Assistant Powered by AIUse the power of AI for quick summarization and note taking, NotebookLM is your powerful virtual research assistant rooted in information you can trust.

**Andrej Bauer** @andrejbauer · Jan 13 *

Jan 13 *

Andrej Bauer @andrejbauer

The first example was the paper https://arxiv.org/abs/2409.17664 on Comodule representations of second-order functionals, co-authored with Danel Ahman. The AI told me that the paper restricted to representations to only finitely-branching trees. When asked to cite the place in the paper where such a restriction is enforced, it said that finite branching is "strongly implied" by the requirement that the trees must be well-founded. Then I confronted it with the fact that the introduction gives the example of countably branching trees, so clearly the authors did not intend finite branching. The response was that the authors misrepresented their work by giving such an example. When forcilby told that it was wrong, the AI eventually admitted its initial summary of the paper was incorrect. (2/3)

arXiv.orgComodule Representations of Second-Order FunctionalsWe develop and investigate a general theory of representations of second-order functionals, based on a notion of a right comodule for a monad on the category of containers. We show how the notion of comodule representability naturally subsumes classic representations of continuous functionals with well-founded trees. We find other kinds of representations by varying the monad, the comodule, and in some cases the underlying category of containers. Examples include uniformly continuous or finitely supported functionals, functionals querying their arguments precisely once, or at most once, functionals interacting with an ambient environment through computational effects, as well as functionals trivially representing themselves. Many of these rely on our construction of a monad on containers from a monad on shapes and a weak Mendler-style monad algebra on the universe for positions. We show that comodule representability on the category of propositional containers, which have positions valued in a universe of propositions, is closely related to instance reducibility in constructive mathematics, and through it to Weihrauch reducibility in computability theory.

**Andrej Bauer** @andrejbauer · Jan 13 *

Jan 13 *

Andrej Bauer @andrejbauer

The second examples was the paper https://arxiv.org/abs/2404.01256 on countable reals, co-authored with James Hanson. Here the AI told me that both exacluded middle and the axiom of choice are needed to carry out Cantor's diagonal argument. When I asked whether it meant "and" or "or", it doubled down and claimed the authors of the paper claim both are needed. I asked for the specific quote from the paper, and received one that used the word "or". I pointed out to the AI that that is clearly an "or", and it responded by blaiming the authors for making the mistake of interpreting "or" as an "and". Again it took a couple more iterations to get things straight.

LLMs may be good for some things, but extracting factually correct summaries from scientific papers isn't one of them. (3/3)

arXiv.orgThe Countable RealsWe construct a topos in which the Dedekind reals are countable. To accomplish this, we first define a new kind of toposes that we call parameterized realizability toposes. They are built from partial combinatory algebras whose application operation depends on a parameter, and in which realizers operate uniformly with respect to a given parameter set. Our topos is the parameterized realizability topos whose realizers are oracle-computable partial maps, with oracles serving as parameters and ranging over the representations of a non-diagonalizable sequence, discovered by Joseph Miller. It is a sequence of reals in $[0,1]$ that is non-diagonalizable in the sense that any real in $[0,1]$ that is oracle-computable, uniformly in oracles representing the sequence, must already appear in the sequence. The Dedekind reals are countable in the topos because the non-diagonalizable sequence appears in it as an epimorphism. The topos is intuitionistic, as it invalidates both the law of excluded middle and the axiom of countable choice. The Cauchy reals are uncountable. The Hilbert cube is countable, from which Brouwer's fixed-point theorem follows as an easy corollary of Lawvere's fixed-point theorem. From the 1-dimensional Brouwer's fixed-point theorem we obtain the intermediate value theorem and the lesser limited principle of omniscience. The Kreisel-Lacombe-Shoenfield-Tseitin theorem stating that all real-valued maps are continuous is valid, because the usual proof is uniform with respect to oracles. Lastly, the closed interval $[0,1]$, being countable, can trivially be covered by a sequence of open intervals whose lengths add up to any prescribed $0 < ε< 1$, and such a cover has no finite subcover. However, we show that any sequence of open intervals with rational endpoints covering $[0,1]$ must has a finite subcover.