Operator precedence

 Operator grammar
 small, but an important class of grammars
 we may have an efficient operator precedence
parser (a shift-reduce parser) for an operator
grammar.
 In an operator grammar, no production rule can have:
  at the right side
 two adjacent non-terminals at the right side.
 Ex:
EAB EEOE EE+E |
Aa Eid E*E |
Bb O+|*|/ E/E | id
not operator grammar not operator grammar operator grammar

Precedence Relations
 In operator-precedence parsing, we define
three disjoint precedence relations between
certain pairs of terminals.
a <. b b has higher precedence than a
a =· b b has same precedence as a
a .> b b has lower precedence than a
 The determination of correct precedence
relations between terminals are based on the
traditional notions of associativity and
precedence of operators. (Unary minus causes a
problem).

Using Operator-Precedence
Relations
 The intention of the precedence relations is
to find the handle of a right-sentential form,
<. with marking the left end,
=· appearing in the interior of the handle,
and
.> marking the right hand.
 In our input string $a1a2...an$, we insert the
precedence relation between the pairs of
terminals

Using Operator -
E  E+E | E-E | E*E | E/E | E^E | (E) | -E | id
The partial operator-precedence
table for this grammar
 Then the input string (id + id*id) with the
precedence relations inserted will be:
$ <. id .> + <. id .> * <. id .> $
id + * $
id .> .> .>
+ <. .> <. .>
* <. .> .> .>
$ <. <. <.

To Find The Handles
1. Scan the string from left end until the first .> is
encountered.
2. Then scan backwards (to the left) over any =· until a <. is
encountered.
3. The handle contains everything to left of the first .> and to
the right of the <. is encountered.
$ <. id .> + <. id .> * <. id .> $ E  id $ id + id * id $
$ <. + <. id .> * <. id .> $ E  id $ E + id * id $
$ <. + <. * <. id .> $ E  id $ E + E * id $
$ <. + <. * .> $ E  E*E $ E + E * .E $
$ <. + .> $ E  E+E $ E + E $
$ $ $ E $

Operator-Precedence Parsing
Algorithm
The input string is w$, the initial stack is $ and a table holds precedence
relations between certain terminals
set p to point to the first symbol of w$ ;
repeat forever
if ( $ is on top of the stack and p points to $ ) then return
else {
let a be the topmost terminal symbol on the stack and let b be the
symbol pointed to by p;
if ( a <. b or a =· b ) then { /* SHIFT */
push b onto the stack;
advance p to the next input symbol;
}
else if ( a .> b ) then /* REDUCE */
repeat pop stack
until ( the top of stack terminal is related by <. to the terminal most
recently popped );
else error();
}

Operator-Precedence Parsing
Algorithm -- Example
stack input action
$ id+id*id$ $ <. id shift
$id +id*id$ id .> + reduceE  id
$ +id*id$ shift
$+ id*id$ shift
$+id *id$ id .> * reduceE  id
$+ *id$ shift
$+* id$ shift
$+*id $id .> $ reduceE  id
$+* $* .> $ reduceE  E*E
$+ $+ .> $ reduceE  E+E
$ $ accept
id + * $
id .> .> .>
+ <. .> <. .>
* <. .> .> .>
$ <. <. <.

How to Create Operator-
 We use associativity and precedence relations among operators.
1. If operator 1 has higher precedence than operator  2,
  1
.>  2 and  2 <.  1
2. If operator  1 and operator  2 have equal precedence,
they are left-associative   1
.>  2 and  2
.>  1
they are right-associative   1 <.  2 and  2 <.  1
3. For all operators ,  <. id, id .> ,  <. (, (<. ,  .> ), ) .> ,  .> $, and
$ <. 
4. Also, let
(=·) $ <. ( id .> ) ) .> $
( <. ( $ <. id id .> $ ) .> )

Operator-Precedence
Relations
+ - * / ^ id ( ) $
+ .> .> <. <. <. <. <. .> .>
- .> .> <. <. <. <. <. .> .>
* .> .> .> .> <. <. <. .> .>
/ .> .> .> .> <. <. <. .> .>
^ .> .> .> .> <. <. <. .> .>
id .> .> .> .> .> .> .>
( <. <. <. <. <. <. <. =·
) .> .> .> .> .> .> .>
$ <. <. <. <. <. <. <.

Handling Unary Minus
 Operator-Precedence parsing cannot handle the unary minus
when we also have the binary minus in our grammar.
 The best approach to solve this problem, let the lexical analyzer
handle this problem.
 The lexical analyzer will return two different tokens for the unary minus
and the binary minus.
 The lexical analyzer will need a lookhead to distinguish the binary
minus from the unary minus.
 Then, we make
 <. unary-minus for any operator
unary-minus .>  if unary-minus has higher precedence than 
unary-minus <.  if unary-minus has lower (or equal)
precedence than 

Precedence Functions
 Compilers using operator precedence parsers do not
need to store the table of precedence relations.
 The table can be encoded by two precedence
functions f and g that map terminal symbols to integers.
 For symbols a and b.
f(a) < g(b) whenever a <. b
f(a) = g(b) whenever a =· b
f(a) > g(b) whenever a .> b

Constructing precedence functions
Method:
1. Create symbols fa and gb for each a that is a terminal or $.
2. Partition the created symbols into as many groups as possible,
in such a way that if a =. b, then fa and gb are in the same group.
3. Create a directed graph whose nodes are the groups found in (2).
For any a and b, if a <.b , place an edge from the group of gb to
the group of fa. Of a .> b, place an edge from the group of fa to
that of gb.
4. If the graph constructed has a cycle, then no precedence
functions exist. If there are no cycle, let f(a) be the length of the
longest path beginning at the group of fa; let g(a) be the length
of the longest path beginning at the group of ga.

Example
f*
f$
fid
f+
gid
g+
g*
g$
+ * Id $
f 2 4 4 0
g 1 3 5 0

Disadvantages of Operator
Precedence Parsing
 Disadvantages:
 It cannot handle the unary minus (the lexical
analyzer should handle the unary minus).
 Small class of grammars.
 Difficult to decide which language is
recognized by the grammar.
 Advantages:
 simple
 powerful enough for expressions in
programming languages

Error Recovery in Operator-
Precedence Parsing
Error Cases:
1. No relation holds between the terminal on
the top of stack and the next input symbol.
2. A handle is found (reduction step), but
there is no production with this handle as a
right side
Error Recovery:
1. Each empty entry is filled with a pointer to
an error routine.
2. Decides the popped handle “looks like”
which right hand side. And tries to recover
from that situation.

Handling Shift/Reduce
Errors
When consulting the precedence matrix to decide whether to
shift or reduce, we may find that no relation holds between the
top stack and the first input symbol.
To recover, we must modify (insert/change)
1. Stack or
2. Input or
3. Both.
We must be careful that we don’t get into an infinite loop.

Example
id ( ) $
id e3 e3 .> .>
( <. <. =. e4
) e3 e3 .> .>
$ <. <. e2 e1
e1: Called when : whole expression is missing
insert id onto the input
issue diagnostic: ‘missing operand’
e2: Called when : expression begins with a right parenthesis
delete ) from the input
issue diagnostic: ‘unbalanced right parenthesis’

Example
id ( ) $
id e3 e3 .> .>
( <. <. =. e4
) e3 e3 .> .>
$ <. <. e2 e1
e3: Called when : id or ) is followed by id or (
insert + onto the input
issue diagnostic: ‘missing operator’
e4: Called when : expression ends with a left parenthesis
pop ( from the stack
issue diagnostic: ‘missing right parenthesis’

Operator precedence

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