diff --git a/info/exercises/ex-03-sol.pdf b/info/exercises/ex-03-sol.pdf
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diff --git a/info/exercises/ex-03.pdf b/info/exercises/ex-03.pdf
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diff --git a/info/exercises/src/ex-03/ex/compute.tex b/info/exercises/src/ex-03/ex/compute.tex
index e7792fd2ce6bfe72241192bbc3935588cdeefe16..41072624c821e6595aa6800bb18b7d9c0ace52ee 100644
--- a/info/exercises/src/ex-03/ex/compute.tex
+++ b/info/exercises/src/ex-03/ex/compute.tex
@@ -66,16 +66,16 @@
\begin{align*}
S &::= Exp~\mathbf{EOF} \\
- Exp &::= Exp_2~ Exp_* \\
- Exp_* &::= +~ Exp_2~ Exp_* \\
- Exp_* &::= -~ Exp_2~ Exp_* \\
- Exp_* &::= \\
- Exp_2 &::= Exp_3~ Exp_{2*} \\
- Exp_{2*} &::= *~ Exp_3~ Exp_{2*} \\
- Exp_{2*} &::= /~ Exp_3~ Exp_{2*} \\
- Exp_{2*} &::= \\
- Exp_3 &::= \mathbf{num} \\
- Exp_3 &::= (Exp) \\
+ Exp &::= Term~ Add \\
+ Add &::= +~ Term~ Add \\
+ Add &::= -~ Term~ Add \\
+ Add &::= \\
+ Term &::= Factor~ Mul \\
+ Mul &::= *~ Factor~ Mul \\
+ Mul &::= /~ Factor~ Mul \\
+ Mul &::= \\
+ Factor &::= \mathbf{num} \\
+ Factor &::= (Exp) \\
\end{align*}
\begin{enumerate}
@@ -104,31 +104,34 @@
\item \(\nullable\):
%
\begin{align*}
+ \nullable(S) &= false \\
\nullable(Exp) &= false \\
- \nullable(Exp_*) &= true \\
- \nullable(Exp_2) &= false \\
- \nullable(Exp_{2*}) &= true \\
- \nullable(Exp_3) &= false
+ \nullable(Add) &= true \\
+ \nullable(Term) &= false \\
+ \nullable(Mul) &= true \\
+ \nullable(Factor) &= false
\end{align*}
\item \(\first\): we have constraints:
%
\begin{align*}
- \first(Exp) &= \first(Exp_2) \\
- \first(Exp_*) &= \{+\} \cup \{-\} \cup \emptyset \\
- \first(Exp_2) &= \first(Exp_3) \\
- \first(Exp_{2*}) &= \{*\} \cup \{/\} \cup \emptyset \\
- \first(Exp_3) &= \{\mathbf{num}\} \cup \{(\}
+ \first(S) &= \first(Exp) \\
+ \first(Exp) &= \first(Term) \\
+ \first(Add) &= \{+\} \cup \{-\} \cup \emptyset \\
+ \first(Term) &= \first(Factor) \\
+ \first(Mul) &= \{*\} \cup \{/\} \cup \emptyset \\
+ \first(Factor) &= \{\mathbf{num}\} \cup \{(\}
\end{align*}
%
which can be solved to get:
%
\begin{align*}
+ \first(S) &= \{\mathbf{num}, (\} \\
\first(Exp) &= \{\mathbf{num}, (\} \\
- \first(Exp_*) &= \{+, -\} \\
- \first(Exp_2) &= \{\mathbf{num}, (\} \\
- \first(Exp_{2*}) &= \{*, /\} \\
- \first(Exp_3) &= \{\mathbf{num}, (\}
+ \first(Add) &= \{+, -\} \\
+ \first(Term) &= \{\mathbf{num}, (\} \\
+ \first(Mul) &= \{*, /\} \\
+ \first(Factor) &= \{\mathbf{num}, (\}
\end{align*}
\item \(\follow\): we have constraints (for each rule, except
empty/terminal rules):
@@ -137,25 +140,25 @@
\begin{align*}
\{\mathbf{EOF}\} &\subseteq \follow(Exp) \\
&\\
- \first(Exp_*) &\subseteq \follow(Exp_2) \\
- \follow(Exp) &\subseteq \follow(Exp_2) \\
- \follow(Exp) &\subseteq \follow(Exp_*) \\
+ \first(Add) &\subseteq \follow(Term) \\
+ \follow(Exp) &\subseteq \follow(Term) \\
+ \follow(Exp) &\subseteq \follow(Add) \\
&\\
- \first(Exp_*) &\subseteq \follow(Exp_2) \\
- \follow(Exp_*) &\subseteq \follow(Exp_2) \\
+ \first(Add) &\subseteq \follow(Term) \\
+ \follow(Add) &\subseteq \follow(Term) \\
&\\
- \first(Exp_*) &\subseteq \follow(Exp_2) \\
- \follow(Exp_*) &\subseteq \follow(Exp_2) \\
+ \first(Add) &\subseteq \follow(Term) \\
+ \follow(Add) &\subseteq \follow(Term) \\
&\\
- \first(Exp_{2*}) &\subseteq \follow(Exp_3) \\
- \follow(Exp_2) &\subseteq \follow(Exp_3) \\
- \follow(Exp_2) &\subseteq \follow(Exp_{2*}) \\
+ \first(Mul) &\subseteq \follow(Factor) \\
+ \follow(Term) &\subseteq \follow(Factor) \\
+ \follow(Term) &\subseteq \follow(Mul) \\
&\\
- \first(Exp_{2*}) &\subseteq \follow(Exp_3) \\
- \follow(Exp_{2*}) &\subseteq \follow(Exp_3) \\
+ \first(Mul) &\subseteq \follow(Factor) \\
+ \follow(Mul) &\subseteq \follow(Factor) \\
&\\
- \first(Exp_{2*}) &\subseteq \follow(Exp_3) \\
- \follow(Exp_{2*}) &\subseteq \follow(Exp_3) \\
+ \first(Mul) &\subseteq \follow(Factor) \\
+ \follow(Mul) &\subseteq \follow(Factor) \\
&\\
\{)\} &\subseteq \follow(Exp) \\
\end{align*}
@@ -165,10 +168,10 @@
\begin{align*}
\follow(S) &= \{\} \\
\follow(Exp) &= \{), \mathbf{EOF}\} \\
- \follow(Exp_*) &= \{), \mathbf{EOF}\} \\
- \follow(Exp_2) &= \{+, -, ), \mathbf{EOF}\} \\
- \follow(Exp_{2*}) &= \{+, -, ), \mathbf{EOF}\} \\
- \follow(Exp_3) &= \{+, -, *, /, ), \mathbf{EOF}\}
+ \follow(Add) &= \{), \mathbf{EOF}\} \\
+ \follow(Term) &= \{+, -, ), \mathbf{EOF}\} \\
+ \follow(Mul) &= \{+, -, ), \mathbf{EOF}\} \\
+ \follow(Factor) &= \{+, -, *, /, ), \mathbf{EOF}\}
\end{align*}
\end{enumerate}
@@ -181,45 +184,45 @@
\hline
\(S\) & 1 & & & & & 1 & &\\
\(Exp\) & 1 & & & & & 1 & &\\
- \(Exp_*\) & & 1 & 2 & & & & 3 & 3 \\
- \(Exp_2\) & 1 & & & & & 1 & & \\
- \(Exp_{2*}\) & & 3 & 3 & 1 & 2 & & 3 & 3 \\
- \(Exp_3\) & 1 & & & & & 2 & & \\
+ \(Add\) & & 1 & 2 & & & & 3 & 3 \\
+ \(Term\) & 1 & & & & & 1 & & \\
+ \(Mul\) & & 3 & 3 & 1 & 2 & & 3 & 3 \\
+ \(Factor\) & 1 & & & & & 2 & & \\
\end{tabular}
\end{center}
\item Parsing the strings:
\begin{enumerate}
\item \((3 + 4) * 5 ~\mathbf{EOF}\) \checkmark
\item \(2 + + ~\mathbf{EOF}\) --- fails on the second \(+\). The
- corresponding error cell in the parsing table is \((Exp_2, +)\).
+ corresponding error cell in the parsing table is \((Term, +)\).
\item \(2 ~\mathbf{EOF}\) \checkmark
\item \(2 * 3 + 4 ~\mathbf{EOF}\) \checkmark
- \item \(2 + 3 * 4 ~\mathbf{EOF}\) fails on the \(*\). Error at \((Exp_*, *)\).
+ \item \(2 + 3 * 4 ~\mathbf{EOF}\) fails on the \(*\). Error at \((Add, *)\).
\end{enumerate}
Example step-by-step LL(1) parsing state for \(2 * 3 + 4\):
\begin{center}
- \begin{tabular}{c c}
- Lookahead token & Stack \\
+ \begin{tabular}{c c c}
+ Lookahead & Stack & Next Rule \\
\hline
- \(2\) & \(S\) \\
- \(2\) & \(Exp ~ \mathbf{EOF}\) \\
- \(2\) & \(Exp_2 ~ Exp_* ~ \mathbf{EOF}\) \\
- \(2\) & \(Exp_3 ~ Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(2\) & \(\mathbf{num} ~ Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(*\) & \(Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(*\) & \(* ~Exp_3 ~ Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(3\) & \(Exp_3 ~ Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(3\) & \(\mathbf{num} ~ Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(+\) & \(Exp_{2*} ~ Exp_* ~ \mathbf{EOF}\) \\
- \(+\) & \(Exp_* ~ \mathbf{EOF}\) \\
- \(+\) & \(+ ~Exp_2 ~Exp_* ~ \mathbf{EOF}\) \\
- \(4\) & \(Exp_2 ~Exp_* ~ \mathbf{EOF}\) \\
- \(4\) & \(Exp_3 ~Exp_{2*} ~Exp_* ~ \mathbf{EOF}\) \\
- \(4\) & \(\mathbf{num} ~Exp_{2*} ~Exp_* ~ \mathbf{EOF}\) \\
- \(\mathbf{EOF}\) & \(Exp_{2*} ~Exp_* ~ \mathbf{EOF}\) \\
- \(\mathbf{EOF}\) & \(Exp_* ~ \mathbf{EOF}\) \\
- \(\mathbf{EOF}\) & \(\mathbf{EOF}\) \\
+ \(2\) & \(S\) & \(S ::= Exp ~\mathbf{EOF}\)\\
+ \(2\) & \(Exp ~ \mathbf{EOF}\) & \(Exp ::= Term~Add\)\\
+ \(2\) & \(Term ~ Add ~ \mathbf{EOF}\) & \(Term ::= Factor~Mul\)\\
+ \(2\) & \(Factor ~ Mul ~ Add ~ \mathbf{EOF}\) & \(Factor ::= \mathbf{num}\)\\
+ \(2\) & \(\mathbf{num} ~ Mul ~ Add ~ \mathbf{EOF}\) & \(match(\mathbf{num})\)\\
+ \(*\) & \(Mul ~ Add ~ \mathbf{EOF}\) & \(Mul ::= *~Factor~Mul\)\\
+ \(*\) & \(* ~Factor ~ Mul ~ Add ~ \mathbf{EOF}\) & \(match(*)\)\\
+ \(3\) & \(Factor ~ Mul ~ Add ~ \mathbf{EOF}\) & \(Factor ::= \mathbf{num}\)\\
+ \(3\) & \(\mathbf{num} ~ Mul ~ Add ~ \mathbf{EOF}\) & \(match(\mathbf{num})\)\\
+ \(+\) & \(Mul ~ Add ~ \mathbf{EOF}\) & \(Mul ::=\)\\
+ \(+\) & \(Add ~ \mathbf{EOF}\) & \(Add ::= +~Term~Add\)\\
+ \(+\) & \(+ ~Term ~Add ~ \mathbf{EOF}\) & \(match(+)\)\\
+ \(4\) & \(Term ~Add ~ \mathbf{EOF}\) & \(Term ::= Factor~Term*\)\\
+ \(4\) & \(Factor ~Mul ~Add ~ \mathbf{EOF}\) & \(Factor ::= \mathbf{num}\)\\
+ \(4\) & \(\mathbf{num} ~Mul ~Add ~ \mathbf{EOF}\) & \(match(\mathbf{num})\)\\
+ \(\mathbf{EOF}\) & \(Mul ~Add ~ \mathbf{EOF}\) & \(Mul ::= \)\\
+ \(\mathbf{EOF}\) & \(Add ~ \mathbf{EOF}\) & \(Add ::= \)\\
+ \(\mathbf{EOF}\) & \(\mathbf{EOF}\) & \(match(\mathbf{EOF})\)\\
\end{tabular}
\end{center}
\end{enumerate}
diff --git a/info/exercises/src/ex-03/ex/table.tex b/info/exercises/src/ex-03/ex/table.tex
index 1e3768b1df53b1d347de1a770aa219f2d4153ed5..aa1de12f5b9c57eddc61dabd02be26459569ead9 100644
--- a/info/exercises/src/ex-03/ex/table.tex
+++ b/info/exercises/src/ex-03/ex/table.tex
@@ -24,21 +24,21 @@
LL(1) grammar.
\begin{equation*}
- E ::= \mathbf{num} + E \mid \mathbf{num} - E
+ E ::= \mathbf{num} + E \mid \mathbf{num} - E \mid \mathbf{num}
\end{equation*}
\begin{solution}
The language is clearly not LL(1), as on seeing a token \(\mathbf{num}\), we
- cannot decide whether to continue parsing it as \(\mathbf{num} + E\) or
- \(\mathbf{num} - E\).
+ cannot decide whether to continue parsing it as \(\mathbf{num} + E\),
+ \(\mathbf{num} - E\), or the end.
- The notable problem is the common prefix between the two rules. We can
- separate this out by introducing a new non-terminal \(T\). This is a
- transformation known as \emph{left factorization}.
+ The notable problem is the common prefix between the rules. We can separate
+ this out by introducing a new non-terminal \(T\). This is a transformation
+ known as \emph{left factorization}.
\begin{align*}
E &::= \mathbf{num} ~T \\
- T &::= + E \mid - E
+ T &::= + E \mid - E \mid \epsilon
\end{align*}
% without changing the terms or the overall "structure" of the grammar, we