Simple precedence parser explained

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. The symbols ⋖, ≐ and ⋗ are used to identify the pivot, and to know when to Shift or when to Reduce.

Implementation

• Compute the Wirth–Weber precedence relationship table for a grammar with initial symbol S.
• Initialize a stack with the starting marker $.
• Append an ending marker $ to the string being parsed (Input).
• Until Stack equals "$ S" and Input equals "$"
  • Search the table for the relationship between Top(stack) and NextToken(Input)
  • if the relationship is ⋖ or ≐
    • Shift:
    • Push(Stack, relationship)
    • Push(Stack, NextToken(Input))
    • RemoveNextToken(Input)
  • if the relationship is ⋗
    • Reduce:
    • SearchProductionToReduce(Stack)
    • Remove the Pivot from the Stack
    • Search the table for the relationship between the nonterminal from the production and first symbol in the stack (Starting from top)
    • Push(Stack, relationship)
    • Push(Stack, Non terminal)

SearchProductionToReduce (Stack)

• Find the topmost ⋖ in the stack; this and all the symbols above it are the Pivot.
• Find the production of the grammar which has the Pivot as its right side.

Example

Given following language, which can parse arithmetic expressions with the multiplication and addition operations:

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; E represents an arithmetic expression, T is a term and F is a factor.

and the Parsing table:

	E	E'	T	T'	F	+		(	)	num	$
E						≐			⋗
E'									≐
T						⋗	≐		⋗		⋗
T'						⋗			⋗		⋗
F						⋗	⋗		⋗		⋗
+			⋖	≐	⋖			⋖		⋖
					≐			⋖		⋖
(	⋖	≐	⋖	⋖	⋖			⋖		⋖
)						⋗	⋗		⋗		⋗
num						⋗	⋗		⋗		⋗
$	⋖		⋖	⋖	⋖			⋖		⋖

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× (F -> num) (T -> F) (T' -> T) (E ->T ') 
$ ⋖ T ≐ * ⋖ (⋖ E            ≐        + 3)$           SHIFT
$ ⋖ T ≐ * ⋖ (⋖ E ≐ +        ⋖        3)$             SHIFT
$ ⋖ T ≐ * ⋖ (⋖ E ≐ + < 3    ⋗)$               REDUCE 3× (F -> num) (T -> F) (T' -> T) 
$ ⋖ T ≐ * ⋖ (⋖ E ≐ + ≐ T    ⋗)$               REDUCE 2× (E -> E + T) (E' -> E)
$ ⋖ T ≐ * ⋖ (≐ E'           ≐)$               SHIFT
$ ⋖ T ≐ * ⋖ (≐ E' ≐)       ⋗        $                REDUCE (F -> (E'))
$ ⋖ T ≐ * ≐ F                ⋗        $                REDUCE (T -> T * F)
$ ⋖ T                        ⋗        $                REDUCE 2× (T' -> T) (E -> T')
$ ⋖ E                                 $                ACCEPT

References

• 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.