Simple precedence parser

In computer science, a Simple precedence parser is a type of bottom-up parser for context-free grammars that can be used only by Simple precedence grammars.

The implementation of the parser is quite similar to the generic bottom-up parser. A stack is used to store a viable prefix of a sentential form from a rightmost derivation. Simbols ${\dot {<}}$, ${\dot {=}}$ and ${\dot {>}}$ are used to identify the pivot, and to know when to Shift or when to Reduce.

Implementation

• Compute the Wirth-Weber precedence relationship table.
• Start with a stack with only the starting marker $.
• Start with the string beeing parsed (Input) ended with an ending marker $.
• While not (Stack equals to $S and Input equals to $) (S = Initial simbol of the grammar)
  • Search in the table the relationship between Top(stack) and NextToken(Input)
  • if the relationship is ${\dot {=}}$ or ${\dot {<}}$
    • Shift:
    • Push(Stack, relationship)
    • Push(Stack, NextToken(Input))
    • RemoveNextToken(Input)
  • if the relationship is ${\dot {>}}$
    • Reduce:
    • SearchProductionToReduce(Stack)
    • RemovePivot(Stack)
    • Search in the table the relationship between the Non terminal from the production and NextToken(Input)
    • Push(Stack, relationship)
    • Push(Stack, Non terminal)

SearchProductionToReduce (Stack)

• search the Pivot in the stack the nearest ${\dot {<}}$ from the top
• search in the productions of the grammar wich one have the same right side than the Pivot

Example

Given the language:
E  --> E + T' | T'
T' --> T
T  --> T * F  | F
F  --> ( E' ) | num
E' --> E

num is a terminal, and the lexer parse any integer as num.

and the Parsing table:

	E	E'	T	T'	F	+	*	(	)	num
E			${\dot {>}}$	${\dot {>}}$	${\dot {>}}$	${\dot {=}}$			${\dot {>}}$	${\dot {>}}$
E'									${\dot {=}}$
T						${\dot {>}}$	${\dot {=}}$		${\dot {>}}$
T'			${\dot {<}}$	${\dot {<}}$		${\dot {>}}$		${\dot {<}}$	${\dot {>}}$	${\dot {<}}$
F						${\dot {>}}$	${\dot {>}}$		${\dot {>}}$
+			${\dot {<}}$	${\dot {=}}$	${\dot {<}}$			${\dot {<}}$		${\dot {<}}$
*					${\dot {=}}$			${\dot {<}}$		${\dot {<}}$
(	${\dot {<}}$	${\dot {=}}$	${\dot {<}}$	${\dot {<}}$	${\dot {<}}$			${\dot {<}}$		${\dot {<}}$
)							${\dot {>}}$		${\dot {>}}$
num							${\dot {>}}$

STACK                   PRECEDENCE    INPUT            ACTION

$                            <        2 * ( 1 + 3 )$   SHIFT
$ < 2                        >        * ( 1 + 3 )$     REDUCE (F -> 2)
$ < F                        >        * ( 1 + 3 )$     REDUCE (T -> F)
$ < T                        =        * ( 1 + 3 )$     SHIFT
$ < T = *                    <        ( 1 + 3 )$       SHIFT
$ < T = * < (                <        1 + 3 )$         SHIFT
$ < T = * < ( < 1            >        + 3 )$           REDUCE 4 times (F -> 1) (T -> F) (T' -> T) (E -> T')
$ < T = * < ( < E            =        + 3 )$           SHIFT
$ < T = * < ( < E = +        <        3 )$             SHIFT
$ < T = * < ( < E = + < 3    >        )$               REDUCE 3 times (F -> 3) (T -> F) (T' -> T) 
$ < T = * < ( < E = + = T    >        )$               REDUCE (E -> E + T)
$ < T = * < ( < E            =        )$               SHIFT
$ < T = * < ( = E = )        >        $                REDUCE (F -> ( E ))
$ < T = * = F                >        $                REDUCE (T -> T * F)
$ < T                        >        $                REDUCE 2 times (T' -> T) (E -> T')
$ < E                        >        $                ACCEPT