4 November 2018
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TODO
In Chapter 44, we examined a set of prototype parsing functions and then used them as patterns for hand-coding of recursive descent parsing functions. We can benefit by generalizing these functions and collecting them into a library.
Consider parseS, one of the prototype parsing functions from a previous section. It parses the grammar rule S ::= A | B, which has two alternatives.
parseS :: String -> (Bool,String)
parseS xs =
case parseA xs of -- try A
(True, ys) -> (True, ys) -- A succeeds
(False, _ ) ->
case parseB xs of -- else try B
(True, ys) -> (True, ys) -- B succeeds
(False, _) -> (False, xs) -- both A,B failNote that parseS and the other prototype parsing functions have the type:
The occurrence of type String in the argument of the function represents the state of the input before evaluation of the function; the second occurrence of String represents the state after evaluation. The type Bool represents the result of the evaluation.
In an imperative program, the state is often left implicit and only the result type is returned. However, in a purely functional program, we must also make both the state change explicit.
Functions that have a type similar to parseS are called state actions or state transitions. We can generalize this parsing state transition as a function type:
In the case of parseS, we specialize this to:
In the case of richer parsing case studies for the prefix and infix parsers, we specialize this type as:
Given the Parser type, we can define a set of combinators that allow us to combine simpler parsers to construct more complex parsers. These combinators can pass along the state implicitly, avoiding some tedious and repetitive work.
We can define a combinator parseAlt that generalizes the parseS prototype function above. It implements a recognizer, so we fix type b to Bool, but leave type argument a general.
parseAlt :: Parser a Bool -> Parser a Bool -> Parser a Bool
parseAlt p1 p2 =
\xs ->
case p1 xs of
(True, ys) -> (True, ys)
(False, _ ) ->
case p2 xs of
(True, ys) -> (True, ys)
(False, _ ) -> (False, xs)Note the use of the anonymous function in the body. Function parseAlt takes two Parser values and then returns a Parser value. The Parser function returned binds in the two component function values. When this function is applied to the parser input (which is the argument of the anonymous function), it applies the two component parsers as needed.
We can easily redefine parseS in terms of the parseAlt combinator and simpler parsers parseA and parseB.
Given parsing input inp, we can invoke the parser with the expression:
Note that this formulation enables us to handle the passing of state among the component parsers implicitly, much as we can in an imperative computation. But it still preserves the nature of purely functional computation.
Now consider the parseA prototype, which implements a two-component sequencing rule A ::= C D.
parseA xs =
case parseC xs of -- try C
(True, ys) -> -- then try D
case parseD ys of
(True, zs) -> (True, zs) -- C D succeeds
(False, _) -> (False, xs) -- both C, D fail
(False, _ ) -> (False,xs) -- C failsAs with parseS, we can generalize parseA as a combinator parseSeq.
parseSeq :: Parser a Bool -> Parser a Bool -> Parser a Bool
parseSeq p1 p2 =
\xs ->
case p1 xs of
(True, ys) ->
case p2 ys of
t@(True, zs) -> t
(False, _ ) -> (False, xs)
(False, _ ) -> (False, xs)Thus we can redefine parseA in terms of the parseSeq combinator and simpler parsers parseC and parseD.
Similarly, we consider the parseB prototype, which implements a repetition rule B ::= { E }.
parseB xs =
case parseE xs of -- try E
(True, ys) -> parseB ys -- try again
(False, ys) -> (True,xs) -- stopAs above, we generalize this as combinator parseStar.
parseStar :: Parser a Bool -> Parser a Bool
parseStar p1 =
\xs ->
case p1 xs of
(True, ys) -> parseStar p1 ys
(False, _ ) -> (True, xs)We can redefine parseB in terms of combinator parseStar and simpler parser parseE.
Finally, consider parsing prototype parseC, which implements an optional rule C ::= [ F ].
We generalize this pattern as parseOpt, as follows.
parseOpt :: Parser a Bool -> Parser a Bool
parseOpt p1 =
\xs ->
case p1 xs of
(True, ys) -> (True, ys)
(False, _ ) -> (True, xs)We can thus redefine parseC in terms of simpler parser parseF and combinator parseOpt.
In this simple example grammar, function parseD is a simple instance of a sequence and parseE and parseF are simple parsers for symbols. These can be directly implemented as basic parsers, as before. However, the technique work if these are more complex parsers built up from combinators.
For convenience and completeness, we include extended alternative and sequencing combinators and parsers that always fail or always succeed.
parseAltList :: [Parser a Bool] -> Parser a Bool
parseSeqList :: [Parser a Bool] -> Parser a Bool
parseFail, parseSucceed :: Parser a BoolThe combinators in this library are in the Haskell module ParserComb.hs. A module that does some testing is TestParserComb.hs.
TODO: Update and document the Parser Combinator library code.
TODO: Expand this library to allow returns of “parse trees” and error messages.
There are a number of relatively standard parsing combinator libraries–e.g. the library Parsec. Readers who wish to develop other parsers may want to study that library.
TODO
I initially developed this prototype set of parsing combinators in Summer 2017 as a part of what is now called the ELI Calculator language case study. I based this case study, in part, on ideas from:
the Scala-based Expression Tree Calculator case study from my Notes on Scala for Java Programmers [Cunningham 2018]: (which was itself adapted from the tutorial Scala for Java Programmers [Schinz 2017] in Spring 2016)
the Lua-based Expression Language 1 and Imperative Core interpreters I developed for the Fall 2016 CSci 450 course
Samuel N. Kamin’s textbook Programming Languages: An Interpreter-based Approach, Addison Wesley, 1990 [Kamin 1990] and my work to implement three (Core, Lisp, and Scheme) of these interpeters in Lua in 2013
the Wikipedia articles on Regular Grammar, Context-Free Grammar, Backus-Naur Form, Lexical Analysis, Parsing, LL Parser, Recursive Descent Parser, and Abstract Syntax [Wikipedia 2018, 2018b].
Chapter 21 (Recursive Descent Parser) of the Martin Fowler and Rebecca Parsons’s book Domain-Specific Languages, Addison Wesley, 2011 [Fowler 2011].
Section 3.2 (Predictive Parsing) of Andrew W. Appel’s textbook Modern Compiler Implementation in ML, Cambridge, 1998
[Appel 1998].
Chapters 6 (Purely Functional State) and 9 (Parser Combinators) from Paul Chiusano and Runar Bjarnason’s Functional Programming in Scala, Manning, 2015.
The parsing combinators were built primarily on the approach of [Fowler 2011], with influences by [Chiusano 2015].
In Fall 2018, I divided the 2017 Expression Language Parsing chapter into two chapters in the 2018 version of the textbook, now titled Exploring Languages with Interpreters and Functional Programming. Sections 11.1-11.4 became Chapter 44, Calculator Parsing, and sections 11.6-11.7 became Chapter 45, Parsing Combinators (this chapter). I merged Section 11.5 into Chapter 43, a new chapter on the modular structure of the ELI Calculator language interpreter.
I maintain these notes as text in Pandoc’s dialect of Markdown using embedded LaTeX markup for the mathematical formulas and then translate the notes to HTML, PDF, and other forms as needed. The HTML version of this document may require use of a browser that supports the display of MathML.
TODO