Editing Operator-precedence parser

{{Short description|Bottom-up parser that interprets an operator-precedence grammar}}
In [[computer science]], an '''operator-precedence parser''' is a [[Bottom-up parsing|bottom-up parser]] that interprets an [[operator-precedence grammar]]. For example, most [[calculator]]s use operator-precedence parsers to convert from the human-readable [[infix notation]] relying on [[order of operations]] to a format that is optimized for evaluation such as [[Reverse Polish notation]] (RPN).

[[Edsger Dijkstra]]'s [[shunting yard algorithm]] is commonly used to implement operator-precedence parsers.

== Relationship to other parsers ==

An operator-precedence parser is a simple [[shift-reduce parser]] that is capable of parsing a subset of [[LR parser|LR(1)]] grammars.  More precisely, the operator-precedence parser can parse all LR(1) grammars where two consecutive [[nonterminal]]s and [[empty string|epsilon]] never appear in the right-hand side of any rule.

Operator-precedence parsers are not used often in practice;  however they do have some properties that make them useful within a larger design.  First, they are simple enough to write by hand, which is not generally the case with more sophisticated right shift-reduce parsers.  Second, they can be written to consult an operator table at [[Run time (program lifecycle phase)|run time]], which makes them suitable for languages that can add to or change their operators while parsing. (An example is [[Haskell (programming language)|Haskell]], which allows user-defined infix operators with custom associativity and precedence; consequently, an operator-precedence parser must be run on the program ''after'' parsing of all referenced modules.)

[[Raku (programming language)|Raku]] sandwiches an operator-precedence parser between two [[recursive descent parser]]s in order to achieve a balance of speed and dynamism. [[GNU Compiler Collection|GCC]]'s C and C++ parsers, which are hand-coded recursive descent parsers, are both sped up by an operator-precedence parser that can quickly examine arithmetic expressions.  Operator-precedence parsers are also embedded within [[compiler-compiler]]-generated parsers to noticeably speed up the recursive descent approach to expression parsing.<ref name="Harwell2008">{{cite web| last=Harwell | first=Sam | date=2008-08-29 | title=Operator precedence parser | url=https://theantlrguy.atlassian.net/wiki/spaces/ANTLR3/pages/2687077/Operator+precedence+parser | publisher=ANTLR3 Wiki | access-date=2017-10-25}}</ref>

== Precedence climbing method ==
The precedence climbing method is a compact, efficient, and flexible algorithm for parsing expressions that was first described by Martin Richards and Colin Whitby-Strevens.<ref name="Richards1979">{{cite book |last1=Richards |first1=Martin |last2=Whitby-Strevens |first2=Colin |title=BCPL — the language and its compiler |year=1979 |publisher=Cambridge University Press |isbn=9780521219655}}</ref>

An infix-notation expression grammar in [[EBNF]] format will usually look like this:

<syntaxhighlight lang="abnf">
expression ::= equality-expression
equality-expression ::= additive-expression ( ( '==' | '!=' ) additive-expression ) *
additive-expression ::= multiplicative-expression ( ( '+' | '-' ) multiplicative-expression ) *
multiplicative-expression ::= primary ( ( '*' | '/' ) primary ) *
primary ::= '(' expression ')' | NUMBER | VARIABLE | '-' primary
</syntaxhighlight>

With many levels of precedence, implementing this grammar with a predictive recursive-descent parser can become inefficient.  Parsing a number, for example, can require five function calls: one for each non-terminal in the grammar until reaching ''primary''.

An operator-precedence parser can do the same more efficiently.<ref name="Harwell2008"/>  The idea is that we can left associate the arithmetic operations as long as we find operators with the same precedence, but we have to save a temporary result to evaluate higher precedence operators.  The algorithm that is presented here does not need an explicit stack;  instead, it uses recursive calls to implement the stack.

The algorithm is not a pure operator-precedence parser like the Dijkstra shunting yard algorithm.  It assumes that the ''primary'' nonterminal is parsed in a separate subroutine, like in a recursive descent parser.

=== Pseudocode ===

The pseudocode for the algorithm is as follows.  The parser starts at function ''parse_expression''.  Precedence levels are greater than or equal to 0.

 <u>parse_expression()</u>
     '''return''' parse_expression_1(parse_primary(), 0)

 <u>parse_expression_1(lhs, min_precedence)</u>
     ''lookahead'' := peek next token
     '''while''' ''lookahead'' is a binary operator whose precedence is >= ''min_precedence''
         ''op'' := ''lookahead''
         advance to next token
         ''rhs'' := ''parse_primary'' ()
         ''lookahead'' := peek next token
         '''while''' ''lookahead'' is a binary operator whose precedence is greater
                  than ''op''<nowiki>'</nowiki>s, or a right-associative operator
                  whose precedence is equal to ''op'''s
             ''rhs'' := ''parse_expression_1'' (''rhs'', precedence of ''op'' + (1 if ''lookahead'' precedence is greater, else 0))
             ''lookahead'' := peek next token
         ''lhs'' := the result of applying ''op'' with operands ''lhs'' and ''rhs''
     '''return''' ''lhs''

Note that in the case of a production rule like this (where the operator can only appear once):

<syntaxhighlight lang="abnf">
equality-expression ::= additive-expression  ( '==' | '!=' ) additive-expression
</syntaxhighlight>

the algorithm must be modified to accept only binary operators whose precedence is > ''min_precedence''.

=== Example execution of the algorithm ===

An example execution on the expression 2 + 3 * 4 + 5 == 19 is as follows.  We give precedence 0 to equality expressions, 1 to additive expressions, 2 to multiplicative expressions.

''parse_expression_1'' (''lhs'' = 2, ''min_precedence'' = 0)
*  the lookahead token is +, with precedence 1.  the outer while loop is entered.
*  ''op'' is + (precedence 1) and the input is advanced
*  ''rhs'' is 3
*  the lookahead token is *, with precedence 2.  the inner while loop is entered.<br>''parse_expression_1'' (''lhs'' = 3, ''min_precedence'' = 2)
:*   the lookahead token is *, with precedence 2.  the outer while loop is entered.
::*    ''op'' is * (precedence 2) and the input is advanced
::*    ''rhs'' is 4
::*    the next token is +, with precedence 1.  the inner while loop is not entered.
::*    ''lhs'' is assigned 3*4 = 12
::*    the next token is +, with precedence 1.  the outer while loop is left.
:*   12 is returned.
*  the lookahead token is +, with precedence 1.  the inner while loop is not entered.
*  ''lhs'' is assigned 2+12 = 14
*  the lookahead token is +, with precedence 1.  the outer while loop is not left.
*  ''op'' is + (precedence 1) and the input is advanced
*  ''rhs'' is 5
*  the next token is ==, with precedence 0.  the inner while loop is not entered.
*  ''lhs'' is assigned 14+5 = 19
*  the next token is ==, with precedence 0.  the outer while loop is not left.
*  ''op'' is == (precedence 0) and the input is advanced
*  ''rhs'' is 19
*  the next token is ''end-of-line'', which is not an operator.  the inner while loop is not entered.
*  ''lhs'' is assigned the result of evaluating 19 == 19, for example 1 (as in the C standard).
*  the next token is ''end-of-line'', which is not an operator.  the outer while loop is left.
1 is returned.

== Pratt parsing ==
{{external links|section|date=August 2023}}
Another precedence parser known as Pratt parsing was first described by [[Vaughan Pratt]] in the 1973 paper "Top Down Operator Precedence",<ref>Pratt, Vaughan. "[https://web.archive.org/web/20151223215421/http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf Top Down Operator Precedence]." ''Proceedings of the 1st Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages'' (1973).</ref> based on [[Recursive descent parser|recursive descent]]. Though it predates precedence climbing, it can be viewed as a generalization of precedence climbing.<ref>{{cite web |last1=Norvell |first1=Theodore |title=Parsing Expressions by Recursive Descent |url=http://www.engr.mun.ca/~theo/Misc/pratt_parsing.htm |website=www.engr.mun.ca |quote=The purpose of this post is to [... start] with precedence climbing and refactoring it to use the command pattern until we arrive at a Pratt parser. [This is the author who coined the term "precedence climbing".]}}</ref>

Pratt designed the parser originally to implement the [[CGOL]] programming language, and it was treated in much more depth in a Masters Thesis under his supervision.<ref>Van De Vanter, Michael L. "[http://publications.csail.mit.edu/lcs/specpub.php?id=715 A Formalization and Correctness Proof of the CGOL Language System]." (Master's Thesis).  MIT Laboratory for Computer Science Technical Report MIT-LCS-TR-147 (Cambridge, Massachusetts).  1975.</ref>

Tutorials and implementations:
* [[Douglas Crockford]] based the JavaScript parser in [[JSLint]] on Pratt parsing.<ref>{{cite web|author=Crockford, D|title=Top Down Operator Precedence|date=2007-02-21|url=http://crockford.com/javascript/tdop/tdop.html}}</ref>
* Comparison between Python implementations of precedence climbing and Pratt parsing: [http://www.oilshell.org/blog/2016/11/01.html "Pratt Parsing and Precedence Climbing Are the Same Algorithm" (2016) by Andy Chu]
* Tutorial using [[Rust (programming language)|Rust]]: [https://matklad.github.io/2020/04/13/simple-but-powerful-pratt-parsing.html "Simple but Powerful Pratt Parsing" (2020) by Aleksey Kladov]
* Tutorial using [[Rust (programming language)|Rust]]: [https://williamr.dev/posts/pratt-parsing/ "The Pratt Parsing Technique" (2024) by William Rågstad]
* Tutorial using [[Python (programming language)|Python]]: [http://effbot.org/zone/simple-top-down-parsing.htm "Simple Top-Down Parsing in Python" (2008) by Fredrik Lundh] {{Webarchive|url=https://web.archive.org/web/20150228044653/http://effbot.org/zone/simple-top-down-parsing.htm |date=2015-02-28 }}
* Tutorial using [[Java (programming language)|Java]]: [http://journal.stuffwithstuff.com/2011/03/19/pratt-parsers-expression-parsing-made-easy/ "Pratt Parsers: Expression Parsing Made Easy" (2011) by Bob Nystrom, author of Crafting Interpreters]
* Implementation in [[C Sharp (programming language)|C#]]: [https://github.com/atifaziz/Gratt "Gratt: A Generic Vaughn Pratt's top-down operator precedence parser for .NET Standard"] (a [[Generic programming|generic]] version inspired by the Java implementation presented by Bob Nystrom in [http://journal.stuffwithstuff.com/2011/03/19/pratt-parsers-expression-parsing-made-easy/ "Pratt Parsers: Expression Parsing Made Easy"])

== Alternative methods ==

There are other ways to apply operator precedence rules.  One is to build a tree of the original expression and then apply tree rewrite rules to it.

Such trees do not necessarily need to be implemented using data structures conventionally used for trees. Instead, tokens can be stored in flat structures, such as tables, by simultaneously building a priority list which states what elements to process in which order.

=== Full parenthesization ===
Another approach is to first fully parenthesize the expression, inserting a number of parentheses around each operator, such that they lead to the correct precedence even when parsed with a linear, left-to-right parser. This algorithm was used in the early [[Fortran#FORTRAN|FORTRAN I]] compiler:<ref name="Padua2000">{{cite journal |last1=Padua |first1=David |title=The Fortran I Compiler |year=2000 |journal=Computing in Science & Engineering |volume=2 |issue=1 |pages=70–75 |url=http://polaris.cs.uiuc.edu/publications/c1070.pdf |doi=10.1109/5992.814661|bibcode=2000CSE.....2a..70P }}</ref>

<blockquote>
The Fortran I compiler would expand each operator with a sequence of parentheses. In a simplified form of the algorithm, it would
* replace <code>+</code> and <code>–</code> with <code>))+((</code> and <code>))-((</code>, respectively;
* replace <code>*</code> and <code>/</code> with <code>)*(</code> and <code>)/(</code>, respectively;
* add <code>((</code> at the beginning of each expression and after each left parenthesis in the original expression; and
* add <code>))</code> at the end of the expression and before each right parenthesis in the original expression.
Although not obvious, the algorithm was correct, and, in the words of [[Donald Knuth|Knuth]], “The resulting formula is properly parenthesized, believe it or not.”<ref name="Knuth1962">{{cite journal |last1=Knuth |first1=Donald E. |title=A HISTORY OF WRITING COMPILERS |year=1962 |publisher=Edmund C. Berkeley |journal=Computers and Automation |volume=11 |issue=12 |pages=8–14 |url=https://archive.org/details/bitsavers_computersA_13990695}}</ref>
</blockquote>
{{missing information|section|why parenthesization works|date=May 2023}}

Example code of a simple C application that handles parenthesisation of basic math operators (<code>+</code>, <code>-</code>, <code>*</code>, <code>/</code>, <code>^</code>, <code>(</code> and <code>)</code><!-- parentheses -->):
<syntaxhighlight lang="c">
#include <stdio.h>
#include <string.h>

// The command-line argument boundary is our lexer.
int main(int argc, char *argv[]) {
  int i;
  printf("((((");
  for (i=1; i!=argc; i++) {
    // strlen(argv[i]) == 2
    if (argv[i] && !argv[i][1]) {
      switch (*argv[i]) {
          case '(': printf("((((");  continue;
          case ')': printf("))))");  continue;
          case '^': printf(")^(");   continue;
          case '*': printf("))*(("); continue;
          case '/': printf("))/(("); continue;
          case '+':
            // unary check: either first or had an operator expecting secondary argument
            if (i == 1 || strchr("(^*/+-", *argv[i-1]))
              printf("+");
            else
              printf(")))+(((");
            continue;
          case '-':
            if (i == 1 || strchr("(^*/+-", *argv[i-1]))
              printf("-");
            else
              printf(")))-(((");
            continue;
      }
    }
    printf("%s", argv[i]);
  }
  printf("))))\n");
  return 0;
}
</syntaxhighlight>

First, you need to compile your program. Assuming your program is written in C and the source code is in a file named program.c, you would use the following command:
<pre>gcc program.c -o program</pre>
The above command tells gcc to compile program.c and create an executable named program.

Command to run the program with parameters, For example; a * b + c ^ d / e
<pre>./program a '*' b + c '^' d / e</pre>

it produces
<pre>((((a))*((b)))+(((c)^(d))/((e))))</pre>
as output on the console.

A limitation to this strategy is that unary operators must all have higher precedence than infix operators. The "negative" operator in the above code has a higher precedence than exponentiation. Running the program with this input
<pre>- a ^ 2</pre>
produces this output
<pre>((((-a)^(2))))</pre>
which is probably not what is intended.

==References==
{{Reflist}}

==External links==
* {{cite web| last=Clarke | first=Keith | date=1992-05-26 | title=Re: compact recursive-descent parsing of expressions | url=http://groups.google.com/group/comp.compilers/browse_thread/thread/aee24b9504d527ca/1210ef8ae529accd?#1210ef8ae529accd | access-date=2012-01-24}}
* [http://groups.google.com/group/comp.compilers/browse_thread/thread/442ca111b2dfb38d/5390237263b3625b?#5390237263b3625b Example C++ code by Keith Clarke for parsing infix expressions using the precedence climbing method]
* {{cite journal| last=Samelson | first=Klaus | author-link=Klaus Samelson |author2=Friedrich L. Bauer |author2-link=Friedrich L. Bauer |date=February 1960 | title=Sequential formula translation | journal=Communications of the ACM | volume=3 | issue=2 | pages=76–83 | doi=10.1145/366959.366968| doi-access=free }}
* [https://web.archive.org/web/20160817143616/http://radovan-augustin.info/techpapers/parser/Parser.pdf Parser for expression with infix notation using precedence lists]

{{Parsers}}

{{DEFAULTSORT:Operator-Precedence Parser}}
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