uiai

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
uiai [2026/03/17 01:39] – [Assumption: Standard setup] pedroortegauiai [2026/03/17 01:40] (current) – [Definition: Counterfactual action] pedroortega
Line 398: Line 398:
 Generate $(\dot{\gamma}_j,\dot{x}_j)_{j \ge k}$ as follows: Generate $(\dot{\gamma}_j,\dot{x}_j)_{j \ge k}$ as follows:
  
-- **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$, $\dot{x}_{\le k-1} := x_{\le k-1}$.+  * **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$, $\dot{x}_{\le k-1} := x_{\le k-1}$.
  
-- **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_k := 1$.+  * **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_k := 1$.
  
-- **Evolve branch chronologically:** For $j \ge k$, first sample the next substrate symbol by $\dot{x}_j \sim \mu(\cdot \mid \underline{a\hat{o}}_{<t} a_t\,w\,\dot{x}_{k:j-1})$, so $\mu$ emits the content of the forced $\mathcal{A}$-block in the branch, conditioned on the shared past and the already-emitted branch block prefix. Then sample the next gate value by+  * **Evolve branch chronologically:** For $j \ge k$, first sample the next substrate symbol by $\dot{x}_j \sim \mu(\cdot \mid \underline{a\hat{o}}_{<t} a_t\,w\,\dot{x}_{k:j-1})$, so $\mu$ emits the content of the forced $\mathcal{A}$-block in the branch, conditioned on the shared past and the already-emitted branch block prefix. Then sample the next gate value by
 $$ $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, \dot{x}_{\le j}).   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, \dot{x}_{\le j}).
Line 426: Line 426:
 To define the world’s $\mathcal{A}$-continuation at $k$, run the following tokenization procedure, initialized from the already-written on-path transcript up to $k-1$. Let $(\dot{\gamma}_j)_{j \ge k}$ be generated as follows: To define the world’s $\mathcal{A}$-continuation at $k$, run the following tokenization procedure, initialized from the already-written on-path transcript up to $k-1$. Let $(\dot{\gamma}_j)_{j \ge k}$ be generated as follows:
  
-- **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.+  * **Shared prefix:** Set $\dot{\gamma}_{\le k-1} := \gamma_{\le k-1}$.
  
-- **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.+  * **Force an $\mathcal{A}$-block start:** Set $\dot{\gamma}_{k} := 1$.
  
-- **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by +  * **Read transcript chronologically:** For $j \ge k$, let $x_j$ be the next substrate symbol generated by the world on-path. Then sample the next gate value by 
 $$ $$
   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, x_{\le j}).   \dot{\gamma}_{j+1} \sim \Gamma(\cdot \mid \dot{\gamma}_{\le j}, x_{\le j}).
  • uiai.1773711582.txt.gz
  • Last modified: 2026/03/17 01:39
  • by pedroortega