text:
# Math Help - Series and complex number

1. ## Series and complex number

This series
$1 + \frac{cos(\theta)}{sin(\theta)} + \frac{cos(2\theta)}{sin^2(\theta)} + ... + \frac{cos(n\theta)}{sin^n(\theta)}$
I think that the summation notation would be $\sum_{i=0}^{n} \frac{cos(i\theta)}{sin^i(\theta)}$.
But anyway, I'm having trouble with the top one. I have tried to transform it so that I can use complex numbers to find the answer. I have not succeed. Does someone know the way (or any method) to do this problem?

2. Originally Posted by vincisonfire
This series
$1 + \frac{cos(\theta)}{sin(\theta)} + \frac{cos(2\theta)}{sin^2(\theta)} + ... + \frac{cos(n\theta)}{sin^n(\theta)}$
I think that the summation notation would be $\sum_{i=0}^{n} \frac{cos(i\theta)}{sin^i(\theta)}$.
But anyway, I'm having trouble with the top one. I have tried to transform it so that I can use complex numbers to find the answer. I have not succeed. Does someone know the way (or any method) to do this problem?
What is your goal :S? What is it you're trying to get to?

PS: Confusing to use i as your index if you're also planning on using complex numbers in the solution!

3. You're right, bad choice of index.
I try to find a "nice" general formula for $S_n$
$
1 + \frac{cos(\theta)}{sin(\theta)} + \frac{cos(2\theta)}{sin^2(\theta)} + ... + \frac{cos(n\theta)}{sin^n(\theta)} = S_n
$

$
S_n =
\sum_{k=0}^{n} \frac{cos(k\theta)}{sin^k(\theta)}
$

4. Hi

Your idea seems to be good

Let $
T_n =
\sum_{k=0}^{n} \frac{sin(k\theta)}{sin^k(\theta)}
$

Then

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{cos(k\theta)+isin(k\theta)}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{e^{ik\theta}}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{({e^{i\theta}})^k}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \left({\frac{e^{i\theta}}{sin(\theta)}}\right)^k
$

5. Originally Posted by running-gag
Hi

Your idea seems to be good

Let $
T_n =
\sum_{k=0}^{n} \frac{sin(k\theta)}{sin^k(\theta)}
$

Then

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{cos(k\theta)+isin(k\theta)}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{e^{ik\theta}}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \frac{({e^{i\theta}})^k}{sin^k(\theta)}
$

$
S_n + i T_n =
\sum_{k=0}^{n} \left({\frac{e^{i\theta}}{sin(\theta)}}\right)^k
$
Also

$
S_n - i T_n =
\sum_{k=0}^{n} \frac{cos(k\theta)-isin(k\theta)}{sin^k(\theta)}
$

$
S_n - i T_n =
\sum_{k=0}^{n} \frac{e^{-ik\theta}}{sin^k(\theta)}
$

$
S_n - i T_n =
\sum_{k=0}^{n} \frac{({e^{-i\theta}})^k}{sin^k(\theta)}
$

$
S_n - i T_n =
\sum_{k=0}^{n} \left({\frac{e^{-i\theta}}{sin(\theta)}}\right)^k
$

Two geometric series!

6. Originally Posted by running-gag
$
S_n + i T_n =
\sum_{k=0}^{n} \left({\frac{e^{i\theta}}{sin(\theta)}}\right)^k
$
I would use $\sum_{k=0}^{n} x^i = \frac{1-x^{n+1}}{1-x}$ and then take $
S_n =
Re (\sum_{k=0}^{n} \left({\frac{e^{i\theta}}{sin(\theta)}}\right)^k)
$

7. Originally Posted by vincisonfire
I would use $\sum_{k=0}^{n} x^i = \frac{1-x^{n+1}}{1-x}$ and then take $
S_n =
Re (\sum_{k=0}^{n} \left({\frac{e^{i\theta}}{sin(\theta)}}\right)^k)
$
Yes
You can also use the hint given by danny arrigo

$S_n = \frac{(S_n + i T_n) + (S_n - i T_n)}{2}$

I did not perform the job, so I don't know which is the easiest way !

8. Answer is $\frac{sin^{n+2}(x)-cos(x)sin^{n+1}(x)-cos((n+1)x)sin(x)+cos(nx)}{sin^n(x)(\frac{3}{2}-\frac{sin(2x+arctan(\frac{1}{2}))}{2sin(arctan(\fr ac{1}{2}))})}$ and both ways of doing it are as long.
To who it may interest. This is the form with the less "sin" and "cos" in it although it looks as complicated as the raw version.
I replaced $*sin^2(x)-2cos(x)sin(x)+1$ by $\frac{3}{2}-\frac{sin(2x+arctan(\frac{1}{2}))}{2sin(arctan(\fr ac{1}{2}))}$


text:
# Sharper Rates and Flexible Framework for Nonconvex SGD with Client and Data Sampling

The new paper Sharper Rates and Flexible Framework for Nonconvex SGD with Client and Data Sampling has been outed.

I was glad to work my peers Alexander Tyurin, Lukang Sun, and prof. Peter Richtarik from King Abdullah University of Science and Technology on the paper “Sharper Rates and Flexible Framework for Nonconvex SGD with Client and Data Sampling”. The arXiv link for the paper: https://arxiv.org/abs/2206.02275.

During numerical experiments for that paper, we heavily exploited FL_PyTorch – an advanced research simulator for Federated Learning:

# Abstract

We revisit the classical problem of finding an approximately stationary point of the average of $$n$$ smooth and possibly nonconvex functions.

The optimal complexity of stochastic first-order methods in terms of the number of gradient evaluations of individual functions is $$\mathcal{O}(n + \sqrt{n}/\varepsilon)$$, attained by the optimal SGD methods SPIDER [Fang et al, NeurIPS 2018] and PAGE [Zhize et al, ICML 2021], for example, where $$\varepsilon$$ is the error tolerance.

However, i) the big-$$\mathcal{O}$$ notation hides crucial dependencies on the smoothness constants associated with the functions, and ii) the rates and theory in these methods assume simplistic sampling mechanisms that do not offer any flexibility. In this work we remedy the situation.

First, we generalize the PAGE algorithm so that it can provably work with virtually any (unbiased) sampling mechanism. This is particularly useful in federated learning, as it allows us to construct and better understand the impact of various combinations of client and data sampling strategies.

Second, our analysis is sharper as we make explicit use of certain novel inequalities that capture the intricate interplay between the smoothness constants and the sampling procedure.

Indeed, our analysis is better even for the simple sampling procedure analyzed in the PAGE paper. However, this already improved bound can be further sharpened by a different sampling scheme which we propose.

In summary, we provide the most general and most accurate analysis of optimal SGD in the smooth nonconvex regime. Finally, our theoretical findings are supposed with carefully designed experiments.

Written on June 7, 2022

format: plain


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# Nontrivial criteria for polynomials to have no common zeros?

When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that the sum of $g_i\cdot f_i$ is $1$, (Nullstellensatz). However, this theorem doesn't seem to have any practical value, consider the following problem:

Is there any nontrivial criterion such that we can determine whether some polynomials (chosen randomly) have common zeros or not by using it? Of course, I don't want to use the Grobner base theory, what I need is just a criterion involving information on the algebraic properties of the given polynomials, something just like Eisensten's criterion on irreducibility.

-
The plural of criterion is criteria. –  Chandan Singh Dalawat Dec 11 '10 at 7:54
Yes - minor editing carried out. –  Gerry Myerson Dec 11 '10 at 8:53
@ougao, well naively speaking, if you consider the chance of a (random) polynomial $f(x)$ sharing a (particular) zero with (another random) polynomial $g(y)$ as being "equally likely" as the chance that these two (random) polynomials do not share any zeroes, then a "trivial" criterion (at least for me) would be: $h(0) = \frac{1}{2}$ where $h(z) = ABS(f(x) - g(y))$. I hope that trivial observation will get to help you with your problem somehow. –  Jose Arnaldo Bebita Dris Dec 11 '10 at 9:17
I don't understand the statement "this theorem doesn't seem to have any practical value". There is an algorithm (the resultant algorithm, as pointed out by Myerson), which has many efficientimplementations and which is used in every known computer algebra system. As for the complexity issues, note that a general polynomial of degree $d$ in $n$ variables has a LOT of terms, so questions about such polynomials are sure to be very hard. If you restrict to SPARSE polynomials, the question becomes interesting. This is covered in the book by R. Zippel (Efficient Polynomial Computation). –  Igor Rivin Dec 11 '10 at 16:08
OK, so it seems the question really is, how can I construct nontrivial examples of polynomials with no common zeros. That of course depends on what "nontrivial" means. Maybe this is trivial: let $f_1,\dots,f_{k-1}$ and $g_1,\dots,g_{k-1}$ be arbitrary, and let $f_k=f_1g_1+\cdots+f_{k-1}g_{k-1}+1$; then $f_1,\dots,f_k$ have no common zeros. –  Gerry Myerson Dec 11 '10 at 21:38

Not known. Your problem is $NP_\mathbf{C}$-complete in the Blum-Shub-Smale complexity model. So anything significantly simpler than Groebner basis would yield significant progress on the $P=NP$ question over $\mathbf{C}$ compared to the current state of the art.

-
Hmm. I thought this was what the resultant was for. Certainly works for $n=1$, $k=2$, and there is such a thing as the resultant in more general cases. –  Gerry Myerson Dec 11 '10 at 5:19
@Gerry: yes, resultants might be suitable depending on the OP's purpose, but it does not change anything to the complexity point of view. Deciding common zeroes is hard. –  Thierry Zell Dec 11 '10 at 5:43
@Thierry: I suppose that deciding whether a given real number is zero is hard, in some model. Yet in practice I rarely have any trouble with this. Perhaps we need ougao to give us a little more context; among other things, how does ougao propose to choose polynomials "randomly"? Reference to "algebraic properties" and Eisenstein criterion suggest maybe integer coefficients in mind. I'm tempted to vote to close as "not a real question" until/unless ougao comes up with the goods. –  Gerry Myerson Dec 11 '10 at 8:51
@Gerry: When I posted my answer, I was hoping that the OP would comment on my answer, and let us know if the complexity point of view was relevant to the original motivation. I wouldn't go as far as saying that we don't have a real question right now, but in the current state, it looks like a question with no definitive answer; the complexity point of view is only one possible angle, and it would be nice to see others. –  Thierry Zell Dec 11 '10 at 11:51
Sorry for my ambiguous words, I just want to say that when I write down some polynomials, I want to detect whether they have no common zeros, if not, then maybe I can go to search the coefficients $g_i$,or at least say something on them. –  Jiang Dec 11 '10 at 17:33


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# Parsec, a fast combinator parser

Daan Leijen

University of Utrecht
Dept. of Computer Science
PO.Box 80.089, 3508 TB Utrecht
The Netherlands
daan@cs.uu.nl, http://www.cs.uu.nl/~daan

4 Oct 2001

## Introduction

Parsec is an industrial strength, monadic parser combinator library for Haskell. It can parse context-sensitive, infinite look-ahead grammars but it performs best on predictive (LL[1]) grammars. Combinator parsing is well known in the literature and offers several advantages to YACC or event-based parsing.
• Combinator parsers are written and used within the same programming language as the rest of the program. There is no gap between the grammar formalism (Yacc) and the actual programming language used (C). Users learn just one language! We can also enjoy all the benefits that exist in the host language: type checking, module system, development environment, etc.
• Parsers are first-class values within the language. They can be put into lists, passed as parameters and returned as values. It is easy extend the available set of parsers with custom made parsers specific for a certain problem, see for example many or buildExpressionParser.
Most ‘real’ parsers are built using bottom-up parser generators like happy. This is mostly for reasons of speed but also because most parser libraries are written for research and inadequate for practical applications. Parsec is designed as a ‘real world’ library from scratch:
• Speed. Most combinator libraries lack the speed necessary to be competetive with bottom-up parser generators. Parsec uses some novel techniques to improve its performance. The library is fast, parsing thousands of lines a second on today's machines (see the benchmark for test results).
• Error reporting. Parsec has extensive error messages; giving position, unexpected input, expected productions and general user messages. The error messages are given in terms of high-level grammar productions and can be localized for different (natural) languages.
• Libraries. Parsec provides a set of basic parsers and provides three extension libraries, a lexical analysis library, a generalized expression parser and a library for parsing permutation phrases.
• Documentation. Parsec is documented with a reference- and user-guide and example grammar implementations.
• Simplicity. Parsec is designed to be as simple as possible, making it accessable to both novice and expert users. Parsec is predictive by default, only backtracking where an explicit try is inserted. This makes the operational behaviour easier to understand than the normal' approach where backtracking is the default behaviour.

This document ships in the following formats:
• Postscript (PS). High quality output, best for printing.
• Acrobat (PDF). High quality output and hyperlinked. Preferred for online browsing.
• HTML. Poor man's PDF – hyperlinked but it lacks an index.
After the introduction, this document contains a simple user guide and a than a full reference guide for the library.

### Compatibility

The core library is written in Haskell98. The extension library ParsecToken uses the forall keyword (to emulate a first-class module) but, to my best knowledge, this extension is supported by every Haskell98 system. The library is successfully compiled with GHC 4.x, GHC 5.x, Hugs98 and nhc98, but probably works with any Haskell98 compliant compiler or interpreter.

#### Compiling with GHC

Parsec is distributed as a package with GHC. This means that you can simply use the -package text switch to use the Parsec library. Unfortunately, Ghc versions up to 5.02 include an old version of Parsec. In this case, you might want to link with the newest Parsec library. Parsec ships with a makefile that creates a Haskell library – just type make at the command prompt.

Suppose that the library is compiled in the directory c:\parsec. When you compile a Haskell source that uses Parsec, you should tell GHC where it can find the interface files (*.hi).

ghc -c myfile -ic:\parsec

When your linking the files together, you need to tell GHC where it can find libraries (-L) and to link with the Parsec library too (-l):

ghc -o myprogram myfile1.o myfile2.o -Lc:\parsec -lparsec

#### Compiling with Hugs

If you use the ParsecToken or ParsecPerm modules, you need to enable the forall extension. Hugs should be started with the -98 option to enable Haskell98 extensions.

### Reporting bugs

Any bugs (or fixes!) can be reported to the author, Daan Leijen (daan@cs.uu.nl). Any feedback on:
• Which parsers are lacking;
• Naming of the parsers;
• Performance, both time and space.
is greatly appreciated too. Besides that, I am also quite interested in any applications, especially in a commercial setting, that are created using this library.

The Parsec homepage can be found at: http://www.cs.uu.nl/~daan/parsec.html. Any useful additions to the library or example grammars will be put on this www-page. If you have written a parser for some language, don't hesitate and send it to author so it can be put on the www-page too.

The library has been tested using the Pbench benchmark. It can be found at http://www.cs.uu.nl/people/daan/pbench.html.

### History

• 4 Oct 2001: Parsec can now work on any input stream which allows one to write a seperate lexer. There is also a user definable state -- for example for creating identifiers from strings. Non-compatible changes:
• brackets is now called angles, while squares is now called brackets.
• The type of many parsers is now more general, ie. GenParser tok st a instead of Parser a.
• The types of Operator and OperatorTable have changed into a more general type.
• The ParsecToken module is now parameterized with a record instead of an imported module.
• The token parser now parses single tokens.
• Added new functionality, like sepEndBy.
• Added an experimental module for parsing permutations (ParsecPerm).

I would like to see this library as widely used as possible. It is therefore distributed with an unrestrictive BSD style license. If there is any problem using this software with this license, I would like to hear about it and see if we can work out a solution.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

• Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
• Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
This software is provided by the copyright holders “as is” and any express or implied warranties, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose are disclaimed. In no event shall the copyright holders be liable for any direct, indirect, incidental, special, exemplary, or consequential damages (including, but not limited to, procurement of substitute goods or services; loss of use, data, or profits; or business interruption) however caused and on any theory of liability, whether in contract, strict liability, or tort (including negligence or otherwise) arising in any way out of the use of this software, even if advised of the possibility of such damage.

## Users guide

All the examples given in this user guide can be found in the parsec/examples/userguide directory of the parser library.

### Running a parser

module Main where

import Parsec

Let's start with a very simple parser. The parser that recognizes a single letter (a character between 'a' and 'z') is written as:
simple :: Parser Char
simple  = letter

The type of simple shows that simple is a parser that will return a Char (the parsed letter) whenever it succeeds. A parser can be executed using the parse function. This function takes three arguments: the parser (simple), the name of the input ("") and the input itself. The name of input is only used for error messages and can be empty in our case. The result of parse is either an error (Left) or a succesful result (Right). We define a simple run function that simply prints either the error value or the result value for a specific parser:
run :: Show a => Parser a -> String -> IO ()
run p input
= case (parse p "" input) of
Left err -> do{ putStr "parse error at "
; print err
}
Right x  -> print x

(The run function is actually defined in the Parsec module as parseTest). When we load this module into hugs, we can directly and interactively test our parsers on different inputs:

Reading file "d:\cvs\parsec\examples\userguide\main.hs":

Hugs session for:
d:\programs\hugs98\lib\Prelude.hs
d:\programs\hugs98\lib\Maybe.hs
d:\programs\hugs98\lib\List.hs
d:\cvs\parsec\ParsecError.hs
d:\programs\hugs98\lib\Char.hs
d:\cvs\parsec\Parsec.hs
d:\cvs\parsec\examples\userguide\Main.hs
Main> run simple "a"
'a'

Main> run simple ""
parse error at (line 1, column 1):
unexpected end of input
expecting letter

Main> run simple "123"
parse error at (line 1, column 1):
unexpected "1"
expecting letter

### Sequence and choice

Two important operations for specifying grammars (and parsers) are sequence and choice. In our case, parsers can be sequenced using the monadic do-notation. For example, the following parser parses an open parenthesis followed by a closing parenthesis ("()"):
openClose :: Parser Char
openClose = do{ char '('
; char ')'
}

Two simple parenthesis are not too interesting but in combination with the choice operator (<|>) we can specify a parser that recognizes matching pairs of parenthesis. Matching parenthesis consist of either an open parenthesis followed by a matching pair of parenthesis, a closing parenthesis and another matching pair of parenthesis, or it is empty; The empty alternative is implemented using the (return x) parser which always succeeds with value x without consuming any input.
parens  :: Parser ()
parens  = do{ char '('
; parens
; char ')'
; parens
}
<|> return ()

Let's run the parens parser on some input to test its behaviour:

Main> run parens "(())()"
()

Main> run parens "(()()"
parse error at (line 1, column 6):
unexpected end of input
expecting "(" or ")"

### Predictive parsers

For reasons of speed, the (<|>) combinator is predictive; it will only try its second alternative if the first parser hasn't consumed any input. Consider the following example parser:

testOr  =   string "(a)"
<|> string "(b)"

The (string s) parser accepts input that equals the string s. Since both strings in the example start with the same initial character, the behaviour is probably not what the user expected:

Main> run testOr "(b)"
parse error at (line 1, column 2):
unexpected 'b'
expecting 'a'

Since the first parser has already consumed some input (the '('), the second alternative is never tried allthough it would succeed! The preferred solution to this problem is to left-factor the grammar by merging common prefixes. The following parser works as expected:

testOr1 = do{ char '('
; char 'a' <|> char 'b'
; char ')'
}

Unfortunately, it is not always possible to left-factor the grammar and in many other cases the grammar becomes more complex to understand. There is a primitive combinator called try that handles these cases in an elegant manner. The parser (try p) behaves just like p except that it pretends that it hasn't consumed any input whenever p fails. In combination with (<|>) this allows for infinite look-ahead in the grammar. Here is the above parser written using try:

testOr2 =   try (string "(a)")
<|> string "(b)"

or an even better version:

testOr3 =   do{ try (string "(a"); char ')'; return "(a)" }
<|> string "(b)"

Allthough the implementation of try is quite efficient there is a penalty when it fails, ie. it shouldn't be used as an alternative to careful grammar design!

Right now, the parens parser simply recognizes matching pairs of parenthesis but doesn't return a useful value. We will now extend the parens parser with semantic' actions which compute the maximal nesting level of the parenthesis. The (<-) operator will be used to bind the intermediate values returned by the parsers.

The nesting parser returns the nesting level of the parenthesis. The empty alternative returns zero. The other alternative returns the maximal nesting level of both other parsers:

nesting :: Parser Int
nesting = do{ char '('
; n <- nesting
; char ')'
; m <- nesting
; return (max (n+1) m)
}
<|> return 0

Here is an example session with nesting:

Main> run nesting "(())()"
2

Main> run nesting "(()(()))"
3

Main> run nesting "(()(())"
parse error at (line 1, column 8):
unexpected end of input
expecting "(" or ")"

### Sequences and seperators

Consider the word parser that parses words: a sequence of one or more letters:
word    :: Parser String
word    = do{ c  <- letter
; do{ cs <- word
; return (c:cs)
}
<|> return [c]
}

After parsing at a letter, it is either followed by a word itself or it returns the single character as a string. Allthough it is not too hard to learn to recognize these patterns, it is not immediately obvious that a sequence of one or more letters is parsed by this grammar. Even widely used parser generators like YACC and JavaCC do not provide any abstraction from these commonly occurring patterns. Luckily, we have the full power of functional programming at our finger tips, it is easy to provide a custom parser combinator that parses a sequence of one or more parsers; it is called many1:

word    :: Parser String
word    = many1 letter

Now it is obvious from the code what word is supposed to do and it is much easier to specifiy; no strange recursive patterns! Beside many1, the library provides the many parser which parses zero or more occurrences of its argument. The combinators skipMany and skipMany1 behave like many and many1 but do not return the values of their argument parser.

Another useful pair of combinators are sepBy and sepBy1 which parse a sequence of parsers separated by some separator. The combinators sepEndBy and sepEndBy1 parse a sequence of parsers that are either all seperated or terminated by some seperator (ie. Haskell declarations). For example, sentences are a sequence of one or more words separated by one or more spaces or commas and ended by a dot, question mark or exclamation mark:

sentence    :: Parser [String]
sentence    = do{ words <- sepBy1 word separator
; oneOf ".?!"
; return words
}

separator   :: Parser ()
separator   = skipMany1 (space <|> char ',')

Let's parse some example sentences:
Main> run sentence "hi,di,hi."
["hi","di","hi"]

Main> run sentence "hi,di hi!"
["hi","di","hi"]

Main> run sentence "hi,123"
parse error at (line 1, column 4):
unexpected "1"
expecting ",", space or letter

### Improving error messages

Allthough the last error message of the previous hugs session is correct, it is not very user friendly. A better message would simply be "expecting word". The error combinator (<?>) attaches a high-level error description to parsers. Errors can now be given in terms of high-level grammar productions instead of at the level of characters. We will first attach a description to the word parser:

word    :: Parser String
word    = many1 letter <?> "word"

Whenever the word parser fails without consuming input, it will now issue an error message "expecting word":

Main> run sentence "hi,123"
parse error at (line 1, column 4):
unexpected "1"
expecting ",", space or word

It is easy to get rid of the comma and space messages; just use an empty description:

separator   :: Parser ()
separator   = skipMany1 (space <|> char ',' <?> "")

Note that the (<|>) combinator binds stronger than the (<?>) operator. The error message only reports the expected word now:

Main> run sentence "hi,123"
parse error at (line 1, column 4):
unexpected "1"
expecting word

Another improvement can be made to the end of sentence' parser:

sentence    :: Parser [String]
sentence    = do{ words <- sepBy1 word separator
; oneOf ".?!" <?> "end of sentence"
; return words
}

Let's test the new parser on an uncompleted sentence:

Main> run sentence "hi,di"
parse error at (line 1, column 6):
unexpected end of input
expecting letter or end of sentence

Allthough the message is again correct, one further improvement can be made. The expected letter is a bit confusing; indeed a letter can be added to the last word but it is probably not what the user intended. Note that the message reports that a letter can be expected and not a word. This is because a message that is attached to a parser using (<?>) will only be issued if the parser can not even be started, i.e. when it hasn't consumed any input. Since the word parser has already consumed two letters (di) the expect message for letter is issued.

What we want to do is to attach an empty message to the letter parser in word but maintain the "word" message on the parser as a whole.

word    :: Parser String
word    = many1 (letter <?> "") <?> "word"

Finally we have a highly customised error messages which behave quite nicely:

Main> run sentence "hi di"
parse error at (line 1, column 6):
unexpected end of input
expecting end of sentence

Main> run sentence "hi di,"
parse error at (line 1, column 7):
unexpected end of input
expecting word

After this section, it might seem that error handling is quite involved, but don't be discouraged; This example grammar was chosen to be tricky and experience shows that for most grammars just a few error descriptions are necessary.

### Expressions

Expressions are probably one of the most common constructs for which a parser is needed. Consider the following standard EBNF grammar for expressions:

expr   ::= expr '+' term | term
term   ::= term '*' factor | factor
factor ::= '(' expr ')' | digit+

digit  ::= '0' | '1' | ... | '9'

The grammar uses different productions to specify priorities ('*' has higher priority than '+') and uses left-recursion to make both star and plus left-associative.

Unfortunately, left-recursive grammars can not be specified directly in a combinator library. If you accidently write a left recursive program, the parser will go into an infinite loop! However, every left-recursive grammar can be rewritten to a non-left-recursive grammar. The library provides combinators which do this automatically for you (chainl and chainl1).

Even when using these combinators to deal with the left-recursion, it is still a nuisance to code the associativity and priorities of the operators in such a primitive way. The parser library provides a standard module ParsecExpr to deal more elegantly with expression style grammars. Using this library, the above expression grammar, extended with minus and divide, is specified using the buildExpressionParser function:

import ParsecExpr

expr    :: Parser Integer
expr    = buildExpressionParser table factor
<?> "expression"

table   = [[op "*" (*) AssocLeft, op "/" div AssocLeft]
,[op "+" (+) AssocLeft, op "-" (-) AssocLeft]
]
where
op s f assoc
= Infix (do{ string s; return f}) assoc

factor  = do{ char '('
; x <- expr
; char ')'
; return x
}
<|> number
<?> "simple expression"

number  :: Parser Integer
number  = do{ ds <- many1 digit
}
<?> "number"

The function buildExpressionParser takes two arguments. The second argument is the basic expression term and the first a table of operators of decreasing priority (the higher in the list, the higher the priority). Operators can be Infix, Prefix or Postfix. Infix operators also have an associativity: AssocLeft, AssocRight or AssocNone.

Here is an example session with the expr parser:

Main> run expr "1+2*3"   -- '*' has higher priority
7

Main> run expr "(1+2)*3"
9

Main> run expr "8/4/2"   -- '/' is left associative
1

Main> run expr "8/(4/2)"
4

Main> run expr "1 + 2"  -- wrong!
1

Main> run expr "1+ 2"   -- wrong!
parse error at (line 1, column 3):
unexpected " "
expecting simple expression

### Lexical analysis

The previous hugs session with expressions shows that the expression parser can not yet deal with whitespace. Most parsers take their input from a lexical analyzer or scanner which filters white space and comments and returns a list tokens for the parser. How Parsec can work on such token streams is described in a later section. Parsec can also merge both passes into one parser specification which allows us to use all the normal parser combinators to deal with lexical analysis too. The standard module ParsecToken deals with issues normally delegated to a seperate scanner: white space, comments, identifiers, reserved words, numbers, strings and other tokens.

Unfortunately, most languages disagree on the syntax of comments, the form of identifiers and for example case sensitivity. The ParsecToken module should be parameterized with those differences, but unfortunately, Haskell doesn't support parameterized modules. The module therefore exports a single function makeTokenParser that takes paremeterizable language features as an argument (LanguageDef) and returns a (rather large) record that contains a set of lexical parsers. For your convenience, the module ParsecLanguage contains a set of common language definitions. For the expression parser, we will use haskell-style comments and identifiers

module Main where
import Parsec
import qualified ParsecToken as P

lexer :: TokenParser ()
lexer  = makeTokenParser
{ reservedOpNames = ["*","/","+","-"]
}

For efficiency, we will bind all the used lexical parsers at toplevel.

whiteSpace= P.whiteSpace lexer
lexeme    = P.lexeme lexer
symbol    = P.symbol lexer
natural   = P.natural lexer
parens    = P.parens lexer
semi      = P.semi lexer
identifier= P.identifier lexer
reserved  = P.reserved lexer
reservedOp= P.reservedOp lexer

Every lexical parser from the ParsecToken module will skip whitespace after each symbol parsed; parsers which skip trailing whitespace are called lexeme parsers (the lexeme combinator can be used to define them). By skipping trailing whitespace, it is garanteed that every parser starts at valid input. For the expression parser, the lexeme parser reservedOp is used instead of the char and string parsers:

import ParsecToken

...

expr    :: Parser Integer
expr    = buildExpressionParser table factor
<?> "expression"

table   = [[op "*" (*) AssocLeft, op "/" div AssocLeft]
,[op "+" (+) AssocLeft, op "-" (-) AssocLeft]
]
where
op s f assoc
= Infix (do{ reservedOp s; return f} <?> "operator") assoc

The ParsecToken module also defines a set of high-level combinators like parens to parse something enclosed in parenthesis and natural to parse natural numbers in decimal, octal or hexadecimal notation:

factor  =   parens expr
<|> natural
<?> "simple expression"

Finally the run function will be adapted to deal with whitespace. The main parser needs to skip any whitespace at the start of the input (since all other parsers only skip trailing whitespace) and needs to end with the eof parser which only succeeds at the end of the input. This parser garantees that all input will be parsed; if it is left out, a parser is allowed to parse only part of the input.

runLex :: Show a => Parser a -> String -> IO ()
runLex p input
= run (do{ whiteSpace
; x <- p
; eof
; return x
}) input

And that's it. In about 30 lines of code we have a fully functional, extensible expression parser with comments, priority and associativity rules and infinite precision numbers in hexadecimal, octal and decimal representation.

Main> runLex expr "1 + 2"
3

Main> runLex expr "1 + {- comment -} 2 * 3 --multiply has higher priority"
7

Main> runLex expr "  0xAA / 0o37 / 2"
2

Main> runLex expr "0xAA / 0o37 2 "
parse error at (line 1, column 13):
unexpected "2"
expecting operator or end of input

### Receipts: Lexeme parsers and Reserved words

Consider the following EBNF grammar for receipts:
receipt     ::= product* total

produkt     ::= "return" price ";"
| identifier price ";"
total       ::= price "total"

price       ::= digit+ "." digit digit

A receipt is a sequence of semicolon terminated produkts and ends with the total price of all products. A produkt is either a name of a produkt with its price or a return produkt in which case the client receives the specified price. We will construct a parser for receipts which checks if the total amount is actually correct.

#### Lexeme parsers

The price parser is a nice exercise in specifying lexeme parsers. The price parser doesn't allow whitespace inside; For example, we can't use the decimal parser to parse the digits in front of the dot since the decimal parser would allow whitespace between those digits and the dot. The appropiate solution is to create a lexeme parser ourselves.

price   :: Parser Int   -- price in cents
price   = lexeme (do{ ds1 <- many1 digit
; char '.'
; ds2 <- count 2 digit
; return (convert 0 (ds1 ++ ds2))
})
<?> "price"
where
convert n []     = n
convert n (d:ds) = convert (10*n + digitToInt d) ds

The implementation of the other productions is almost a literal translation of the grammar rules:

receipt :: Parser Bool
receipt = do{ ps <- many produkt
; p  <- total
; return (sum ps == p)
}

produkt = do{ symbol "return"
; p <- price
; semi
; return (-p)
}
<|> do{ identifier
; p  <- price
; semi
; return p
}
<?> "produkt"

total   = do{ p <- price
; symbol "total"
; return p
}

Let's try the parser on a few receipts:

Main> runLex receipt "book 12.00; plant 2.55; 14.55 total"
True

Main> runLex receipt "book 12.00; plant 2.55; 12.55 total"
False

Main> runLex receipt "book 12.00; plant 2; 12.55 total"
parse error at (line 1, column 20):
unexpected ";"
expecting digit or "."

Main> runLex receipt "book 12.00; return 2.00; plant 2.55; 12.55 total"
True

#### Reserved words

Unfortunately, the parser will incorrectly return an error on the following input:

Main> runLex receipt "book 12.00; reader 2.00; plant 1.00; 15.00 total"
parse error at (line 1, column 13):
unexpected "a"
expecting "return"

What happens here? The produkt parser will try to recognize the keyword return when it starts scanning "reader". When the "a" is encountered an error is returned. The parser will not try the second alternative which starts with an identifier since the it has already consumed input (i.e. "re").

This is a good example of a grammar which is hard to left-factor. The use of the try combinator is the appropiate way of solving this problem:

produkt = do{ try (symbol "return")
; p <- price
; semi
; return (-p)
}
<|> do{ identifier
; p  <- price
; semi
; return p
}
<?> "produkt"

The test run now gives the expected result:

Main> runLex receipt "book 12.00; reader 2.00; plant 1.00; 15.00 total"
True

However, we are still not done as the following example shows:

Main> runLex receipt "book 12.00; returns 2.00; plant 1.00; 15.00 total"
parse error at (line 1, column 19):
unexpected "s"
expecting price

The current parser recognises the keyword return now and it complains about the letter 's' following it. A keyword however shouldn't be recognised as a keyword when it is only a prefix of an identifier. The parser notFollowedBy can be used to make sure that a keyword is not followed by a legal identifier character. Just like checking that a keyword is not a prefix of an identifier, we should also check that an identifier doesn't make up a keyword, in order to make the parser independent of the order of choices.

Since adding the try combinator in the right places, using notFollowedBy and checking for reserved words is quite subtle, the ParsecToken module implements all of this complexity for you; simply add the reserved words to your token definition:

lexer :: TokenParser ()
lexer  = makeTokenParser
{ reservedNames   = ["return","total"]
, reservedOpNames = ["*","/","+","-"]
}

Instead of symbol, the combinator reserved is used to scan reserved words. The try combinator is not necessary anymore since the reserved combinator already takes care of it.

receipt :: Parser Bool
receipt = do{ ps <- many produkt
; p  <- total
; return (sum ps == p)
}

produkt = do{ reserved "return"
; p <- price
; semi
; return (-p)
}
<|> do{ identifier
; p  <- price
; semi
; return p
}
<?> "produkt"

total   = do{ p <- price
; reserved "total"
; return p
}

Finally, everything works as expected:

Main> runLex receipt "book 12.00; returns 2.00; plant 1.00; 15.00 total"
True

Main> runLex receipt "book 12.00; total 2.00; plant 1.00; 15.00 total"
parse error at (line 1, column 13):
unexpected reserved word "total"
expecting produkt or price

Main> runLex receipt "book 12.00; totals 2.00; return 1.00; 13.00 total"
True

### Examples

The following example parsers are shipped with Parsec:
• Tiger and While. Both parsers are written by Jurriaan Hage, who used the languages during CS cources at Utrecht University. The Tiger language is the example language in the compiler book by Andrew Appel (1997). The While language is the example language in the semantics book by Nielson, Nielson and Hankin (1999).
• Henk 2000, based on Pure Type Systems. Written by Jan-Willem Roorda and Daan Leijen. A parser for the Henk language as descibed by Peyton Jones and Meijer (1997). The master thesis of Jan-Willem is online available together with a full interpreter for pure-type-systems.
• Mondrian, the internet programming language (Mondrian, 2000). An example of a Java-style language.

In the previous examples we used an integrated approach – the lexer was specified using normal parser combinators. This is not always desirable, sometimes there is already a scanner available or sometimes the input needs pre-processing. Another advantage of writing a scanner by hand is that the performance of the parser generally improves about 30% (see the benchmark for test results).

Suppose that we have written a seperate scanner for receipts. The scanner function returns a list of tokens and a list of error messages.

type Token  = (SourcePos,Tok)
data Tok    = Identifier String
| Reserved   String
| Symbol     String
| Price      Int
deriving Show

scanner :: [Char] -> ([Token],[String])

The parsers should now work on these token streams instead of the normal chararacter streams. This is reflected in the type. The type of general parsers is GenParser tok st a, where tok is the type of the tokens, st the type of the local user state and a is the type of the result. Indeed, Parser is just a type synonym for parsers that work on characters and have no state:

type Parser a   = GenParser Char () a

The token parser is used to define a parser that scans a single token. The function takes three arguments: a function to show tokens, a function to extract the source position of a token and finally a test function that determines if the token is accepted or not. The first two arguments are the same for all our tokens so we define a little abstraction:

type MyParser a   = GenParser Token () a

mytoken :: (Tok -> Maybe a) -> MyParser a
mytoken test
= token showToken posToken testToken
where
showToken (pos,tok)   = show tok
posToken  (pos,tok)   = pos
testToken (pos,tok)   = test tok

Now, it is easy to define the basic parser that work on tokens.

identifier :: MyParser String
identifier
= mytoken (\tok -> case tok of
Identifier name -> Just name
other           -> Nothing)

reserved :: String -> MyParser ()
reserved name
= mytoken (\tok -> case tok of
Reserved s   | s == name  -> Just ()
other        -> Nothing)

...

For an extended example, you might want to look at the implementation of the ParsecChar module.

Since Parsec is already a state monad, it is easy to include a user definable state. This state is very useful when building a symbol table – strings can immediately be converted to identifiers [footnote: I have to warn the reader though that experience with the HaskellLight compiler has shown that it hardly pays off in practice to use special identifier representations instead of normal strings].

Parsec provides three combinators to handle state. The function getState gets the state, setState st sets the state to st, and updateState f updates the state by applying function f to it. As an example, we show how we can count the number of identifiers in a parser.

run input
= case runParser parser 0 "" input of
Right n  -> putStrLn ("counted " ++ show n ++ " identifiers!")
Left err -> do{ putStr "parse error at "
; print err
}

parser :: CharParser Int Int
parser
= do{ ...
; n <- getState
; return n
}

...

myIdentifier :: CharParser Int String
myIdentifier
= do{ x <- identifier
; return x
}

A permutation phrase is sequence of elements of possibly different types in which each element occurs at most once and the order is irrelevant. Parsec provides the module ParsecPerm to parse such free-order constructs. Possible applications include parsers for XML attributes and Haskell style records.

Since each element can be of a different type, it is hard to find a typable approach and we have to resort to judicious use of permutation parser combinators. Parsers are combined into a permutation parser with the (<||>) operator. At the end, all the elements are combined using a combination function f that is applied using the (<$$>) combinator. The whole permutation parser is than converted into a normal parser using the permute function. Suppose we want to parse a permutation of the characters a, b, and c. We can write this as: perm0 = permute (f (<$$>) char 'a'
(<||>) char 'b'
(<||>) char 'c')
where
f a b c  = [a,b,c]

Let's try the parser against some real input!

Main> run perm0 "abc"
"abc"

Main> run perm0 "cba"
"abc"

Main> run perm0 "b"
parse error at (line 1, column 2):
unexpected end of input
expecting "c" or "a"

There is an implicit restriction that the parsers in a permutation should never succeed on empty input – ie. you can't use (many (char 'a')) to parse an optional string of 'a' characters. If there are optional elements in the permutation, you need to use the special (<|?>) operator that takes both a parser and a default value. If the first parser can be optional, you can use the (<$?>) operator. For example, suppose we parse a permutation of: an optional string of a's, the character b and an optional c. This can be described by: perm1 :: Parser (String,Char,Char) perm1 = permute (tuple <$?> ("",many1 (char 'a'))
<||> char 'b'
<|?> ('_',char 'c'))
where
tuple a b c  = (a,b,c)

At the command prompt we can see how error correction works with permutations.

Main> run perm1 "caaaaab"
("aaaaa",'b','c')

Main> run perm1 "cb"
("",'b','c')

Main> run perm1 "b"
("",'b','_')

Main> run perm1 ""
parse error at (line 1, column 1):
unexpected end of input
expecting "c", "b" or "a"

Main> run perm1 "c"
parse error at (line 1, column 2):
unexpected end of input
expecting "b" or "a"

Main> run perm1 "ca"
parse error at (line 1, column 3):
unexpected end of input
expecting "a" or "b"

## Reference guide

Parsec consists of the following main modules:
• Parsec implements the core parser library.
• ParsecExpr is an extension library for parsing expression grammars.
• ParsecToken and ParsecLanguage are extension libraries for parsing lexical tokens.
• ParsecPerm is an extension library for parsing permutation phrases.

### Module Parsec

This module is the core parser library. It exports functions from following primitive modules:
• ParsecPos. Source positions.
• ParsecError. Error messages.
• ParsecPrim. The main parser type and primitive parser combinators, for example parse.
• ParsecCombinator. General polymorphic parser combinators, for example many.
• ParsecChar. A range of parsers for characters, for example digit.

### Module ParsecPrim

Parser a
A type synonym for GenParser Char () a, i.e. a parser for character streams without a user state.
GenParser tok st a
The General Parser GenParser tok st a data type represents a parser that parses tokens of type tok with a user supplied state st and returns a value of type a on success. GenParser is an instance of the Functor, Monad and MonadPlus classes.
runParser  :: GenParser tok st a -> st -> FilePath -> [tok] -> Either ParseError a
The most general way to run a parser. (runParser p state filePath input) runs parser p on the input list of tokens input, obtained from source filePath with the initial user state st. The filePath is only used in error messages and may be the empty string. Returns either a ParseError (Left) or a value of type a (Right).

parseFromFile p fname   = do{ input <- readFile fname       ; return (runParser p () fname input)       }

parse  :: Parser a -> FilePath -> String -> Either ParseError a
(parse p filePath input) runs a character parser p without user state. The filePath is only used in error messages and may be the empty string. Returns either a ParseError (Left) or a value of type a (Right).

main    = case (parse numbers "" "11, 2, 43") of            Left err  -> print err            Right xs  -> print (sum xs) numbers = commaSep integer

parseFromFile  :: Parser a -> FilePath -> IO (Either ParseError a)
(parseFromFile p filePath) runs a character parser p on the input read from filePath. Returns either a ParseError (Left) or a value of type a (Right).

main    = do{ result <- parseFromFile numbers "digits.txt"             ; case (result) of                 Left err  -> print err                 Right xs  -> print (sum xs)             }

parseTest  :: Show a => GenParser tok () a -> [tok] -> IO ()
The expression parseTest p input applies a parser p against input input and prints the result to stdout. Used for testing parsers.

parseTest p input         = case (runParser p () "" input) of             Left err -> do{ putStr "parse error at "                           ; print err                           }             Right x  -> print x

return  :: a -> GenParser tok st a
The parser (return x) always succeeds with value x without consuming any input.
(>>=)  :: GenParser tok st a -> (a -> GenParser tok st b) -> GenParser tok st binfixl 1
This parser is called bind and implements sequencing. The parser (p >>= f) first applies parser p. It than applies f to the returned value of p and applies the resulting parser.

The do-notation of Haskell is syntactic sugar that automatically uses this combinator. Using this notation, we can write do{ x <- p; q } instead of the more cumbersome (p >>= (\x -> q)). The do-notation has a natural operational reading: first apply parser p, binding its result to value x, and than apply the parser q.

Since the second parser can depend on the result of the first parser, we are able to parse context sensitive grammars. Examples of context sensitive parsing are XML tags, variable definition before use and environments in TEX. A simple example of parsing TEX environments is:

environment  = do{ name <- envBegin                  ; environment                  ; envEnd name                  }              <|> return () envBegin :: Parser String envBegin     = do{ reserved "\\begin"                  ; braces (many1 letter)                  } envEnd :: String -> Parser () envEnd name  = do{ reserved "\\end"                  ; braces (string name)                  }

(<|>)  :: GenParser tok st a -> GenParser tok st a -> GenParser tok st ainfixr 1
This combinator implements choice. The parser (p <|> q) first applies p. If it succeeds, the value of p is returned. If p fails without consuming any input, parser q is tried. This combinator is defined equal to the mplus member of the MonadPlus class.

The parser is called predictive since q is only tried when parser p didn't consume any input (i.e.. the look ahead is 1). This non-backtracking behaviour allows for both an efficient implementation of the parser combinators and the generation of good error messages.

fail  :: String -> GenParser tok st a
The parser (fail msg) always fails with message-error msg without consuming any input.

The return, (>>=) and fail combinators are the members of the Monad class. The Parser is called monadic since it both provides an implementation for these three combinators, making it an instance of the Monad class, and it satisfies the monad-laws.

pzero  :: GenParser tok st a
pzero always fails without consuming any input. pzero is defined equal to the mzero member of the MonadPlus class.
try  :: GenParser tok st a -> GenParser tok st a
The parser (try p) behaves like parser p, except that it pretends that it hasn't consumed any input when an error occurs.

This combinator is used whenever arbitrary look ahead is needed. Since it pretends that it hasn't consumed any input when p fails, the (<|>) combinator will try its second alternative even when the first parser failed while consuming input.

The try combinator can for example be used to distinguish identifiers and reserved words. Both reserved words and identifiers are a sequence of letters. Whenever we expect a certain reserved word where we can also expect an identifier we have to use the try combinator. Suppose we write:

expr        = letExpr <|> identifier <?> "expression" letExpr     = do{ string "let"; ... } identifier  = many1 letter

If the user writes "lexical", the parser fails with: unexpected 'x', expecting 't' in "let". Indeed, since the (<|>) combinator only tries alternatives when the first alternative hasn't consumed input, the identifier parser is never tried (because the prefix "le" of the (string "let") parser is already consumed). The right behaviour can be obtained by adding the try combinator:

expr        = letExpr <|> identifier <?> "expression" letExpr     = do{ try (string "let"); ... } identifier  = many1 letter

Since the use of the try combinator with lexical tokens is quite tricky, the ParsecToken module can be used to parse lexical tokens. This module automatically uses the try combinator in the appropiate places.

token  :: (tok -> String) -> (tok -> SourcePos) -> (tok -> Maybe a) -> GenParser tok st a
The parser token showTok posFromTok testTok accepts a token t with result x when the function testTok t returns Just x. The source position of the t should be returned by posFromTok t and the token can be shown using showTok t.

This combinator is expressed in terms of tokenPrim. It is used to accept user defined token streams. For example, suppose that we have a stream of basic tokens tupled with source positions. We can than define a parser that accepts single tokens as:

mytoken :: Show t => t -> GenParser ((Int,Int),t) () t mytoken x   = token showTok posFromTok testTok   where     showTok (pos,t)     = show t     posFromTok (pos,t)  = pos     testTok (pos,t)     = if (x == t) then Just t else Nothing

tokenPrim  :: (tok -> String) -> (SourcePos -> tok -> [tok] -> SourcePos) -> (tok -> Maybe a) -> GenParser tok st a
The parser (token showTok nextPos testTok) accepts a token t with result x when the function testTok t returns Just x. The token can be shown using showTok t. The position of the next token should be returned when nextPos is called with the current source position pos, the current token t and the rest of the tokens toks, (nextPos pos t toks).

This is the most primitive combinator for accepting tokens. For example, the char parser could be implemented as:

char :: Char -> GenParser Char st Char char c   = tokenPrim showChar nextPos testChar   where     showChar x       = "'" ++ x ++ "'"     testChar x        = if (x == c) then Just x else Nothing     nextPos pos x xs  = updatePosChar pos x

(<?>)  :: GenParser tok st a -> String -> GenParser tok st ainfix 0
The parser p <?> msg behaves as parser p, but whenever the parser p fails without consuming any input, it replaces expect error messages with the expect error message msg.

This is normally used at the end of a set alternatives where we want to return an error message in terms of a higher level construct rather than returning all possible characters. For example, if the expr parser from the try example would fail, the error message is: ...: expecting expression. Without the (<?>) combinator, the message would be like ...: expecting "let" or letter, which is less friendly.

unexpected  :: String -> GenParser tok st a
The parser (unexpected msg) always fails with an unexpected error message msg without consuming any input.

The parsers fail, (<?>) and unexpected are the three parsers used to generate error messages. Of these, only (<?>) is commonly used. For an example of the use of unexpected, see the definition of notFollowedBy.

getState  :: GenParser tok st st
Returns the current user state.
setState  :: st -> GenParser tok st ()
(setState st) set the user state to st.
updateState  :: (st -> st) -> GenParser tok st ()
(updateState f) applies function f to the user state. Suppose that we want to count identifiers in a source, we could use the user state as:

expr :: GenParser Char Int Expr expr  = do{ x <- identifier           ; updateState (+1)           ; return (Id x)           }

getPosition  :: GenParser tok st SourcePos
Returns the current source position. See also SourcePos
setPosition  :: SourcePos -> GenParser tok st ()
setPosition pos sets the current source position to pos.
getInput  :: GenParser tok st [tok]
Returns the current input
setInput  :: [tok] -> GenParser tok st ()
setInput input continues parsing with input. The getInput and setInput functions can for example be used to deal with #include files.

### Module ParsecCombinator

many  :: GenParser tok st a -> GenParser tok st [a]
(many p) applies the parser p zero or more times. Returns a list of the returned values of p.

identifier  = do{ c  <- letter                 ; cs <- many (alphaNum <|> char '_')                 ; return (c:cs)                 }

many1  :: GenParser tok st a -> GenParser tok st [a]
(many p) applies the parser p one or more times. Returns a list of the returned values of p.

word  = many1 (letter)

skipMany  :: GenParser tok st a -> GenParser tok st ()
(skipMany p) applies the parser p zero or more times, skipping its result.

spaces  = skipMany space

skipMany1  :: GenParser tok st a -> GenParser tok st ()
(skipMany1 p) applies the parser p one or more times, skipping its result.
sepBy  :: GenParser tok st a -> GenParser tok st sep -> GenParser tok st [a]
(sepBy p sep) parses zero or more occurrences of p, separated by sep. Returns a list of values returned by p.

commaSep p  = p sepBy (symbol ",")

sepBy1  :: GenParser tok st a -> GenParser tok st sep -> GenParser tok st [a]
(sepBy1 p sep) parses one or more occurrences of p, separated by sep. Returns a list of values returned by p.
endBy  :: GenParser tok st a -> GenParser tok st end -> GenParser tok st [a]
(endBy p sep) parses zero or more occurrences of p, seperated and ended by sep. Returns a list of values returned by p.

cStatements  = cStatement endBy semi

endBy1  :: GenParser tok st a -> GenParser tok st end -> GenParser tok st [a]
(endBy1 p sep) parses one or more occurrences of p, seperated and ended by sep. Returns a list of values returned by p.
sepEndBy  :: GenParser tok st a -> GenParser tok st sep -> GenParser tok st [a]
(sepEndBy p sep) parses zero or more occurrences of p, separated and optionally ended by sep, ie. haskell style statements. Returns a list of values returned by p.

haskellStatements  = haskellStatement sepEndBy semi

sepEndBy1  :: GenParser tok st a -> GenParser tok st sep -> GenParser tok st [a]
(sepEndBy1 p sep) parses one or more occurrences of p, separated and optionally ended by sep. Returns a list of values returned by p.
count  :: Int -> GenParser tok st a -> GenParser tok st [a]
(count n p) parses n occurrences of p. If n is smaller or equal to zero, the parser equals to (return []). Returns a list of n values returned by p.
between  :: GenParser tok st open -> GenParser tok st close -> GenParser tok st a -> GenParser tok st a
(between open close p) parses open, followed by p and close. Returns the value returned by p.

braces  = between (symbol "{") (symbol "}")

option  :: a -> GenParser tok st a -> GenParser tok st a
(option x p) tries to apply parser p. If p fails without consuming input, it returns the value x, otherwise the value returned by p.

priority :: Parser Int priority  = option 0 (do{ d <- digit                         ; return (digitToInt d)                          })

choice  :: [GenParser tok st a] -> GenParser tok st a
(choice ps) tries to apply the parsers in the list ps in order, until one of them succeeds. Returns the value of the succeeding parser. choice can be defined as:

choice ps  = foldl (<|>) pzero ps

manyTill  :: GenParser tok st a -> GenParser tok st end -> GenParser tok st [a]
(manyTill p end) applies parser p zero or more times until parser end succeeds. Returns the list of values returned by p . This parser can be used to scan comments:

simpleComment   = do{ string "<!--"                     ; manyTill anyChar (try (string "-->"))                     }

Note the overlapping parsers anyChar and string "<!--", and therefore the use of the try combinator.

chainl  :: GenParser tok st a -> GenParser tok st (a->a->a) -> a -> GenParser tok st a
(chainl p op x) parser zero or more occurrences of p, separated by op. Returns a value obtained by a left associative application of all functions returned by op to the values returned by p. If there are zero occurrences of p, the value x is returned. .
chainl1  :: GenParser tok st a -> GenParser tok st (a->a->a) -> GenParser tok st a
(chainl1 p op x) parser one or more occurrences of p, separated by op Returns a value obtained by a left associative application of all functions returned by op to the values returned by p. . This parser can for example be used to eliminate left recursion which typically occurs in expression grammars.

expr    = term   chainl1 mulop term    = factor chainl1 addop factor  = parens expr <|> integer mulop   =   do{ symbol "*"; return (*)   }         <|> do{ symbol "/"; return (div) } addop   =   do{ symbol "+"; return (+) }         <|> do{ symbol "-"; return (-) }

chainr  :: GenParser tok st a -> GenParser tok st (a->a->a) -> a -> GenParser tok st a
(chainr p op x) parser zero or more occurrences of p, separated by op Returns a value obtained by a right associative application of all functions returned by op to the values returned by p. If there are no occurrences of p, the value x is returned. .
chainr1  :: GenParser tok st a -> GenParser tok st (a->a->a) -> GenParser tok st a
(chainr1 p op x) parser one or more occurrences of p, separated by op Returns a value obtained by a right associative application of all functions returned by op to the values returned by p. .
eof  :: Show tok => GenParser tok st ()
This parser only succeeds at the end of the input. This is not a primitive parser but it is defined using notFollowedBy.

eof  = notFollowedBy anyToken <?> "end of input"

Wow, I really like this definition!

notFollowedBy  :: Show tok => GenParser tok st tok -> GenParser tok st ()
(notFollowedBy p) only succeeds when parser p fails. This parser does not consume any input. This parser can be used to implement the 'longest match' rule. For example, when recognizing keywords (for example let), we want to make sure that a keyword is not followed by a legal identifier character, in which case the keyword is actually an identifier (for example lets). We can program this behaviour as follows:

keywordLet  = try (do{ string "let"                      ; notFollowedBy alphaNum                       }                     )

Surprisingly, this parser is not primitive and can be defined as:

notFollowedBy p  = try (do{ c <- p; unexpected (show [c]) }                        <|> return ()                        )

anyToken  :: Show tok => GenParser tok st tok
The parser anyToken accepts any kind of token. It is for example used to implement eof. Returns the accepted token.

### Module ParsecChar

CharParser st a
A type synonym for GenParser Char st a, i.e. a parser for character streams with user state st.
oneOf  :: [Char] -> CharParser st Char
(oneOf cs) succeeds if the current character is in the supplied list of characters cs. Returns the parsed character. See also satisfy.

vowel  = oneOf "aeiou"

noneOf  :: [Char] -> CharParser st Char
As the dual of oneOf, (noneOf cs) succeeds if the current character not in the supplied list of characters cs. Returns the parsed character.

consonant = noneOf "aeiou"

char  :: Char -> CharParser st Char
(char c) parses a single character c. Returns the parsed character (i.e. c).

semiColon  = char ';'

string  :: String -> CharParser st String
(string s) parses a sequence of characters given by s. Returns the parsed string (i.e. s).

divOrMod    =   string "div"              <|> string "mod"

anyChar  :: CharParser st Char
This parser succeeds for any character. Returns the parsed character.
upper  :: CharParser st Char
Parses an upper case letter (a character between 'A' and 'Z'). Returns the parsed character.
lower  :: CharParser st Char
Parses a lower case character (a character between 'a' and 'z'). Returns the parsed character.
letter  :: CharParser st Char
Parses a letter (an upper case or lower case character). Returns the parsed character.
alphaNum  :: CharParser st Char
Parses a letter or digit (a character between '0' and '9'). Returns the parsed character.
digit  :: CharParser st Char
Parses a digit. Returns the parsed character.
hexDigit  :: CharParser st Char
Parses a hexadecimal digit (a digit or a letter between 'a' and 'f' or 'A' and 'F'). Returns the parsed character.
octDigit  :: CharParser st Char
Parses an octal digit (a character between '0' and '7'). Returns the parsed character.
newline  :: CharParser st Char
Parses a newline character ('\n'). Returns a newline character.
tab  :: CharParser st Char
Parses a tab character ('\t'). Returns a tab character.
space  :: CharParser st Char
Parses a white space character (any character in " \v\f\t\r\n"). Returns the parsed character.
spaces  :: CharParser st ()
Skips zero or more white space characters. See also skipMany.
satisfy  :: (Char -> Bool) -> CharParser st Char
The parser (satisfy f) succeeds for any character for which the supplied function f returns True. Returns the character that is actually parsed.

digit     = satisfy isDigit oneOf cs  = satisfy (\c -> c elem cs)

### Module ParsecPos

SourcePos
The abstract data type SourcePos represents source positions. It contains the name of the source (i.e. file name), a line number and a column number. SourcePos is an instance of the Show, Eq and Ord class.
Line
A type synonym for Int
Column
A type synonym for Int
sourceName  :: SourcePos -> FilePath
Extracts the name of the source from a source position.
sourceLine  :: SourcePos -> Line
Extracts the line number from a source position.
sourceColumn  :: SourceColumn -> Column
Extracts the column number from a source position.
incSourceLine  :: SourcePos -> Int -> SourcePos
Increments the line number of a source position.
incSourceColumn  :: SourcePos -> Int -> SourcePos
Increments the column number of a source position.
setSourceLine  :: SourcePos -> SourceLine -> SourcePos
Set the line number of a source position.
setSourceColumn  :: SourcePos -> SourceColumn -> SourcePos
Set the column number of a source position.
setSourceName  :: SourcePos -> SourceName -> SourcePos
Increments the line number of a source position.
updatePosChar  :: SourcePos -> Char -> SourcePos
Update a source position given a character. If the character is a newline ('\n') or carriage return ('\r') the line number is incremented by 1. If the character is a tab ('\t') the column number is incremented to the nearest 8'th column, ie. (column + 8 - ((column-1) mod 8)). In all other cases, the column is incremented by 1.
updatePosString  :: SourcePos -> String -> SourcePos
The expression (updatePosString pos s) updates the source position pos by calling updatePosChar on every character in s, ie. (foldl updatePosChar pos string).

### Module ParsecError

ParseError
The abstract data type ParseError represents parse errors. It provides the source position (SourcePos) of the error and a list of error messages (Message). A ParseError can be returned by the function parse. ParseError is an instance of the Show class.
errorPos  :: ParseError -> SourcePos
Extracts the source position from the parse error
errorMessages  :: ParseError -> [Message]
Extracts the list of error messages from the parse error
Message
This abstract data type represents parse error messages. There are four kinds of messages:

data Message = SysUnExpect String              | UnExpect String              | Expect String              | Message String

The fine distinction between different kinds of parse errors allows the system to generate quite good error messages for the user. It also allows error messages that are formatted in different languages. Each kind of message is generated by different combinators:

• A SysUnExpect message is automatically generated by the satisfy combinator. The argument is the unexpected input.
• A UnExpect message is generated by the unexpected combinator. The argument describes the unexpected item.
• A Expect message is generated by the <?> combinator. The argument describes the expected item.
• A Message message is generated by the fail combinator. The argument is some general parser message.

messageString  :: Message -> String
Extract the message string from an error message
messageCompare  :: Message -> Message -> Ordering
Compares two error messages without looking at their content. Only the constructors are compared where:

SysUnExpect < UnExpect < Expect < Message

messageEq  :: Message -> Message -> Bool
(messageEq m1 m2) equals True if (messageCompare m1 m2) equals EQ, in all other cases it equals False
showErrorMessages  :: [Message] -> String
The standard function for showing error messages. Formats a list of error messages in English. This function is used in the Show instance of ParseError. The resulting string will be formatted like:

unexpected {The first UnExpect or a SysUnExpect message}; expecting {comma separated list of Expect messages}; {comma separated list of Message messages}

### Module ParsecToken

The extension module ParsecToken is used to parse lexical tokens (for example: identifiers, comments and numbers). Since programming languages all have slightly different grammar rules for lexical tokens, the module should be parameterized with the most common differences between languages (for example: valid identifier characters, reserved words etc). Since Haskell doesn't have parametrized modules, we resort to a trick. The module exports a single function (makeTokenParser) that returns a rather large record that contains all the lexical parse functions.

Unfortunately, some of those functions (like parens) are polymorphic which implies that we need to use the forall keyword to give their type signature inside the record. This keyword is not part of the Haskell98 standard but almost all Haskell systems support it nowadays. You might need to add some flags to your compiler options to make this work, for example, GHC needs -fglasgow-exts and Hugs the -98 option.

All lexical features that are not parameterizable are implemented according to the Haskell lexical rules (for example: escape codes and floating point numbers). Luckily most language grammars match the Haskell grammar on these tokens quite closely (if not completely).

TokenParser st
The type of the record that holds lexical parsers that work on character streams with state st. Each function of this record is seperately described as if it was a toplevel function in a later section.

data TokenParser st  = TokenParser{       identifier  :: CharParser st String    , reserved    :: String -> CharParser st ()    , operator    :: CharParser st String    , reservedOp  :: String -> CharParser st ()               , charLiteral :: CharParser st Char    , stringLiteral :: CharParser st String    , natural     :: CharParser st Integer    , integer     :: CharParser st Integer    , float       :: CharParser st Double    , naturalOrFloat:: CharParser st (Either Integer Double)    , decimal     :: CharParser st Integer    , hexadecimal :: CharParser st Integer    , octal       :: CharParser st Integer    , symbol      :: String -> CharParser st String    , lexeme      :: forall a. CharParser st a -> CharParser st a    , whiteSpace  :: CharParser st ()         , parens      :: forall a. CharParser st a -> CharParser st a     , braces      :: forall a. CharParser st a -> CharParser st a    , brackets    :: forall a. CharParser st a -> CharParser st a    , squares     :: forall a. CharParser st a -> CharParser st a    , semi        :: CharParser st String    , comma       :: CharParser st String    , colon       :: CharParser st String    , dot         :: CharParser st String    , semiSep     :: forall a . CharParser st a -> CharParser st [a]    , semiSep1    :: forall a . CharParser st a -> CharParser st [a]    , commaSep    :: forall a . CharParser st a -> CharParser st [a]    , commaSep1   :: forall a . CharParser st a -> CharParser st [a]                    }

LanguageDef st
The LanguageDef type is a record that contains all parameterizable features of the ParsecToken module. The module ParsecLanguage contains some default definitions. The members of LanguageDef are described in a later section.

data LanguageDef st     = LanguageDef    { commentStart   :: String   , commentEnd     :: String   , commentLine    :: String   , nestedComments :: Bool                     , identStart     :: CharParser st Char   , identLetter    :: CharParser st Char   , opStart        :: CharParser st Char   , opLetter       :: CharParser st Char   , reservedNames  :: [String]   , reservedOpNames:: [String]   , caseSensitive  :: Bool   }

makeTokenParser  :: LanguageDef st -> TokenParser st
The expression (makeTokenParser language) creates a TokenParser record that contains lexical parsers that are defined using the definitions in the language record.

The use of this function is quite stylized – one imports the appropiate language definition and selects the lexical parsers that are needed from the resulting TokenParser.

module Main where import Parsec import qualified ParsecToken as P import ParsecLanguage (haskellDef) -- The parser ... expr  =   parens expr       <|> identifier       <|> ...       -- The lexer lexer       = P.makeTokenParser haskellDef          parens      = P.parens lexer braces      = P.braces lexer identifier  = P.identifier lexer reserved    = P.reserved lexer ...

### The members of TokenParser

The following functions are all members of the TokenParser record but they are described as if they are top-level functions since this how they are normally used (as shown in the example of makeTokenParser).

whiteSpace  :: CharParser st ()
Parses any white space. White space consists of zero or more occurrences of a space, a line comment or a block (multi line) comment. Block comments may be nested. How comments are started and ended is defined in the LanguageDef that is passed to makeTokenParser.
lexeme  :: CharParser st a -> CharParser st p
(lexeme p) first applies parser p and than the whiteSpace parser, returning the value of p. Every lexical token (lexeme) is defined using lexeme, this way every parse starts at a point without white space. Parsers that use lexeme are called lexeme parsers in this document.

The only point where the whiteSpace parser should be called explicitly is the start of the main parser in order to skip any leading white space.

mainParser  = do{ whiteSpace                 ; ds <- many (lexeme digit)                 ; eof                 ; return (sum ds)                 }

symbol  :: String -> CharParser st String
Lexeme parser (symbol s) parses string s and skips trailing white space.
parens  :: CharParser st a -> CharParser st a
Lexeme parser (parens p) parses p enclosed in parenthesis, returning the value of p. It can be defined as:

parens p  = between (symbol "(") (symbol ")") p

braces  :: CharParser st a -> CharParser st a
Lexeme parser (braces p) parses p enclosed in braces ('{' and '}'), returning the value of p.
brackets  :: CharParser st a -> CharParser st a
Lexeme parser (brackets p) parses p enclosed in brackets ('<' and '>'), returning the value of p.
squares  :: CharParser st a -> CharParser st a
Lexeme parser (squares p) parses p enclosed in square brackets ('[' and ']'), returning the value of p.
semi  :: CharParser st String
Lexeme parser semi parses the character ';' and skips any trailing white space. Returns the string ";".
comma  :: CharParser st String
Lexeme parser comma parses the character ',' and skips any trailing white space. Returns the string ",".
colon  :: CharParser st String
Lexeme parser colon parses the character ':' and skips any trailing white space. Returns the string ":".
dot  :: CharParser st String
Lexeme parser dot parses the character '.' and skips any trailing white space. Returns the string ".".
semiSep  :: CharParser st a -> CharParser st [a]
Lexeme parser (semiSep p) parses zero or more occurrences of p separated by semi. Returns a list of values returned by p. This combinator can be defined as:

semiSep p  = sepBy p semi

semiSep1  :: CharParser st a -> CharParser st [a]
Lexeme parser (semiSep1 p) parses one or more occurrences of p separated by semi. Returns a list of values returned by p.
commaSep  :: CharParser st a -> CharParser st [a]
Lexeme parser (commaSep p) parses zero or more occurrences of p separated by comma. Returns a list of values returned by p.
commaSep1  :: CharParser st a -> CharParser st [a]
Lexeme parser (commaSep1 p) parses one or more occurrences of p separated by comma. Returns a list of values returned by p.
charLiteral  :: CharParser st Char
This lexeme parser parses a single literal character. Returns the literal character value. This parsers deals correctly with escape sequences. The literal character is parsed according to the grammar rules defined in the Haskell report (which matches most programming languages quite closely).
stringLiteral  :: CharParser st String
This lexeme parser parses a literal string. Returns the literal string value. This parsers deals correctly with escape sequences and gaps. The literal string is parsed according to the grammar rules defined in the Haskell report (which matches most programming languages quite closely).
decimal  :: CharParser st Integer
Parses a positive whole number in the decimal system. Returns the value of the number.
hexadecimal  :: CharParser st Integer
Parses a positive whole number in the hexadecimal system. The number should be prefixed with "0x" or "0X". Returns the value of the number.
octal  :: CharParser st Integer
Parses a positive whole number in the octal system. The number should be prefixed with "0o" or "0O". Returns the value of the number.
natural  :: CharParser st Integer
This lexeme parser parses a natural number (a positive whole number). Returns the value of the number. The number can be specified in decimal, hexadecimal or octal. The number is parsed according to the grammar rules in the Haskell report.
integer  :: CharParser st Integer
This lexeme parser parses an integer (a whole number). This parser is like natural except that it can be prefixed with sign (i.e. '-' or '+'). Returns the value of the number. The number can be specified in decimal, hexadecimal or octal. The number is parsed according to the grammar rules in the Haskell report.
float  :: CharParser st Double
This lexeme parser parses a floating point value. Returns the value of the number. The number is parsed according to the grammar rules defined in the Haskell report.
naturalOrFloat  :: CharParser st (Either Integer Double)
This lexeme parser parses either natural or a float. Returns the value of the number. This parsers deals with any overlap in the grammar rules for naturals and floats. The number is parsed according to the grammar rules defined in the Haskell report.
identifier  :: CharParser st String
This lexeme parser parses a legal identifier. Returns the identifier string. This parser will fail on identifiers that are reserved words. Legal identifier (start) characters and reserved words are defined in the LanguageDef that is passed to makeTokenParser. An identifier is treated as a single token using try.
reserved  :: String -> CharParser st String
The lexeme parser (reserved name) parses (symbol name), but it also checks that the name is not a prefix of a valid identifier. A reserved word is treated as a single token using try.
operator  :: CharParser st String
This lexeme parser parses a legal operator. Returns the name of the operator. This parser will fail on any operators that are reserved operators. Legal operator (start) characters and reserved operators are defined in the LanguageDef that is passed to makeTokenParser. An operator is treated as a single token using try.
reservedOp  :: String -> CharParser st String
The lexeme parser (reservedOp name) parses (symbol name), but it also checks that the name is not a prefix of a valid operator. A reservedOp is treated as a single token using try.

### The members of LanguageDef

The following functions are all members of the LanguageDef record but they are described as if they are top-level values. These values determine how lexers that are constructed using makeTokenParser will behave.

commentStart  :: String
Describes the start of a block comment. Use the empty string if the language doesn't support block comments. For example "/*".
commentEnd  :: String
Describes the end of a block comment. Use the empty string if the language doesn't support block comments. For example "*/".
commentLine  :: String
Describes the start of a line comment. Use the empty string if the language doesn't support line comments. For example "//".
nestedComments  :: Bool
Set to True if the language supports nested block comments.
identStart  :: CharParser st Char
This parser should accept any start characters of identifiers. For example (letter <|> char "_").
identLetter  :: CharParser st Char
This parser should accept any legal tail characters of identifiers. For example (alphaNum <|> char "_").
opStart  :: CharParser st Char
This parser should accept any start characters of operators. For example (oneOf ":!#$%&*+./<=>?@\\^|-~") opLetter  :: CharParser st Char This parser should accept any legal tail characters of operators. Note that this parser should even be defined if the language doesn't support user-defined operators, or otherwise the reservedOp parser won't work correctly. reservedNames  :: [String] The list of reserved identifiers. reservedOpNames  :: [String] The list of reserved operators. caseSensitive  :: Bool Set to True if the language is case sensitive. ### Module ParsecLanguage This module defines some default LanguageDef definitions. These definitions provide easy starting points for defining your own LanguageDef definitions. emptyDef  :: LanguageDef st This is the most minimal token definition. It is recommended to use this definition as the basis for other definitions. emptyDef has no reserved names or operators, is case sensitive and doesn't accept comments, identifiers or operators. emptyDef = LanguageDef { commentStart = "" , commentEnd = "" , commentLine = "" , nestedComments = True , identStart = letter <|> char '_' , identLetter = alphaNum <|> oneOf "_'" , opStart = opLetter emptyDef , opLetter = oneOf ":!#$%&*+./<=>?@\\^|-~"    , reservedOpNames= []    , reservedNames  = []    , caseSensitive  = True    }

javaStyle  :: LanguageDef st
This is a minimal token definition for Java style languages. It defines the style of comments, valid identifiers and case sensitivity. It does not define any reserved words or operators.

javaStyle        = emptyDef     { commentStart   = "/*"     , commentEnd     = "*/"     , commentLine    = "//"     , nestedComments = True     , identStart     = letter     , identLetter    = alphaNum <|> oneOf "_'"     , reservedNames  = []     , reservedOpNames= []     , caseSensitive  = False        }

haskellStyle  :: LanguageDef st
This is a minimal token definition for Haskell style languages. It defines the style of comments, valid identifiers and case sensitivity. It does not define any reserved words or operators.

haskellStyle     = emptyDef     { commentStart   = "{-"     , commentEnd     = "-}"     , commentLine    = "--"     , nestedComments = True     , identStart     = letter     , identLetter    = alphaNum <|> oneOf "_'"     , opStart        = opLetter haskell     , opLetter       = oneOf ":!#$%&*+./<=>?@\\^|-~" , reservedOpNames= [] , reservedNames = [] , caseSensitive = True }  mondrianDef  :: LanguageDef st The language definition for the language Mondrian. mondrian = javaStyle { reservedNames = [ "case", "class", "default" , "extends" , "import", "in", "let" , "new", "of", "package" ] , caseSensitive = True }  haskell98Def  :: LanguageDef st The language definition for the language Haskell98. haskell98Def = haskellStyle { reservedOpNames= ["::","..","=","\\","|" ,"<-","->","@","~","=>" ] , reservedNames = ["let","in","case","of" ,"if","then","else" ,"data","type", ,"class","default","deriving" ,"do","import", ,"infix","infixl","infixr" ,"instance","module" ,"newtype","where" ,"primitive" -- "as","qualified","hiding" ] }  haskellDef  :: LanguageDef st The language definition for the Haskell language. haskellDef = haskell98Def { identLetter = identLetter haskell98Def <|> char '#' , reservedNames = reservedNames haskell98Def ++ ["foreign","import","export","primitive" ,"_ccall_","_casm_" ,"forall" ] }  mondrian  :: TokenParser st A lexer for the mondrian language. mondrian = makeTokenParser mondrianDef  haskell  :: TokenParser st A lexer for the haskell language. mondrian = makeTokenParser haskellDef  ### Module ParsecExpr ParsecExpr is a small extension module for parsing expression grammars. It imports the Parsec module. This data type specifies the associativity of operators: left, right or none. data Assoc = AssocNone | AssocLeft | AssocRight  Operator tok st a This data type specifies operators that work on values of type a. An operator is either binary infix or unary prefix or postfix. A binary operator has also an associated associativity. data Operator tok st a = Infix (GenParser tok st (a -> a -> a)) Assoc | Prefix (GenParser tok st (a -> a)) | Postfix (GenParser tok st (a -> a))  OperatorTable tok st a An (OperatorTable tok st a) is a list of (Operator tok st a) lists. The list is ordered in descending precedence. All operators in one list have the same precedence (but may have a different associativity). type OperatorTable tok st a = [[Operator tok st a]]  buildExpressionParser  :: OperatorTable tok st a -> GenParser tok st a -> GenParser tok st a (buildExpressionParser table term) builds an expression parser for terms term with operators from table, taking the associativity and precedence specified in table into account. Prefix and postfix operators of the same precedence can only occur once (i.e. --2 is not allowed if - is prefix negate). Prefix and postfix operators of the same precedence associate to the left (i.e. if ++ is postfix increment, than -2++ equals -1, not -3). The buildExpressionParser takes care of all the complexity involved in building expression parser. Here is an example of an expression parser that handles prefix signs, postfix increment and basic arithmetic. expr = buildExpressionParser table term <?> "expression" term = parens expr <|> natural <?> "simple expression" table :: OperatorTable Char () Integer table = [ [prefix "-" negate, prefix "+" id ] , [postfix "++" (+1)] , [binary "*" (*) AssocLeft, binary "/" (div) AssocLeft ] , [binary "+" (+) AssocLeft, binary "-" (-) AssocLeft ] ] binary name fun assoc = Infix (do{ reservedOp name; return fun }) assoc prefix name fun = Prefix (do{ reservedOp name; return fun }) postfix name fun = Postfix (do{ reservedOp name; return fun })  ### Module ParsecPerm This module contains combinators for building parsers that parse permutation phrases. It uses a typed approach that is descibed in (Baars et al., 2001). This library is still experimental and might change. PermParser tok st a The type (PermParser tok st a) denotes a permutation parser that, when converted by the permute function, parses token streams of type tok with user state st and returns a value of type a on success. Normally, a permutation parser is first build with special operators like (<||>) and than transformed into a normal parser using permute. permute  :: PermParser tok st a -> GenParser tok st a The parser (permute perm) parses a permutation of parser described by perm. For example, suppose we want to parse a permutation of: an optional string of a's, the character b and an optional c. This can be described by: test :: Parser (String,Char,Char) test = permute (tuple <$?> ("",many1 (char 'a'))                        <||> char 'b'                         <|?> ('_',char 'c'))       where         tuple a b c  = (a,b,c)

(<$$>)  :: (a -> b) -> GenParser tok st a -> PermParser tok st binfixl 2 The expression (f <$$> p) creates a fresh permutation parser consisting of parser p. The the final result of the permutation parser is the function f applied to the return value of p. The parser p is not allowed to accept empty input – use the optional combinator (<$?>) instead. If the function f takes more than one parameter, the type variable b is instantiated to a functional type which combines nicely with the adds parser p to the (<||>) combinator. This results in stylized code where a permutation parser starts with a combining function f followed by the parsers. The function f gets its parameters in the order in which the parsers are specified, but actual input can be in any order. permute$ f <> parse field x   <||> parse field y where   f x y = ...

(<||>)  :: PermParser tok st (a -> b) -> GenParser tok st a -> PermParser tok st binfixl 1
The expression (perm <||> p) adds parser p to the permutation parser perm. The parser p is not allowed to accept empty input – use the optional combinator (<|?>) instead. Returns a new permutation parser that includes p.
(<$?>)  :: PermParser tok st (a -> b) -> (a,GenParser tok st a) -> PermParser tok st binfixl 1 The expression (f <$?> (x,p)) creates a fresh permutation parser consisting of parser p. The the final result of the permutation parser is the function f applied to the return value of p. The parser p is optional – if it can not be applied, the default value x will be used instead.
(<|?>)  :: PermParser tok st (a -> b) -> (a,GenParser tok st a) -> PermParser tok st binfixl 1
The expression (perm <||> (x,p)) adds parser p to the permutation parser perm. The parser p is optional – if it can not be applied, the default value x will be used instead. Returns a new permutation parser that includes the optional parser p`.


text:
In a square in which the houses are evenly spaced around the outside, numbers $3$ and $10$ are opposite each other.

What is the smallest number of houses in the square?

What is the largest possible number of houses in the square?


text:
In molecular electronics, metallic contacts are made at two ends of the molecule, and electronic current is expected to be carried by electrons tunneling from the metal with energies in the band gap region. The branch point plays an important role in the analysis of the electron transport properties. The branch point energy, E bp , is an estimate for the lineup of the metal Fermi level (E F ) with the molecular levels, and β bp ) β(E bp ) provides the tunneling decay rate. We now use this information to obtain a simple estimate of the conductance of a molecule. According to Landauer theory, 51 when E F crosses a propagating molecular state (such as the highest occupied molecular orbital) in the real k region of the complex band structure, the low-voltage conductance of the molecule will be approximately equal to the quantum of conductance, G 0 ≈ 77 µS. However, E F will lie in the band gap, and the conductance, g, will be reduced from its value of G 0 by the approximate factor e -β bp L , where β bp is the β value of the most penetrating gap state at the Fermi level, E F , and L is the length of the molecule. Using these approxima- tions, the estimated conductance is given by g ≈ G 0 e -β bp L
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