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. 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:

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.

Parsing algorithms

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