1 Lexical Versus Syntactic Analysis
2 Eliminating Ambiguity
3 Elimination of Left Recursion
4 Left Factoring
5 Non-Context-Free Language Constructs
6 Exercises for Section 4.3

**Writing a Grammar**

*1 Lexical Versus Syntactic
Analysis*

*2 Eliminating Ambiguity*

*3 Elimination of Left Recursion*

*4 Left Factoring*

*5 Non-Context-Free Language
Constructs*

*6 Exercises for Section 4.3*

Grammars are capable of describing most, but not all, of the syntax of
pro-gramming languages. For instance, the requirement that identifiers be
declared before they are used, cannot be described by a context-free grammar.
Therefore, the sequences of tokens accepted by a parser form a superset of the
program-ming language; subsequent phases of the compiler must analyze the
output of the parser to ensure compliance with rules that are not checked by
the parser.

This section begins with a discussion of how to divide work between a
lexical analyzer and a parser. We then consider several transformations that
could be applied to get a grammar more suitable for parsing. One technique can
elim-inate ambiguity in the grammar, and other techniques — left-recursion
elimi-nation and left factoring — are useful for rewriting grammars so they
become suitable for top-down parsing. We conclude this section by considering
some programming language constructs that cannot be described by any grammar.

1. **Lexical
Versus Syntactic Analysis**

As we
observed in Section 4.2.7, everything that can be described by a regular
expression can also be described by a grammar. We may therefore reasonably ask:
"Why use regular expressions to define the lexical syntax of a
language?" There are several reasons.

Separating the syntactic
structure of a language into lexical and non-lexical parts provides a
convenient way of modularizing the front end of

a compiler into two manageable-sized components.

The lexical rules of a language
are frequently quite simple, and to describe them we do not need a notation as
powerful as grammars.

Regular expressions generally
provide a more concise and easier-to-under-stand notation for tokens than
grammars.

More efficient lexical analyzers
can be constructed automatically from regular expressions than from arbitrary
grammars.

There are no firm guidelines as to what to put into the lexical rules,
as op-posed to the syntactic rules. Regular expressions are most useful for
describing the structure of constructs such as identifiers, constants,
keywords, and white space. Grammars, on the other hand, are most useful for
describing nested structures such as balanced parentheses, matching
begin-end's, corresponding if-then-else's, and so on. These nested structures
cannot be described by regular expressions.

2. **Eliminating
Ambiguity**

Sometimes an ambiguous grammar can be rewritten to eliminate the
ambiguity. As an example, we shall eliminate the ambiguity from the following
"dangling-else" grammar:

Here
"other" stands for any other statement. According to this grammar,
the compound conditional statement

In all
programming languages with conditional statements of this form, the first parse
tree is preferred. The general rule is, "Match each **else** with the closest unmatched **then . "** ** ^{2}** This
disambiguating rule can theoretically be in-corporated directly into a grammar,
but in practice it is rarely built into the productions.

**Example 4.16 :** We can
rewrite the dangling-else grammar (4.14) as the fol-lowing unambiguous grammar.
The idea is that a statement appearing between a t h e n and an else must be
"matched"; that is, the interior statement must not end with an
unmatched or open then . A matched statement is either an if-then-else
statement containing no open statements or it is any other kind of
unconditional statement. Thus, we may use the grammar in Fig. 4.10. This
grammar generates the same strings as the dangling-else grammar (4.14), but it
allows only one parsing for string (4.15); namely, the one that associates each
else with the closest previous unmatched then . •

3. **Elimination
of Left Recursion**

A grammar
is *left recursive* if it has a
nonterminal *A* such that there is a
derivation *A* 4 *A*a for some string a*.*
Top-down parsing methods cannot handle left-recursive grammars, so a
transformation is needed to eliminate left recursion. In Section 2.4.5, we
discussed *immediate left recursion,*
where there is a production of the form *A
->• A*a*.* Here, we study the general case. In Section 2.4.5, we showed how
the left-recursive pair of productions *A
-»• A*a | b could be replaced by
the non-left-recursive productions:

without changing the strings derivable from *A.* This rule by itself suffices for many grammars.

**Example 4.17 :** The non-left-recursive expression grammar (4.2), repeated here,

is obtained by eliminating immediate left recursion from the expression
gram-mar (4.1). The left-recursive pair of productions *E -» E + T | T* are replaced by *E
-> T E'* and *E'* -»• + T *E' |* e. The new productions for *T* and *T'* are obtained similarly by eliminating immediate left recursion.
•

Immediate left recursion can be eliminated by the following technique,
which works for any number of A-productions. First, group the productions as

where no bi begins with an *A.* Then, replace the A-productions by

This procedure
eliminates all left recursion from the *A* and *A'* pro-ductions
(provided no is e), but it does not eliminate left recursion involving
derivations of two or more steps. For example, consider the grammar

The nonterminal *S* is
left recursive because *S* =>*Aa=>* *Sda, *but it is not
immediately left recursive.

Algorithm 4.19, below,
systematically eliminates left recursion from a gram-mar. It is guaranteed to
work if the grammar has no cycles (derivations of the form *A=>* A)* or
e-productions (productions of the form *A —>* e). Cycles can
be eliminated systematically from a grammar, as can e-productions (see
Exercises 4.4.6 and 4.4.7).

Algorithm 4. 1 9
: Eliminating left recursion.

**INPUT : **Grammar** ***G*** **with
no cycles or e-productions.

**OUTPUT : **An equivalent grammar with no left recursion.

**METHOD : **Apply the algorithm in Fig. 4.11 to** ***G.* Note
that the resulting non-left-recursive grammar may have e-productions.

Figure 4.11: Algorithm
to eliminate left recursion from a grammar

The procedure in Fig.
4.11 works as follows. In the first iteration for % = 1, the outer for-loop of
lines (2) through (7) eliminates any immediate left recursion among *A1*-productions.
Any remaining *A1* productions of the form *A1 —> A1a *must
therefore have / > 1. After the* i — *1st iteration of the
outer for-loop, all nonterminals *Ak,* where *k < i,* are
"cleaned"; that is, any production *Ak —> Aia, *must
have* I > k. *As a result, on the* ith. *iteration,
the inner loop of lines (3) through (5) progressively raises the lower limit in any
production Ai -> A ma, until we have m > i. Then, eliminating immediate left
recursion for the Ai productions at line (6) forces m to be greater than i.

Example 4.20 : Let us apply
Algorithm 4.19 to the grammar (4.18).
Technically, the algorithm is not guaranteed to work, because of the
e-production, but in this case, the production A -» e turns out to be harmless.

We order the nonterminals S, A. There is no immediate left recursion
among the 5-productions, so nothing happens during the outer loop for i = 1.
For i = 2, we substitute for S in A —> S d to obtain the following
^.-productions.

A-> Ac | Aa d | b d | e

Eliminating the immediate left recursion among these ^-productions
yields the following grammar.

S->Aa | b

A->bdA' | A'

A'-> cA' | ad A' | e

**4. Left Factoring**

Left factoring is a grammar transformation that is useful for producing
a gram-mar suitable for predictive, or top-down, parsing. When the choice
between two alternative ^-productions is not clear, we may be able to rewrite
the pro-ductions to defer the decision until enough of the input has been seen
that we can make the right choice.

For
example, if we have the two productions

on seeing the input if, we cannot immediately tell which production to choose to expand stmt. In general, if A ->• a b1 | a b2 are two A-productions, and the input begins with a nonempty string derived from a, we do not know whether to expand A to a b1 or ab2 . However, we may defer the decision by expanding A to aA'. Then, after seeing the input derived from a, we expand A' to b1 or to b2. That is, left-factored, the original productions become

Algorithm 4.21 : Left factoring a grammar.

INPUT : Grammar G.

OUTPUT : An equivalent left-factored grammar.

Thus, we may expand 5 to iEtSS' on input i, and wait until i EtS has
been seen to decide whether to expand S' to eS or to e. Of course, these
grammars are both ambiguous, and on input e, it will not be clear which
alternative for S' should be chosen. Example 4.33 discusses a way out of this
dilemma. •

**5. Non-Context-Free Language
Constructs**

A few syntactic constructs found in typical programming languages cannot
be specified using grammars alone. Here, we consider two of these constructs,
using simple abstract languages to illustrate the difficulties.

Example
4.25 : The language in this example abstracts the problem of checking that
identifiers are declared before they are used in a program. The language
consists of strings of the form wcw, where the first w represents the
declaration of an identifier w, c represents an intervening program fragment,
and the second w represents the use of the identifier.

The
abstract language is L1 — {wcw 1 w is in (a|b)*} . L1 consists of all words
composed of a repeated string of a's and 6's separated by c, such as aabcaab.
While it is beyond the scope of this book to prove it, the non-context-freedom
of Li directly implies the non-context-freedom of programming languages like C
and Java, which require declaration of identifiers before their use and which
allow identifiers of arbitrary length.

For this
reason, a grammar for C or Java does not distinguish among identi-fiers that
are different character strings. Instead, all identifiers are represented by a
token such as id in the grammar. In a compiler for such a language, the
semantic-analysis phase checks that identifiers are declared before they are
used. •

Example
4.26 : The non-context-free language in this example abstracts the problem of
checking that the number of formal parameters in the declaration of a function
agrees with the number of actual parameters in a use of the function.

The
language consists of strings of the form anbmcndm. (Recall an means a written n
times.) Here an and bm could represent the formal-parameter lists of two
functions declared to have n and m arguments, respectively, while cn and dm
represent the actual-parameter lists in calls to these two functions.

The
abstract language is L2 = {a^{n}b^{m}c^{n}d^{m}
1 n >= 1 and m >= 1}. That is, L2
consists of strings in the language generated by the regular expression
a*b*c*d* such that the number of a's and c's are equal and the number of 6's
and d's are equal. This language is not context free.

Again,
the typical syntax of function declarations and uses does not concern itself
with counting the number of parameters. For example, a function call in C-like
language might be specified by

with
suitable productions for expr. Checking
that the number of parameters in a call is correct is usually done during the
semantic-analysis phase.

**6. Exercises for Section 4.3**

**Exercise 4 . 3 . 1 **: The following is a grammar for regular expressions over sym-bols *a* and *b* only, using + in place of | for union, to avoid conflict with the
use of vertical bar as a metasymbol in grammars:

Left factor this grammar.

Does left factoring make the grammar
suitable for top-down parsing?

In addition to left factoring,
eliminate left recursion from the original grammar.

Is the resulting grammar suitable for
top-down parsing?

**Exercise 4 . 3 . 2:
**Repeat Exercise** 4**.3.1 on the following grammars:

The grammar of Exercise 4.2.1.

The grammar of Exercise 4.2.2(a).

The grammar of Exercise 4.2.2(c).

The grammar of Exercise 4.2.2(e).

The grammar of Exercise 4.2.2(g).

**
****Exercise **4 . 3 . 3 : The following grammar
is proposed to remove the "dangling-else ambiguity" discussed in
Section 4.3.2:

Show that
this grammar is still ambiguous.

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