Parsing all context-free grammars using Earley's algorithm.

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Stackage Nightly 2017-08-20:
LTS Haskell 9.1:
Stackage Nightly 2017-08-20:
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BSD3 licensed by Olle Fredriksson

Earley Build Status Hackage

Go to the API documentation on Hackage.

This (Text.Earley) is a library consisting of a few main parts:


An embedded context-free grammar (CFG) domain-specific language (DSL) with semantic action specification in applicative style.

An example of a typical expression grammar working on an input tokenised into strings is the following:

   expr :: Grammar r (Prod r String String Expr)
   expr = mdo
     x1 <- rule $ Add <$> x1 <* namedToken "+" <*> x2
               <|> x2
               <?> "sum"
     x2 <- rule $ Mul <$> x2 <* namedToken "*" <*> x3
               <|> x3
               <?> "product"
     x3 <- rule $ Var <$> (satisfy ident <?> "identifier")
               <|> namedToken "(" *> x1 <* namedToken ")"
     return x1
       ident (x:_) = isAlpha x
       ident _     = False


An implementation of (a modification of) the Earley parsing algorithm.

To invoke the parser on the above grammar, run e.g. (here using words as a stupid tokeniser):

   fullParses (parser expr) $ words "a + b * ( c + d )"
   = ( [Add (Var "a") (Mul (Var "b") (Add (Var "c") (Var "d")))]
     , Report {...}

Note that we get a list of all the possible parses (though in this case there is only one).

Another invocation, which shows the error reporting capabilities (giving the last position that the parser reached and what it expected at that point), is the following:

   fullParses (parser expr) $ words "a +"
   = ( []
     , Report { position   = 2
              , expected   = ["(","identifier","product"]
              , unconsumed = []


Functionality to generate the members of the language that a grammar generates.

To get the language of a grammar given a list of allowed tokens, run e.g.:

   language (generator romanNumeralsGrammar "VIX")
   = [(0,""),(1,"I"),(5,"V"),(10,"X"),(20,"XX"),(11,"XI"),(15,"XV"),(6,"VI"),(9,"IX"),(4,"IV"),(2,"II"),(3,"III"),(19,"XIX"),(16,"XVI"),(14,"XIV"),(12,"XII"),(7,"VII"),(21,"XXI"),(25,"XXV"),(30,"XXX"),(31,"XXXI"),(35,"XXXV"),(8,"VIII"),(13,"XIII"),(17,"XVII"),(26,"XXVI"),(29,"XXIX"),(24,"XXIV"),(22,"XXII"),(18,"XVIII"),(36,"XXXVI"),(39,"XXXIX"),(34,"XXXIV"),(32,"XXXII"),(23,"XXIII"),(27,"XXVII"),(33,"XXXIII"),(28,"XXVIII"),(37,"XXXVII"),(38,"XXXVIII")]

The above example shows the language generated by a Roman numerals grammar limited to the tokens 'V', 'I', and 'X'.


Helper functionality for creating parsers for expressions with mixfix identifiers in the style of Agda.

How do I write grammars?

As hinted at above, the grammars are written inside Grammar, which is a Monad and MonadFix. For the library to be able to tame the recursion in the grammars, we have to use the rule function whenever a production is recursive.

Whenever you would write e.g.

p = foo <|> bar <*> p

in a conventional combinator parser library, you instead write the following:

grammar = mdo
  p <- rule $ foo <|> bar <*> p

Apart from making it possible to do recursion (even left-recursion), rules have an additional benefit: they control where work is shared, by the rule that any rule is only ever expanded once per position in the input string. If a rule is encountered more than once at a position, the work is shared.

Compared to parser generators and combinator libraries

This library differs from the main methods that are used to write parsers in the Haskell ecosystem:

  • Compared to parser generators (YACC, Happy, etc.) it requires very little pre-processing of the grammar. It also allows you to stay in the host language for both grammar and parser, i.e. there is no use of a separate tool. This also means that you are free to use the abstraction facilities of Haskell when writing a grammar. Currently the library requires a linear traversal of the grammar's rules before use, which is usually fast enough to do at run time, but precludes infinite grammars.

  • The grammar language is similar to that of many parser combinators (Parsec, Attoparsec, parallel parsing processes, etc.), providing an applicative interface, but the parser gracefully handles all finite CFGs, including those with left-recursion. On the other hand, its productions are not monadic meaning that it does not support context-sensitive or infinite grammars, which are supported by many parser combinator libraries.

    Note: The Grammar type is a Monad (used to provide observable sharing) but it lives a layer above productions. It cannot be used to decide what production to use depending on the result of a previous production, i.e. it does not give us monadic parsing.

The parsing algorithm

The parsing algorithm that this library uses is based on Earley's parsing algorithm. The algorithm has been modified to produce online parse results, to give good error messages, and to allow garbage collection of the item sets. Essentially, instead of storing a sequence of sets of items like in the original algorithm, the modified algorithm just stores pointers back to sets of reachable items.

The worst-case run time performance of the Earley parsing algorithm is cubic in the length of the input, but for large classes of grammars it is linear. It should however be noted that this library will likely be slower than most parser generators and parser combinator libraries.

The parser implements an optimisation similar to that presented in Joop M.I.M Leo's paper A general context-free parsing algorithm running in linear time on every LR(k) grammar without using lookahead, which removes indirections in sequences of non-ambiguous backpointers between item sets.

For more in-depth information about the internals of the library, there are implementation notes currently being written.


Olle Fredriksson - https://github.com/ollef



  • Update 'base' dependency bounds

  • Add the Generator module for generating grammar members
  • Change (simplify) the type returned by parser, introducing a Parser type synonym for it, and change the signature of allParses, fullParses, and report to accept a Parser
  • The Text.Earley.Internal module is now Text.Earley.Parser.Internal

  • Add missing modules to Cabal file

  • Add IsString Prod instance
  • Change the signature of Terminal to take a function a -> Maybe b, and add a new operator terminal
  • Move satisfy to the Derived module
  • Add the token, namedToken, and list operators
  • Deprecate the symbol, namedSymbol, and word operators (use the above instead)
  • Add the listLike operator

  • Fix bug concerning nullable rules (#14)
  • Add runGrammar

  • Add changelog


  • Remove Args, and use Results instead
  • Make parser function not take input directly
  • Remove redundant type parameter to Grammar


  • Optimise handling of nullable non-terminals
  • Pass a record of arguments in the parse routine
  • Add support for consecutive mixfix holes
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