parsers

# Stream parsers

We describe here the syntax and the semantics of the parsers of streams of Camlp5. Streams are kinds of lazy lists. The parsers of these streams use recursive descendent method without backtracking, which is the most natural one in functional languages. In particular, parsers are normal functions.

Notice that the parsers have existed in OCaml since many years (the beginning of the 90ies), but some new features have been added in 2007 (lookahead, "no error" optimization, let..in statement and left factorization) in Camlp5 distribution. This chapter describes them also.

## Introduction

Parsers apply to values of type "Stream.t" defined in the module "Stream" of the standard library of OCaml. Like the type "list", the type "Stream.t" has a type parameter, indicating the type of its elements. They differ from the lists that they are lazy (the elements are evaluated as long as the parser need them for its actions), and imperative (parsers deletes their first elements when they take their parsing decisions): notice that purely functional parsers exist in Camlp5, where the corresponding streams are lazy and functional, the analyzed elements remaining in the initial stream and the semantic action returning the resulting stream together with the normal result, which allow natural limited backtrack but have the drawback that it is not easy to find the position of parsing errors when they happen.

Parsers of lazy+imperative streams, which are described here, use a method named "recursive descendent": they look at the first element, they decide what to do in function of its value, and continue the parsing with the remaining elements. Parsers can call other parsers, and can be recursive, like normal functions.

Actually, parsers are just pure syntactic sugar. When writing a parser in the syntax of the parser, Camlp5 transforms them into normal call to functions, use of patterns matchings and try..with statements. The pretty printer of Camlp5, by default, displays this expanded result, without syntax of parsers. A pretty printing kit, when added, can rebuild the parsers in their initial syntax and display it.

## Syntax

The syntax of the parsers, when loading "pa_rp.cmo" (or already included in the command "camlp5r"), is the following:

                expression ::= parser
                             | match-with-parser
                    parser ::= "parser" pos-opt "[" parser-cases "]"
                             | "parser" pos-opt parser-case
         match-with-parser ::= "match" expression "with" parser
              parser-cases ::= parser-cases parser-case
                             | <nothing>
               parser-case ::= "[:" stream-pattern ":]" pos-opt "->" expression
            stream-pattern ::= stream-patt-comp
                             | stream-patt-comp ";" stream-patt-cont
                             | "let" LIDENT "=" expression "in" stream-pattern
                             | <nothing>
          stream-patt-cont ::= stream-patt-comp-err
                             | stream-patt-comp-err ";" stream-patt-cont
                             | "let" LIDENT "=" expression "in" stream-patt-cont
      stream-patt-comp-err ::= stream-patt-comp
                             | stream-patt-comp "?" expression
                             | stream-patt-comp "!"
          stream-patt-comp ::= "`" pattern
                             | "`" pattern "when" expression
                             | "?=" lookaheads
                             | pattern "=" expression
                             | pattern
                lookaheads ::= lookaheads "|" lookahead
                             | lookahead
                 lookahead ::= "[" patterns "]"
                  patterns ::= patterns pattern
                             | pattern
                   pos-opt ::= pattern
                             | <nothing>

## Streams

The parsers are functions taking streams as parameter. Streams are are values of type "`Stream.t a`" for some type "`a`". It is possible to build streams using the functions defined in the module "`Stream`":

### Stream.from

"`Stream.from f`" returns a stream built from the function "`f`". To create a new stream element, the function "`f`" is called with the current stream count, starting with zero. The user function "`f`" must return either "`Some <value>`" for a value or "`None`" to specify the end of the stream.

### Stream.of_list

Return a stream built from the list in the same order.

### Stream.of_string

Return a stream of the characters of the string parameter.

### Stream.of_channel

Return a stream of the characters read from the input channel parameter.

## Semantics of parsers

### Parser

A parser, defined with the syntax "parser" above, is of type "`Stream.t a -> b`" where "a" is the type of the elements of the streams and "b" the type of the result. The parser cases are tested in the order they are defined until one of them applies. The result is the semantic action of the parser case which applies. If no parser case applies, the exception "`Stream.Failure`" is raised.

When testing a parser case, if the first stream pattern component matches, all remaining stream pattern components of the stream pattern must match also. If one does not match, the parser raises the exception "`Stream.Error`" which has a parameter of type string: by default, this string is the empty string, but if the stream pattern component which does not match is followed by a question mark and an expression, this expression is evaluated and given as parameter to "`Stream.Error`".

In short, a parser can return with three ways:

- A normal result, of type "`b`" for a parser of type "`Stream.t a -> b`".
- Raising the exception "`Stream.Failure`".
- Raising the exception "`Stream.Error`".

Fundamentally, the exception "`Stream.Failure`" means "this parser does not apply and no element have been removed from the initial stream". This is a normal case when parsing: the parser locally fails, but the parsing can continue.

Conversely, the exception "`Stream.Error`" means that "this parser encountered a syntax error and elements have probably been removed from the stream". In this case, there is no way to recover the parsing, and it definitively fails.

### Left factorization

In parsers, consecutive rules starting with the same components are left factorized. It means that they are transformed into one only rule starting with the common path, and continuing with a call to a parser separating the two cases. The order is kept, except that the possible empty rule is inserted at the end.

For example, the parser:

      parser
      [ [: `If; e1 = expr; `Then; e2 = expr; `Else; e3 = expr :] -> f e1 e2 e3
      | [: `If; e1 = expr; `Then; e2 = expr :] -> g e1 e2 ]

is transformed into:

      parser
        [: `If; e1 = expr; `Then; e2 = expr;
           a =
             parser
             [ [: `Else; e3 = expr :] -> f e1 e2 e3
             | [: :] -> g e1 e2 ] :] -> a

The version where rules are inverted:

      parser
      [ [: `If; e1 = expr; `Then; e2 = expr :] -> g e1 e2
      | [: `If; e1 = expr; `Then; e2 = expr; `Else; e3 = expr :] -> f e1 e2 e3 ]

is transformed into the same parser.

Notice that:

- Only consecutive rules are left factorized. In the following parser:

        parser
        [ [: `If; e1 = expr; `Then; e2 = expr; `Else; e3 = expr :] -> ...
        | [: a = b :] -> a
        | [: `If; e1 = expr; `Then; e2 = expr :] -> ... ]

  the two rules starting with "`If`" are not left factorized, and the second "`If`" rule will never work.
- The components which are not identical are not factorized. In the following parser:

        parser
        [ [: `If; e1 = expr; `Then; e2 = expr; `Else; e3 = expr :] -> ...
        | [: `If; e4 = expr; `Then; e2 = expr :] -> ... ]

  only the first component, "`If`" is factorized, the second one being different because of different patterns ("`e1`" and "`e4`").

### Match with parser

The syntax "match expression with parser" allows to match a stream against a parser. It is, for "parser", the equivalent of "match expression with" for "fun". The same way we could say:

      match expression with ...

could be considered as an equivalent to:

      (fun ...) expression

we could consider that:

      match expression with parser ...

is an equivalent to:

      (parser ...) expression

### Error messages

A "`Stream.Error`" exception is raised when a stream pattern component does not match and that it is not the first one of the parser case. This exception has a parameter of type string, useful to specify the error message. By default, this is the empty string. To specify an error message, add a question mark and an expression after the stream pattern component. A typical error message is "that stream pattern component expected". Example with the parser of "if..then..else.." above:

      parser
        [: `If; e1 = expr ? "expression expected after 'if'";
           `Then ? "'then' expected";
           e2 = expr ? "expression expected after 'then'";
           a =
             parser
             [ [: `Else; e3 = expr ? "expression expected" :] -> f e1 e2 e3
             | [: :] -> g e1 e2 ] :] -> a

Notice that the expression after the question mark is evaluated only in case of syntax error. Therefore, it can be a complicated call to a complicated function without slowing down the normal parsing.

### Stream pattern component

In a stream pattern (starting with "`[:`" and ending with "`:]`"), the stream pattern components are separated with the semicolon character. There are three cases of stream pattern components with some sub-cases for some of them, and an extra syntax can be used with a "let..in" construction. The three cases are:

- A direct test of one or several stream elements (called terminal symbol), in three ways:
  1. The character "backquote" followed by a pattern, meaning: if the stream starts with an element which is matched by this pattern, the stream pattern component matches, and the stream element is removed from the stream.
  2. The character "backquote" followed by a pattern, the keyword "when" and an expression of type "`bool`", meaning: if the stream starts with an element which is matched by this pattern and if the evaluation of the expression is "`True`", the stream pattern component matches, and the first element of the stream is removed.
  3. The character "question mark" followed by the character "equal" and a lookahead expression (see further), meaning: if the lookahead applies, the stream pattern component matches. The lookahead may unfreeze one or several elements on the stream, but does not remove them.
- A pattern followed by the "equal" sign and an expression of type "`Stream.t x -> y`" for some types "`x`" and "`y`". This expression is called a non terminal symbol. It means: call the expression (which is a parser) with the current stream. If this sub-parser:
  1. Returns an element, the pattern is bound to this result and the next stream pattern component is tested.
  2. Raises the exception "`Stream.Failure`", there are two cases:
     - if the stream pattern component is the first one of the stream case, the current parser also fails with the exception "`Stream.Failure`".
     - if the stream pattern component is not the first one of the stream case, the current parser fails with the exception "`Stream.Error`".
     In this second case:
     - If the stream pattern component is followed by a "question mark" and an expression (which must be of type "`string`"), the expression is evaluated and given as parameter of the exception "`Stream.Error`".
     - If the expression is followed by an "exclamation mark", the test and conversion from "`Stream.Failure`" to "`Stream.Error`" is not done, and the parser just raises "`Stream.Failure`" again. This is an optimization which must be assumed by the programmer, in general when he knows that the sub-parser called never raises "`Stream.Failure`" (for example if the called parser ends with a parser case containing an empty stream pattern). See "no error optionization" below.
     - Otherwise the exception parameter is the empty string.
- A pattern, which is bound to the current stream.

Notice that patterns are bound immediately and can be used in the next stream pattern component.

### Let statement

Between stream pattern components, it is possible to use the "let..in" construction. This is not considered as a real stream pattern component, in the fact that is is not tested against the exception "`Stream.Failure`" it may raise. It can be useful for intermediate computation. In particular, it is used internally by the lexers (see chapter about lexers as character stream parsers).

Example of use, when an expression have to be used several times (in the example, "`d a`", which is bound to the variable "`c`"):

      parser
        [: a = b;
           let c = d a in
           e =
             parser
             [ [: f = g :] -> h c
             | [: :] -> c ] :] -> e

### Lookahead

The lookahead feature allows to look at several terminals in the stream without removing them, in order to take decisions when more than one terminal is necessary.

For example, when parsing the normal syntax of the OCaml language, there is a problem, in recursing descendent parsing, for the cases where to treat and differentiate the following inputs:

      (-x+1)
      (-)

The first case is treated in a rule, telling: "a left parenthesis, followed by an expression, and a right parenthesis". The second one is "a left parenthesis, an operator, a right parenthesis". Programming it like this (left factorizing the first parenthesis):

      parser
        [: `Lparen;
           e =
             parser
             [ [: e = expr; `Rparen :] -> e
             | [: `Minus; `Rparen :] -> minus_op ] :] -> e

does not work if the input is "`(-)`" because the rule "`e = expr`" accepts the minus sign as expression start, removing it from the input stream and fails as parsing error, while encountering the right parenthesis.

Conversely, writing it this way:

      parser
        [: `Lparen;
           e =
             parser
             [ [: `Minus; `Rparen :] -> minus_op
             | [: e = expr; `Rparen :] -> e ] :] -> e

does not help, because if the input is "`(-x+1)`" the rule above starting with "``Minus`" is accepted and the exception "`Stream.Error`" is raised while encountering the variable "`x`" since a right parenthesis is expected.

In general, this kind of situation is best resolved by a left factorization of the parser cases (see the section "Semantics" above), but that is not possible in this case. The solution is to test whether the character after the minus sign is a right parenthesis:

      parser
        [: `Lparen;
           e =
             parser
             [ [: ?= [ _ Rparen ]; `Minus; `Rparen :] -> minus_op
             | [: e = expr; `Rparen :] -> e ] :] -> e

It is possible to put several lists of patterns separated by a vertical bar in the lookahead construction, but with a limitation (due to the implementation): all lists of patterns must have the same number of elements.

### No error optimization

The "no error optimization" is the fact to end a stream pattern component of kind "non-terminal" ("pattern" "equal" "expression") by the character "exclamation mark". Like said above, this inhibits the transformation of the exception "`Stream.Failure`", possibly raised by the called parser, into the exception "`Stream.Error`".

The code:

      parser [: a = b; c = d ! :] -> e

is equivalent to:

      parser [: a = b; s :] -> let c = d s in e

One interest of the first syntax is that it shows to readers that "`d`" is indeed a syntactic sub-parser. In the second syntax, it is called in the semantic action, which makes the parser case not so clear, as far as readability is concerned.

If the stream pattern component is at end of the stream pattern, this allow possible tail recursion by the OCaml compiler, in the following case:

      parser [: a = b; c = d ! :] -> c

since it is equivalent (with the fact that "`c`" is at the same time the pattern of the last case and the expression of the parser case semantic action) to:

      parser [: a = b; s :] -> d s

The call to "`d s`" can be a tail recursive call. Without the use of the "exclamation mark" in the rule, the equivalent code is:

      parser [: a = b; s :] ->
        try d s with [ Stream.Failure -> raise (Stream.Error "") ]

which is not tail recursive (due to the "try..with" construction pushes a context), preventing the compiler to optimize its code. This can be important when many recursive calls happen, since it can overflow the OCaml stack.

### Position

The optional "pattern" before and after a stream pattern is bound to the current stream count. Indeed, streams internally contain a count of their elements. At the beginning the count is zero. When an element is removed, the count is incremented. The example:

      parser [: a = b :] ep -> c

is equivalent to:

      parser [: a = b; s :] -> let ep = Stream.count s in c

There is no direct syntax equivalent to the optional pattern at beginning of the stream pattern:

      parser bp [: a = b :] -> c

These optional patterns allow disposal of the stream count at the beginning and at the end of the parser case, allowing to compute locations of the rule in the source. In particular, if the stream is a stream of characters, these counts are the source location in number of characters.

### Semantic action

In a parser case, after the stream pattern, there is an "arrow" and an expression, called the "semantic action". If the parser case is matched the parser returns with the evaluated expression whose environment contains all values bound in the stream pattern.

## Remarks

### Simplicity vs Associativity

This parsing technology has the advantage of simplicity of use and understanding, but it does not treat the associativity of operators. For example, if you write a parser like this (to compute arithmetic expressions):

      value rec expr =
        parser
        [ [: e1 = expr; `'+'; e2 = expr :] -> e1 + e2
        | [: `('0'..'9' as c) :] -> Char.code c - Char.code '0' ]

this would loop endlessly, exactly as if you wrote code starting with:

      value rec expr e =
        let e1 = expr e in
        ...

One solution is to treat the associativity "by hand": by reading a sub-expression, then looping with a parser which parses the operator and another sub-expression, and so on.

An alternative solution is to write parsing "combinators". Indeed, parsers being normal functions, it is possible to make a function which takes a parser as parameter and returning a parser using it. For example, left and right associativity parsing combinators:

      value rec left_assoc op elem =
        let rec op_elem x =
          parser
          [ [: t = op; y = elem; r = op_elem (t x y) :] -> r
          | [: :] -> x ]
        in
        parser [: x = elem; r = op_elem x :] -> r
      ;

      value rec right_assoc op elem =
        let rec op_elem x =
          parser
          [ [: t = op; y = elem; r = op_elem y :] -> t x r
          | [: :] -> x ]
        in
        parser [: x = elem; r = op_elem x :] -> r
      ;

which can be used, e.g. like this:

      value expr =
        List.fold_right (fun op elem -> op elem)
          [left_assoc (parser [: `'+' :] -> fun x y -> x +. y);
           left_assoc (parser [: `'*' :] -> fun x y -> x *. y);
           right_assoc (parser [: `'^' :] -> fun x y -> x ** y)]
          (parser [: `('0'..'9' as c) :] -> float (Char.code c - Char.code '0'))
      ;

and tested, e.g. in the toplevel, like that:

      expr (Stream.of_string "2^3^2+1");

The same way, it is possible to parse non-context free grammars, by programming parsers returning other parsers.

A third solution, to resolve the problem of associativity, is to use the grammars of Camlp5, which have the other advantage that they are extensible.

### Lexing vs Parsing

In general, while analyzing a language, there are two levels:

- The level where the input, considered as a stream of characters, is read to make a stream of tokens (for example "words", if it is a human language, or punctuation). This level is generally called "lexing".
- The level where the input is a stream of tokens where grammar rules are parsed. This level is generally called "parsing".

The "parser" construction described here can be used for both, thanks to the polymorphism of OCaml:

- The lexing level is a "parser" of streams of characters returning tokens.
- The parsing level is a "parser" of streams of tokens returning syntax trees.

By comparison, the programs "lex" and "yacc" use two different technologies. With "parser"s, it is possible to use the same one for both.

### Lexer syntax vs Parser syntax

For "lexers", i.e. for the specific case of parsers when the input is a stream of characters, it is possible to use a shorter syntax. See the chapter on lexers. They have another syntax, shorter and adapted for the specific type "`char`". But they still are internally parsers of streams with the same semantics.

### Purely functional parsers

This system of parsers is imperative: while parsing, the stream advances and the already parsed terminals disappear from the stream structure. This is useful because it is not necessary to return the remaining stream together with the normal result. This is the reason there is this "`Stream.Error`" exception: when it is raised, it means that some terminals have been consummed from the stream, which are definitively lost, and therefore that are no more possible parser cases to try.

An alternative is to use functional parsers which use a new stream type, lazy but not destructive. Their advantage is that they use a limited backtrack: the case of "if..then..else.." and the shorter "if..then.." work without having to left factorize the parser cases, and there is no need to lookahead. They have no equivalent to the exception "`Stream.Error`": when all cases are tested, and have failed, the parsers return the value "`None`". The drawback is that, when a parsing error happens, it is not easily possible to know the location of the error in the input, as the initial stream has not been modified: the system would indicate a failure at the first character of the first line: this is a general drawback of backtracking parsers. See the solutions found to this problem in the chapter about purely functional parsers.

A second alternative is to use the backtracking parsers. They use the same stream type as the functional parsers, but they test more cases than them. They have the same advantages and drawbacks than the functional parsers.