frame-wta-1.tex

% SPDX-License-Identifier: CC-BY-4.0
% Copyright 2018 Toni Dietze
\begin{columns}[T]
\column{0.5\linewidth}
	\begin{align*}
		P_1 & = (G_1, ι_1, δ_1)
	\end{align*}
	\begin{align*}
		ι_1\colon
		\nt{S} & ↦ 1 / 2
	\\	\nt{C} & ↦ 1 / 2
	\end{align*}
	\begin{alignat*}{2}
		δ_1\colon
		\nt{S} & → σ(\nt{A}, \nt{C}) && ↦ 1
	\\	\nt{C} & → σ(\nt{B}, \nt{C}) && ↦ 1/3
	\\	\nt{C} & → σ(\nt{B}, \nt{B}) && ↦ 2/3
	\\	\nt{A} & → α && ↦ 1
	\\	\nt{B} & → β && ↦ 1
	\end{alignat*}
\column{0.5\linewidth}
	\begin{overprint}
	\onslide<1>
		\begin{block}{probabilistic rtg (prtg)}
			tuple \(P = (G, ι, δ)\) where
			\begin{itemize}
			\item
				\(G = (N, Σ, I, Δ)\) is an rtg
			\item
				\(ι\colon I → [0, 1]\) \hfill (\emph{initial weights})
			\item
				\(δ\colon Δ → [0, 1]\) \hfill (\emph{rule weights})
			\end{itemize}
		\end{block}
	\onslide<2>
		\begin{center}
		\begin{tikzpicture}[anchor=base, level distance=3em]
			\node (t) {}
				child { node {\(σ\)}
					child { node {\(β\)}
						edge from parent node[left] {\(\nt{B}\)}
					}
					child { node {\(β\)}
						edge from parent node[right] {\(\nt{B}\)}
					}
					edge from parent node[right] {\(\nt{C}\)}
					edge from parent [draw=none]
				};
			\node[left=1em of t-1] {\(t_2\colon\)};
		\end{tikzpicture}
		\end{center}
		\begin{align*}
			\semantics{P_1}(t_2) = {}& ι_1(\nt{C})
		\\	{} ⋅ {}& δ_1(\nt{C} → δ(\nt{B}, \nt{B}))
		\\	{} ⋅ {}& δ_1(\nt{B} → β)
		\\	{} ⋅ {}& δ_1(\nt{B} → β)
		\end{align*}
	\onslide<3>
		\begin{block}{proper prtg}
			A prtg is called \emph{proper}, if
			\begin{itemize}
			\item
				\(\sum_{A \in I} ι(A) = 1\), and
			\item
				\(\forall A \in N\colon\)
				\\\(\sum_{r \in Δ\colon \lhs{r} = A} δ(r) = 1\).
			\end{itemize}
		\end{block}
	\end{overprint}
\end{columns}%