Simple precedence parser

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

SearchProductionToReduce (Stack)

• search the Pivot in the stack the nearest ${\displaystyle \lessdot }$ from the top
• search in the productions of the grammar which one have the same right side than the Pivot

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						${\displaystyle {\dot {=}}}$			${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
E'									${\displaystyle {\dot {=}}}$
T						${\displaystyle \gtrdot }$	${\displaystyle {\dot {=}}}$		${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
T'						${\displaystyle \gtrdot }$			${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
F						${\displaystyle \gtrdot }$	${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
+			${\displaystyle \lessdot }$	${\displaystyle {\dot {=}}}$	${\displaystyle \lessdot }$			${\displaystyle \lessdot }$		${\displaystyle \lessdot }$
*					${\displaystyle {\dot {=}}}$			${\displaystyle \lessdot }$		${\displaystyle \lessdot }$
(	${\displaystyle \lessdot }$	${\displaystyle {\dot {=}}}$	${\displaystyle \lessdot }$	${\displaystyle \lessdot }$	${\displaystyle \lessdot }$			${\displaystyle \lessdot }$		${\displaystyle \lessdot }$
)						${\displaystyle \gtrdot }$	${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
num						${\displaystyle \gtrdot }$	${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$		${\displaystyle \gtrdot }$
$	${\displaystyle \lessdot }$		${\displaystyle \lessdot }$	${\displaystyle \lessdot }$	${\displaystyle \lessdot }$			${\displaystyle \lessdot }$		${\displaystyle \lessdot }$

STACK                   PRECEDENCE    INPUT            ACTION

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

• Alfred V. Aho, Jeffrey D. Ullman (1977). Principles of Compiler Design. 1st Edition. Addison–Wesley.
• William A. Barrett, John D. Couch (1979). Compiler construction: Theory and Practice. Science Research Associate.
• Jean-Paul Tremblay, P. G. Sorenson (1985). The Theory and Practice of Compiler Writing. McGraw–Hill.