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Message definition (Diffusion Models Are Real-Time Game Engines ) An ''Interactive Environment'' <math>\mathcal{E}</math> consists of a space of latent states <math>\mathcal{S}</math>, a space of partial projections of the latent space <math>\mathcal{O}</math>, a partial projection function <math>V: \mathcal{S} \rightarrow \mathcal{O}</math>, a set of actions <math>\mathcal{A}</math>, and a transition probability function <math>p \left( s \,|\, a, s^{\prime} \right)</math> such that <math>s, s^{\prime} \in \mathcal{S}, a \in \mathcal{A}</math>.
An Interactive Environment
E
{\displaystyle {\mathcal {E}}}
S
{\displaystyle {\mathcal {S}}}
O
{\displaystyle {\mathcal {O}}}
V
:
S
→
O
{\displaystyle V:{\mathcal {S}}\rightarrow {\mathcal {O}}}
A
{\displaystyle {\mathcal {A}}}
p
(
s
|
a
,
s
′
)
{\displaystyle p\left(s\,|\,a,s^{\prime }\right)}
s
,
s
′
∈
S
,
a
∈
A
{\displaystyle s,s^{\prime }\in {\mathcal {S}},a\in {\mathcal {A}}}
