2 Sample Problems

The problems in this section are all concerned with the following grammar:

\[ \begin{align} P \rightarrow & S \\ L \rightarrow & S L | S\\ S \rightarrow & s ; | \{ L \} \\ \end{align}
\]

A discussion of what this grammar “means” and how to read it:

2.1 Derivations

1. Give a derivation for the string: s;

   Reveal

   \[ P \rightarrow S \rightarrow s ; \]

2. Give a derivation for the string {s;}

   Reveal

   \[ P \rightarrow S \rightarrow \{ L \} \rightarrow \{ S \} \rightarrow \{ s ; \} \]

3. Give a (left-most) derivation for the string {s;s;}

   Reveal

   \[ P \rightarrow S \rightarrow \{ L \} \rightarrow \{ S L \} \rightarrow \{ s ; L \} \rightarrow \{ s ; S \} \rightarrow \{ s ; s ; \} \]

   Using recursion in context-free grammars to describe lists of things is a very common pattern, one that you should try to keep in mind both when reading grammars and when writing your own.

4. Give a right-most derivation for the string {s;s;}

   Reveal

   \[ P \rightarrow S \rightarrow \{ L \} \rightarrow \{ S L \} \rightarrow \{ S S \} \rightarrow \{ S s ; \} \rightarrow \{ s ; s ; \} \]

2.2 Parse Trees

1. Give a parse tree for the string: s;

   Reveal

   A reminder that our derivation was:

   \[ P \rightarrow S \rightarrow s ; \]

    

   

2. Give a parse tree for the string {s;}

   Reveal

   \[ P \rightarrow S \rightarrow \{ L \} \rightarrow \{ S \} \rightarrow \{ s ; \} \]

    

   

3. Give a parse tree for the string {s;s;}

   Reveal

   \[ P \rightarrow S \rightarrow \{ L \} \rightarrow \{ S L \} \rightarrow \{ s ; L \} \rightarrow \{ s ; S \} \rightarrow \{ s ; s ; \} \]

    

   

The grammar we have used in these examples was intended to be reminiscent of the idea of a program (P) consisting of either a single statement (S) or a list of statements (L) enclosed in brackets. We can actually expand on this idea to make rhe relationship clearer by replacing the simple idea that a “statement” is simply a single token (s) by the idea that s could go on to derive any of a number of more familiar forms of programming language constructs.

3 Sample Problems – Ambiguity

We’ll continue working with the grammar:

\[ \begin{align} P \rightarrow & S \\ L \rightarrow & S L \; | \; S\\ S \rightarrow & s ; \\ & | \, \{ L \} \\ & | \, i \, ( \, E \, ) \, S \\ & | \, i \, ( \, E \, ) \, S \, e \, S \\ E \rightarrow & x | y \end{align}
\]

This grammar can generate the possible sequences of if and else (represented by i and e, respectively).

1. Show that the string ‘$i (x) \, i (y) \, x = y; e \, y = x;$’ is in the language described by this grammar.

   Reveal

    

   

2. Give a left-most derivation for that string.

   Reveal

   \[ \begin{align} P & \rightarrow S \\ & \rightarrow i \, ( \, E \, ) \, S \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, S \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, E \, ) \, S \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, S \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, E \, = \, E \, ; \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, x \, = \, E \, ; \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, x \, = \, y \, ; \, e \, S \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, x \, = \, y \, ; \, e \, E \, = \, E \, ; \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, x \, = \, y \, ; \, e \, y \, = \, E \, ; \\ & \rightarrow i \, ( \, x \, ) \, i \, ( \, y \, ) \, x \, = \, y \, ; \, e \, y \, = \, x \, ; \end{align} \]

3. Using that same string, show that the grammar is ambiguous

   Reveal

   To show ambiguity, it suffices to show two different parse trees for some string, or two different left derivations for it, or two different right derivations.

    

   

Some discussion of why ambiguity is both important and can be hard to prove/disprove.