The Department of Computer Science & Engineering 
STUART C. SHAPIRO: CSE
305

Issue: How to describe a piece of source code, or other computer command or piece of language in general?
Notice submit instructions on course home
page: e.g. submit_cse305r1 file
The submit_cse305r1
part is to be entered as a command
exactly as shown, but the file
part is to be
replaced by an actual file name.
We can consider submit_cse305r1 file
to be a
command that is not yet finishedsome of it needs to be replaced by
a final (or terminal) string. The submit_cse305r1
part
is already a terminal string; in fact, a terminal symbol, but the
file
part is a nonterminal symbol.
The first component of BNF notation is a way to distinguish
terminal symbols from nonterminal symbols. In submit_cse305r1
file
I have used different fonts, this font
for terminal symbols, and this font
for
nonterminal symbols. The text uses roman or bold font for terminal
symbols, and <roman font inside angle brackets> for nonterminal
symbols. That is also very common. When you read a BNF grammar
someone else has written, there should be a mention of the system
they use, but it should be pretty obvious also.
Consider a small programming language that has two kinds of statements: an assignment statement; and an if statement. We want to say
The way BNF specifies how a nonterminal can be replaced is by a rule, with a lefthand side (LHS) which is always a nonterminal, and a righthand side (RHS), which is a sequence of terminals or nonterminals, separated by a rewrite symbol. The text uses > as a rewrite symbol; the symbol ::= is also often used. So we can rewrite and complete the specification of the statements of this small language as:
To use this grammar to generate a statement of this language, we start with <statement>, and derive a statement by steps. In each step, we choose a nonterminal of the string and replace it by the RHS of any rule in which that nonterminal is the LHS. For example,<statement> > <assignment> <statement> > <if> <assignment> > <variable> = <expression> <if> > if (<expression>) <statement> <expression> > <variable> <expression> > <variable> + <variable> <expression> > <variable>  <variable> <variable> > x <variable> > y <variable> > z
Notice that this grammar is recursive, <statement> derives a string in which <statement> occurs. Recursive grammars can generate infinitely long sentences.<statement> <if> if (<expression>) <statement> if (<variable>) <statement> if (y) <statement> if (y) if (<expression>) <statement> if (y) if (<variable>) <statement> if (y) if (z) <statement> if (y) if (z) <assignment> if (y) if (z) <variable> = <expression> if (y) if (z) x = <expression> if (y) if (z) x = <variable> + <variable> if (y) if (z) x = y + <variable> if (y) if (z) x = y + z
This derivation can also be represented by a parse tree:
The sentence, itself, is formed by the leaves of the parse tree, read
left to right. They are in blue for easier reading.
Notice that the tree makes the structure of the constituents of the
sentence clear. It shows that y + z
is an expression
that forms the righthand side of the assignment statement, and that
if (z) x = y + z
is the statement that is under the control
of the if (y)
condition.
When multiple rules of a grammar have the same LHS, they may be combined by connecting their RHSs with the or symbol, . The grammar above becomes
A grammar may be ambiguous. Consider this grammar for expressions:<statement> > <assignment>  <if> <assignment> > <variable> = <expression> <if> > if (<expression>) <statement> <expression> > <variable>  <variable> + <variable>  <variable>  <variable> <variable> > x  y  z
<expression> > <variable>  <expression> * <expression>  <expression> + <expression> <variable> > x  y  z
x * y + z
has two derivations, as shown by
these two parse trees:
The grammar can be made unambiguous by some small rewriting. We could have two versions:
<expression> > <variable>  <variable> * <expression>  <variable> + <expression> <variable> > x  y  z
<expression> > <variable>  <expression> * <variable>  <expression> + <variable> <variable> > x  y  z
Using grammar 1, which is right recursive, the expression x *
y + z
only has the parse tree:
Using grammar 2, which is left recursive, the expression x *
y + z
only has the parse tree:
Notice that the left recursive grammar gives a left associative expression, while a right recursive grammar gives a right associative expression.
Of course, we really want * to always have higher precedence than +. For a grammar that does that, see Example 3.5 on page 129 of the text. Within a single precedence level, we usually want a left associative expression, and, therefore a left associative grammar. This, and the right associativity of **, is also illustrated by Example 3.5.
Sometimes, the grammar bottoms out at some nonterminals which are described, not in BNF, but in informal text. For example,
<variable> > <identifier>An<identifier>
is a string of characters, the first of which can be an upper or lowercase alphabetic character, and the rest of which can be upper or lowercase alphabetic characters, decimal digits, or the _ character.
<if> > if (<expression>) <statement> [else <statement>]
<decl> > <type> <variable> {, <variable>};
Sometimes, instead, the the section of RHS is enclosed in parentheses (if more than one symbol), and followed by a Kleene *, indicating 0 or more occurrences, or a Kleene +, indicating 1 or more occurrence. For example,
<decl> > <type> <variable> (, <variable>)*;
<decl> > <type> <variable> [, <variable>]+;
<expression> > <variable>  <expression> (*  +) <variable> <variable> > x  y  z
<if> > if (<expression>) <statement> [else <statement>]