what's the format of inline formula? #1370
Description
for the paragraph that include inline formula, what's the prompt? OCR or formula recognition? for example: "Let us start formalizing the framework for our work. First, we consider binary boolean models $ \mathcal{M}\colon\{0,1\}^{d}\to\{0,1\} $. Despite our domain being binary, we will need a third value, $ \bot $, to denote “unknown” values. For example, we may represent a person who does have a car, does not have a house, and for whom we do not know if they have a pet or not, as $ \left(1,\,0,\,\bot\right) $. We say elements of $ \{0,1,\bot\}^{d} $ are partial instances, while elements of $ \{0,1\}^{d} $ are simply instances. To illustrate, in Example 1 we used the partial instance $ {\bm{y}}=\left(1,\,1,\,\bot,\,1\right) $ to explain $ \mathcal{M}({\bm{x}})=1 $. We use the notation $ {\bm{y}}\subseteq{\bm{x}} $ to denote that the (partial) instance $ {\bm{x}} $ “fills in” values of the partial instance $ {\bm{y}} $; more formally, we use $ {\bm{y}}\subseteq{\bm{x}} $ to mean that $ y_{i}=\bot\lor y_{i}=x_{i} $ for every $ i\in[d] $. Finally, for any partial instance $ {\bm{y}} $ we denote by $ \operatorname{Comp}({\bm{y}}) $ the set of instances $ {\bm{x}} $ such that $ {\bm{y}}\subseteq{\bm{x}} $, thinking of $ \operatorname{Comp}({\bm{y}}) $ as the set of completions of $ {\bm{y}} $. One can define sufficient reasons as follows with this notation."