Vanessa Kosoy comments on Vanessa Kosoy’s Shortform

[Vanessa Kosoy]

Here is a modification of the IBP framework which removes the monotonicity principle, and seems to be more natural in other ways as well.

First, let our notion of “hypothesis” be $\Theta \in {\square }^c(\Gamma \times 2^{\Gamma })$. The previous framework can be interpreted in terms of hypotheses of this form satisfying the condition

$${pr}_{\Gamma \times 2^{\Gamma }}Br\left(\Theta \right)=\Theta$$

(See Proposition 2.8 in the original article.) In the new framework, we replace it by the weaker condition

$$Br\left(\Theta \right)\supseteq ({id}_{\Gamma }\times {diag}_{2^{\Gamma }}{)}_{\ast }\Theta$$

This can be roughly interpreted as requiring that (i) whenever the output of a program P determines whether some other program Q will run, program P has to run as well (ii) whenever programs P and Q are logically equivalent, program P runs iff program Q runs.

The new condition seems to be well-justified, and is also invariant under (i) mixing hypotheses (ii) taking joins/​meets of hypotheses. The latter was not the case for the old condition. Moreover, it doesn’t imply that $\Theta$ is downward closed, and hence there is no longer a monotonicity principle^[1].

The next question is, how do we construct hypotheses satisfying this condition? In the old framework, we could construct hypotheses of the form $\Xi \in {\square }^c(\Gamma \times \Phi )$ and then apply the bridge transform. In particular, this allows a relatively straightforward translation of physics theories into IBP language (for example our treatment of quantum theory). Luckily, there is an analogous construction in the new framework as well.

First notice that our new condition on $\Theta$ can be reformulated as requiring that

• $supp\Theta \subseteq {el}^{\Gamma }$

• For any $s:\Gamma \to \Gamma$ define ${\tau }_s:{\Delta }^c{el}^{\Gamma }\to {\Delta }^c{el}^{\Gamma }$ by ${\tau }_s\theta :={\chi }_{{el}^{\Gamma }}(s\times {id}_{2^{\Gamma }}{)}_{\ast }$. Then, we require ${\tau }_s\Theta \subseteq \Theta$.

For any $\Phi$, we also define ${\tau }_s^{\Phi }:{\Delta }^c({el}^{\Gamma }\times \Phi )\to {\Delta }^c({el}^{\Gamma }\times \Phi )$ by

$${\tau }_s^{\Phi }\theta :={\chi }_{{el}^{\Gamma }\times \Phi }(s\times {id}_{2^{\Gamma }\times \Phi }{)}_{\ast }$$

Now, for any $\Xi \in {\square }^c(\Gamma \times \Phi )$, we define the “conservative bridge transform^[2]” $CBr\left(\Xi \right)\in {\square }^c(\Gamma \times 2^{\Gamma }\times \Phi )$ as the closure of all ${\tau }_s^{\Phi }\theta$ where $\theta$ is a maximal element of $Br\left(\Xi \right).$ It is then possible to see that $\Theta \in {\square }^c(\Gamma \times 2^{\Gamma })$ is a valid hypothesis if and only if it is of the form ${pr}_{\Gamma \times 2^{\Gamma }}CBr\left(\Xi \right)$ for some $\Phi$ and $\Xi \in {\square }^c(\Gamma \times \Phi )$.

1. ^

   I still think the monotonicity principle is saying something about the learning theory of IBP which is still true in the new framework. Namely, it is possible to learn that a program is running but not possible to (confidently) learn that a program is not running, and this limits the sort of frequentist guarantees we can expect.

2. ^

   Intuitively, it can be interpreted as a version of the bridge transform where we postulate that a program doesn’t run unless $\Xi$ contains a reason while it must run.