Parsing Theory: Context-Free Grammars

CFG $G = (N,T,P,S)$ : non-terminals, terminals, productions and start symbol ( $S \in N$ ). Let $\Sigma = N \cup T$ be the set of all symbols. The productions relate nonterminals to strings of terminals and non-terminals: $P \subseteq N \times \Sigma^*$ .
Derivation is defined as a relation $\{ \alpha A \beta \Rightarrow \alpha \gamma \beta \mid A \in N, (A,\gamma) \in P \}$ The transitive closure of $\Rightarrow$ if $\stackrel{*}{\Rightarrow}$ .
$L(G) = \{{w \in T^* \mid S \stackrel{*}{\Rightarrow}w}\}$
CFL ${} = \{L \mid \exists G=(N,T,P,S), L=L(G) \}$

Parsing Theory: Push-Down Automata

PDA $M=(Q,T,\Gamma,\delta,q_0,Z_0,F)$ : states, symbols, stack symbols, transitions ( $\delta \subseteq (Q \times T \cup \{ \epsilon \} \times \Gamma) \times (Q \times \Gamma^*)$ ), initial state ( $q_0 \in Q$ ), initial stack ( $Z_0 \in \Gamma$ ), final states ( $F \subseteq Q$ ).
Deterministic: $((q,a,Z),(p,\alpha)),((q,a,Z),(p',\alpha')) \in \delta (a \in T) \Longrightarrow p=p', \alpha=\alpha', \not\exists ((q,\epsilon,Z),(p',\alpha)) \in \delta$
Instantaneous description (ID ${} \in Q \times T^* \times \Gamma^*$ )
Movement: $(q,aw,Z\alpha) \vdash (p,w,\beta\alpha)$ if $((q,a,Z),(p,\beta)) \in \delta$ where $a \in T \cup \{{\epsilon}\}$ . The transitive closure of $\vdash$ is $\stackrel{\raisebox{-1.0ex}[0pt][0pt]{\tiny {\mbox{{$\,$} *{}}}}}{\vdash}$ .
$L(M) = \{{w \in T^* \mid (q_0,w,Z_0) \stackrel{\raisebox{-1.0ex}[0pt][0pt]{\tiny {\mbox{{$\,$} *{}}}}}{\vdash} (p,\epsilon,\epsilon), p \in F}\}$

CFL = $\{{L(M)\mid M=(\ldots)}\}$
But nondeterministic PDAs are impractical, so we want to use DPDAs which are not equivalent in power to NDPDAs.
DCFL = $\{{L(M)\mid M=(\ldots), M \textrm{ deterministic}}\}$ .

What does a DCFL look like ?

Bottom-up left-to-right parsing: Rightmost derivation, in reverse.
Rightmost derivation: $\{ \alpha A w \stackrel{\textstyle{\Rightarrow}}{\scriptscriptstyle{\mathbf{rm}}}\alpha \gamma w \mid A \in N, (A,\gamma) \in P, w \in T^* \}$
Shift-reduce parsing: A special kind of DPDA (but equivalent in power).

Handle of $\psi$ is $\alpha\beta$ where

$\begin{displaymath}S \stackrel{*}{\stackrel{\textstyle{\Rightarrow}}{\scriptscri... ...}}}\delta\alpha\beta w = \psi, \quad A \rightarrow \alpha\beta \end{displaymath}$
A viable prefix $\gamma$ of $\psi$ is a (perhaps trivial) prefix of $\delta\alpha\beta$ .
An item $A \rightarrow \alpha \cdot \beta$ is valid for a viable prefix $\gamma$ if

$\begin{displaymath} S \stackrel{*}{\stackrel{\textstyle{\Rightarrow}}{\scriptscr... ...criptstyle{\mathbf{rm}}}\delta \alpha \beta w = \gamma \beta w \end{displaymath}$
A complete item is of form $A \rightarrow \alpha \cdot$ .

John Tang Boyland
2004-09-09