LL Parsing Terminology (1)

LL = ``Predictive'' = ``top-down'' parsing;

• How can a derivation start out? (used for LL$(k)$):
  $$\begin{eqnarray*} \mathrm{First}_k(A) &=& \{{w \in T^{0..k} \mid A \stackrel{*... ...ckrel{*}{\Rightarrow}wx, (x = \epsilon \vee \vert w\vert = k)}\} \end{eqnarray*}$$
  
  
  
  
  

LL Parsing Terminology (2)

When the First sets don't contain long enough strings, we need to know what may follow the nonterminal.

• What strings may come after a nonterminal in a valid yield?
  
  $$\mathrm{Follow}_k(A) = \{{w \in {T\cup\{{\$}\}}^{0..k} \mi... ...ghtarrow}\alpha A wx, (\vert w\vert = k \vee x = \epsilon)}\}$$
  
  
  In particular except for useless nonterminals.
• Prediction sets (improved)
  
  $$\mathrm{Predict}_k(A \rightarrow \alpha) = \{{yz \mid % \in ... ...rst}_k(\alpha), z \in \mathrm{Follow}_{k-\vert y\vert}(A)}\}$$
  
  

Computing First$_k$

• For each $A \to \alpha$:
  $$\begin{eqnarray*} \mathrm{First}_0(.) &=& \{{\epsilon}\} \\ \mathrm{First}_k(A)... ...rm{First}_k(s), x \in \mathrm{First}_{k-\vert w\vert}(\alpha)}\} \end{eqnarray*}$$
  
  
  
  
  

Computing Follow$_k$

• For each $A \to \alpha B \beta$:
  $$\begin{eqnarray*} \mathrm{Follow}_k(B) &\supseteq& \{{ wx \mid w \in \textrm{First}_k(\beta), x \in \textrm{Follow}_{k-\vert w\vert}(A) }\} \end{eqnarray*}$$