# Lexing mathematical expressions with F#

05 Aug 2015
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Part 1: Lexing mathematical expressions with F#
Part 2: Parsing mathematical expressions with Shunting Yard, F#, and C#
Part 3: Recursive descent parsing of mathematical expressions with C#

(Code for this post is available on GitHub.)

In this post I show how to parse a list of characters making up a mathematical expression, such as "-12+34*56", into a list of tokens, such as UnaryMinOp, Integer 12, BinPlusOp, Integer 34, BinMulOp, and Integer 56. In order to transform characters into tokens, I define a couple of low-level parser functions for elements like numbers and operators and combine these into a larger parser for the entire expression.

Parsing input left to right, returning a list of tokens, suffices when no or only a simple evaluation step follows parsing. Operator precedence and associativity, however, cause evaluation of mathematical expressions to become non-trivial, non-left to right.

## Defining types of tokens

Parsing at the character level is generally referred to as tokenization or lexing of a string. The idea is that instead of dealing with single characters, related characters (by some definition of related) are grouped into higher-level data structures called tokens. These tokens may then be fed into another parser, which in turn may return an even higher-level data structure, such as a list or tree or perhaps only a single value. This multi-stage process makes the overall task more tractable.

Perhaps unsurprisingly, the exact types of tokens required depend heavily on the desired outcome. Evaluating mathematical expressions, I could model each digit as a separate token. But at no point during parsing or evaluation do I need to conditionally branch on the actual value, and thus a single token with an associated integer value suffices. Operators and parentheses, however, affect evaluation order and thus I create separate tokens for those.

In principle I could make due with a single token for operators and parentheses by associating a string (+, -, *, /, (, ), or ^), a lexeme, with the token. But for type safety, I create unique cases, ending up with the following tokens:

type Token =
| Integer of Int32
| BinPlusOp
| BinMinOp
| BinMulOp
| BinDivOp
| BinExpOp
| UnaryPlusOp
| UnaryMinOp
| LParen
| RParen


Given this discriminated union of token types, parsing "-12+34*56" would therefore yields the following list of tokens: [UnaryMinOp, Integer 12, BinPlusOp, Integer 34, BinMulOp, Integer 56].

## Parsing characters into tokens

To kick off the tokenization process, I start by defining a helper function to deconstruct an input string into a list of characters. In effect, I apply an initial, low-level form of parsing to the input. Following this and subsequent functions, I include a couple of Unquote tests to illustrate their behavior.

// val toArray : s:string -> char list
let toArray (s: string) = s.ToCharArray() |> List.ofArray

test <@ toArray "" = [] @>
test <@ toArray "1" = ['1'] @>
test <@ toArray "1+2" = ['1'; '+'; '2'] @>


A common approach to parsing by hand is to create a set of small parsers. Each such parser would attempt to consume (group together) characters of the input from left to right until either enough characters have been matched to constitute a token or the parser determines that no match is possible. Upon a successful match, both the token and the remaining characters of the input are returned. The process is then repeated, passing the remaining input into another parser, collecting the tokens returned, until there's no more input to consume.

Thus, the order in which each smaller parser gets invoked defines what makes up a valid expression.

A small parser capable of parsing only a single digit is shown below. I could've implemented a more general regular expression parser instead of multiple specific ones, but chose not to in order to keep things as straightforward as possible:

// val parseDigit : _arg1:char list -> (int * char list) option
let parseDigit = function
| hd :: tl ->
match hd with
|'0'|'1'|'2'|'3'|'4'
|'5'|'6'|'7'|'8'|'9' as digit ->
Some (Char.GetNumericValue(digit) |> int, tl)
| _ -> None
| _ -> None

test <@ parseDigit ("" |> toArray) = None @>
test <@ parseDigit ("." |> toArray) = None @>
test <@ parseDigit ("2" |> toArray) = Some(2, []) @>
test <@ parseDigit ("12" |> toArray) = Some(1, ['2']) @>
test <@ parseDigit ("123" |> toArray) = Some(1, ['2'; '3']) @>


Parsing a number of multiple digits means defining another parser which calls parseDigit one or more times until it returns None. This repeated application of a parser until it fails is such a common occurrence that it could've been implemented by a general parser accepting another parser as input. For simplicity, however, I created a specific one:

// val parseInteger : input:char list -> (Token * char list) option
let parseInteger input =
let rec parse' input digits =
match parseDigit input with
| Some (digit, rest) -> parse' rest (digit :: digits)
| None -> (digits, input)

let (digits, tl) = parse' input []
let s = digits |> List.rev |> List.fold (fun acc d -> acc + d.ToString()) ""
let (isInteger, integer) = Int32.TryParse(s)
if isInteger then Some(Integer(integer), tl) else None

test <@ parseInteger ("" |> toArray) = None @>
test <@ parseInteger ("1" |> toArray) = Some(Integer 1, []) @>
test <@ parseInteger ("01" |> toArray) = Some(Integer 1, []) @>
test <@ parseInteger ("12" |> toArray) = Some(Integer 12, []) @>
test <@ parseInteger ("12.3" |> toArray) = Some(Integer 12, ['.'; '3']) @>


Next, I focus on parsing parentheses. Coming from digits, they're relatively easy to deal with:

// val parseParen : _arg1:char list -> (Token * char list) option
let parseParen = function
| hd :: tl ->
match hd with
| '(' -> Some(LParen, tl)
| ')' -> Some(RParen, tl)
| _ -> None
| _ -> None

test <@ parseParen ("" |> toArray) = None @>
test <@ parseParen ("1" |> toArray) = None @>
test <@ parseParen ("(" |> toArray) = Some(LParen, []) @>
test <@ parseParen (")" |> toArray) = Some(RParen, []) @>
test <@ parseParen (")+1" |> toArray) = Some(RParen, ['+'; '1']) @>


Now, because I went with separate tokens for unary minus and unary plus (over storing the sign as part of the Integer), I need a parser for those. Problem is that I cannot discriminate UnaryOp from BinOp without a priori knowledge of the last matched token, if any. No such context is passed to parseUnaryOp and given its nature it shouldn't have to care about binary operators. Instead, disambiguation happens at the call site inside the larger parseExpression.

// val parseUnaryOp : input:char list -> (Token * char list) option
let parseUnaryOp input =
match input with
| hd :: tl ->
match hd with
| '-' -> Some(UnaryMinOp, tl)
| '+' -> Some(UnaryPlusOp, tl)
| _ -> None
| _ -> None

test <@ parseUnaryOp ("" |> toArray) = None @>
test <@ parseUnaryOp ("-" |> toArray) = Some(UnaryMinOp, []) @>
test <@ parseUnaryOp ("+" |> toArray) = Some(UnaryPlusOp, []) @>
test <@ parseUnaryOp ("+1" |> toArray) = Some(UnaryPlusOp, ['1']) @>


In retrospect it might have been simpler to include parsing the sign as part of the integer parser. Nonetheless, separate parsers show how sometimes additional work is required to determine which parser to call.

Parsing binary operators follow the same familiar model:

// val parseBinOp : _arg1:char list -> (Token * char list) option
let parseBinOp = function
| hd :: tl ->
match hd with
| '+' -> Some(BinPlusOp, tl)
| '-' -> Some(BinMinOp, tl)
| '*' -> Some(BinMulOp, tl)
| '/' -> Some(BinDivOp, tl)
| '^' -> Some(BinExpOp, tl)
| _ -> None
| _ -> None

test <@ parseBinOp ("" |> toArray) = None @>
test <@ parseBinOp ("1" |> toArray) = None @>
test <@ parseBinOp ("+" |> toArray) = Some(BinPlusOp, []) @>
test <@ parseBinOp ("^12" |> toArray) = Some(BinExpOp, ['1'; '2']) @>


Finally, combining calls to the previously defined parsers, it's time for the top-level expression parser:

// val parseExpression : input:string -> Token list
let parseExpression input =
// some types of tokens requires input to be consumed more
// greedy than others. Try the most greedy one first to
// resolve any ambiguity that might arise.

// val parse' : Token list -> char list -> Token list
let rec parse' tokens rest =
match parseInteger rest with
| Some (integer, tl) -> parse' (integer :: tokens) tl
| _ ->
match parseUnaryOp rest with
| Some (unaryOp, tl) ->
let isPrevTokenBinaryOp token =
[BinPlusOp; BinMinOp; BinMinOp; BinDivOp; BinExpOp; LParen]
|> List.exists (fun o -> o = token)
if tokens = List.empty ||
isPrevTokenBinaryOp (List.head tokens) then
parse' (unaryOp :: tokens) tl
else
// no? Must be BinPlusOp or BinMinOp instead then
match unaryOp with
| UnaryMinOp -> parse' (BinMinOp :: tokens) tl
| UnaryPlusOp -> parse' (BinPlusOp :: tokens) tl
| _ -> failwith "Should never happen"
| _ ->
match parseBinOp rest with
| Some (binOp, tl) -> parse' (binOp :: tokens) tl
| _ ->
match parseParen rest with
| Some (paren, tl) -> parse' (paren :: tokens) tl
| _ -> tokens

parse' [] (input |> toArray) |> List.rev

test <@ parseExpression "" = [] @>
test <@ parseExpression "1" = [Integer 1] @>
test <@ parseExpression "1+2" = [Integer 1; BinPlusOp; Integer 2] @>
test <@ parseExpression "1-2" = [Integer 1; BinMinOp; Integer 2] @>

// is this legal math syntax? PowerShell can only parse "1-(-2)"
// whereas LibreOffice parses and evaluates "1--2"
test <@ parseExpression "1--2" =
[Integer 1; BinMinOp; UnaryMinOp; Integer 2] @>
test <@ parseExpression "-1" = [UnaryMinOp; Integer 1] @>
test <@ parseExpression "+1" = [UnaryPlusOp; Integer 1] @>
test <@ parseExpression "1+-2" =
[Integer 1; BinPlusOp; UnaryMinOp; Integer 2] @>
test <@ parseExpression "-(1+2*3/4^5)" =
[UnaryMinOp; LParen; Integer 1; BinPlusOp; Integer 2; BinMulOp;
Integer 3; BinDivOp; Integer 4; BinExpOp; Integer 5; RParen] @>


Focusing on the inner parse' function, say I wish to parse "1+2". parse' is a state machine which assumes an expression either starts with an integer, a unary operator, or any parenthesis (even though only ')' is legal). I could've extended the tokenizer with a start left parenthesis state, but it generally isn't the tokenizer's job to make the list of tokens make overall sense.

Having consumed the first integer, parse' calls itself with the tokens matched so far and the remaining "+2" input. Question is whether "+" means unary or binary addition. I start out assuming it means unary and then attempt to prove this hypothesis. It can only hold true if the parser hasn't matches any tokens thus far (as in "-1") or the last matchd token was a binary operator (as in "1+-2").

## Conclusion

In practice, I would rarely write my own parser but fall back on a parser combinator library such as FParsec, Sprache, or Irony. These libraries come with more advanced versions of the parsers defined above and support for common parsing patterns such as applying a parser multiple times until it fails. Never having written a parser by hand, though, it's hard to fully understand and appreciate what a parser combinator library brings to the table and how to use it effectively.