Comment by Tomás Orozco (tomas-orozco) on Definitions of Causal Abstraction: Reviewing Beckers & Halpern · 2020-04-29T19:04:06.560Z · LW · GW

Dear John Wentworth:

I have a doubt regarding the implications of Beckers' paper on abstractions. I am a lawyer by profession so I'm venturing pretty far afield here, and I hope my question will not be too trivial.

Given that every constructive abstraction is also a $\tau$ - abstraction, there must be some surjective function ${\tau }_U$ that is compatible with $\tau$ . Hence, for constructive abstractions, must there also be mappings ${\tau }_U^1,\dots ,{\tau }_U^n$ such that, ${\tau }_U\left({\overrightarrow{u}}_L\right)={\tau }_U^1\left({\overrightarrow{z}}_U^1\right)\cdot \dots \cdot {\tau }_U^n\left({\overrightarrow{z}}_U^n\right)$ , where ${\overrightarrow{z}}_U^i$ is the projection of ${\overrightarrow{u}}_L$ onto the variables in ${\overrightarrow{Z}}_U^i$ ? In other words, must there also be a partition of the low-level exogenous variables where each partition is mapped to a distinct high-level variable? I missed ${\tau }_U$ in the definition of constructive abstraction..

Thank you!