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The forward tracking quantifier modifier

Published 2012-01-16 C# , Howto , Java , Programming , Python , Regex , Regexp , Regular Expressions , Ruby , Scala , Technology Leave a Comment
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Because of backtracking, we might sometimes match more than we hoped for. The task below is to catch all the div tags in an HTML document and put them in a vector. Our naïve solution gives the wrong answer:

  • '<div>a</div><span>c</span><div>b</div>'.scan /<div>.*<\/div>/ #=> ["<div>a</div><span>c</span><div>b</div>"]

Quantifiers are greedy in regex. They devour as much as they can. The dot in the idiom .* matches anything except newline (remember Barbapapa?). And the asterisk means that this anything is repeated as many times as possible. Since the string ends with a “</div>” the entire substring “a</div><span>c</span><div>b” will be consumed by .*.

There are many stories about greedy people who claim more than they need. As Louis Blanc wrote already 1840 in The Organization of Work: "From each according to his abilities, to each according to his needs." We don't think that the asterisk and the dot in the above example needs to consume more than the substring "<div>a</div>". Alexander Pushkin describes in The Tale of the Fisherman and the Fish how a magic fish promises to fulfill any of the fisherman's whishes. The fisherman's wife starts asking the fish for bigger and better things -- and she gets them -- until she eventually wants to become Ruler of the Sea. Then the magic fish takes back everything he gave the fisherman's wife.

As a complement to backtracking, regex also provides forward tracking. The quantifier tries to take as little as it can and then holds to see if the whole expression can be matched. If the entire expression can't be matched, then the quantifier will capture one more letter and holds to see if it's now possible to match everything. The method is repeated until either we can match everything or we can confirm that there's no possible way to create a match. Does it matter if we use forward tracking instead of backtracking? Well, the strings that can be matched with backtracking can also be matched with forward tracking and vice versa. But, when there are multiple possible matches, forward tracking will sometimes choose a different match than backtracking does.

There is no special symbol for forward tracking quantifiers. Instead, there's modifier symbol -- written as a question mark -- that can be used right after any quantifier. While * means repeat as many times as possible, *? means repeat as few times as possible. Similarly, we can modify all the other quantifiers:

  • At least one, as few as possible: +?
  • Zero or one, preferably zero: ??
  • Between 3 and 5, as few as possible: {3.5}?

Note that the question mark that modifies quantifiers isn't the same question mark that's used as the conditional quantifier. They can even be used in conjunction, as ?? shows above.

Here are some examples. The first one is the solution to the div tag problem:

  • '<div>a</div><span>c</span><div>b</div>'.scan /<div>.*?<\/div>/ #=> ["<div>a</div>", "<div>b</div>"]
  • 'aa'[/a?/] #=> "a"
  • 'aa'[/a??/] #=> ""
  • 'aaaaa'[/a{2,4}/] #=> "aaaa"
  • 'aaaaa'[/a{2,4}?/] #=> "aa"
  • 'aaaaa'[/a{2,}/] #=> "aaaaa"
  • 'aaaaa'[/a{2,}?/] #=> "aa"
  • 'aaaaa'[/a{,4}/] #=> "aaaa"
  • 'aaaaa'[/a{,4}?/] #=> ""

Pomodoro Technique Illustrated -- New book from The Pragmatic Programmers, LLC

Regex memorizing — here’s the pushdown automaton

Published 2011-06-20 finite automata , Howto , Programming , Pushdown Automata , Regex , Regexp , Regular Expressions Leave a Comment
Pushdown automaton that matches all strings with the same number of white and blue dots.

Pushdown automaton that matches all strings with
the same number of white and blue dots.

Formal regular expressions can be described by a finite automaton, but modern regex engines support un-regular operators. The problem with finite automata is that they don’t have any memory. Once they are in a state, they have no idea, how they got there. Wise guys invented the stack. When we add a stack to a finite automaton it becomes very powerful, but also quite complex. And of course, it’s not a finite automaton anymore – it’s a pushdown automaton. Why do I then describe the regex engines as finite automata when pushdown automata is a superset of finite automata?

The finite automaton reads a tape serially. Every new symbol read from the tape, initiates a transition in the automaton from the current state to a new state (the new state may be the same state as the current state). If we are in an accept state when we have read all the tape, then we have a match. In pushdown automata, we add a stack. For every iteration, the pushdown automaton may read a symbol from the tape, pop a symbol from the stack, or both. Depending on the current state, the symbol read from the tape and/or the symbol popped from the stack, the pushdown automaton can now initiate a state transition, push a symbol to the stack, or both. If we end up in a accept state when all the tape is read and the stack is at that time empty then we have a match.

The figure above shows a pushdown automaton that matches any input with the same number of blue and white dots – something that is impossible to describe with a finite automaton. This automaton has two states: a start state and an accept state. The four icons in the middle represents that a dot is popped or pushed from the stack. Suppose that the input is blue-white-white-blue:

  1. Read blue dot and make a transition from the start state, through the third stack icon from the top – i.e. a blue dot is pushed to the stack – and back to the start state.
  2. Read white dot and make a transition from the start state, through the first stack icon – i.e. a blue dot is popped from the stack – and back to the start state. Now the stack is empty.
  3. Read white dot and make a transition from the start state, through the second stack icon from the top – i.e. a white dot is pushed to the stack – and back to the start state.
  4. Read blue dot and make a transition from the start state, through the fourth stack icon – i.e. a white dot is popped from the stack – and to the accept state. Now the stack is empty again and all the input is read. It’s a match.

You can probably feel that the pushdown automaton gives us a tool for all kind of recursive implementations. Now that you understand how, I’ll go back to finite automata in all explanations where possible. It’s important for you to understand e.g. what backtracking and greediness does to performance and what input that will be matched. You don’t need a complicated pushdown automaton to learn that. We’ll stick to finite automaton in this book and at the same time remember that many operators in modern regex engines rely on a stack.

By the way: Did you notice the corner case bug in the pushdown automaton above? Doesn’t an empty input string have the same number of blue and white dots?

Complicated logic calls for regex

Published 2011-05-30 Howto , Java , Programming , Regex , Regexp , Regular Expressions 1 Comment
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When is regex the proper tool? One use case is when you try to verify a text, and the verification rules have complicated logic. I found the following Java method in real production code. It verifies that a time zone designator adheres to the ISO 8601 standard:

public boolean isTimeDesignator(String designator) {
  if (designator.equals("Z")) {
    return true;
  } else if (designator.startsWith(" ") && designator.trim().equals("UTC")) {
    return true;
  } else if (designator.startsWith("+") || designator.startsWith("-")) {
    try {
      int hour = Integer.parseInt(designator.substring(1,3));
      int minute;
      if (3 == designator.length()) {
        minute = 0;
      } else if (5 == designator.length()) {
        minute = Integer.parseInt(designator.substring(3,5));
      } else if (6 == designator.length() && ':' == designator.charAt(3)) {
        minute = Integer.parseInt(designator.substring(4,6));
      } else {
        return false;
      }
      return 0 <= hour && 24 >= hour && 0 <= minute && 59 >= minute;
    } catch (NumberFormatException e) {
      return false;
    }
  }
    
  return false;
}

Only counting the if and else, there are 7 ways the control flow can traverse this code. Adding the NumberFormatException there’s another 3. These ten routes need to be covered by tests.

In ISO 8601, global time is always represented as an offset to Coordinated Universal Time, abbreviated UTC. UTC is informally equivalent to the Greenwich Mean Time, also known as GMT. The designator is an offset in hours and minutes to UTC. Hours are mandatory, but minutes and a colon in-between minutes and hours are optional. Offset zero can be represented as a space and then ‘UCT’ or as ‘Z’. Complicated?

In regex — still in Java — it’s rather crisp to verify:

public boolean isTimeDesignator(String designator) {
  return designator.matches("Z| +UTC|[+-]([01]\\d|2[0-4])(:?[0-5]\\d)?");
}

The latter solution may seem cryptic, for people without experience in regex. But if you think about it, the first solution is cryptic for anyone not proficient in imperative coding. And if you ask me, the first solution is hard to follow for anybody. To understand a regex, you need to understand regex. That’s a tautology. With regex in your repertoire, some problem classes can be solved much more smoothly and the solutions are easier to maintain.

Regex syntax and semantics varies

Published 2011-05-4 Howto , Java , Programming , Regex , Regexp , Regular Expressions , Ruby , Technology 1 Comment

Regex engines do differ in syntax and semantics. This is one reason why you can’t just find an expression with Google and use it in your code – without fully understand it.

Try for example the following in JIRB, the interactive JRuby tool:

irb(main):001:0> require 'java'
=> true
irb(main):002:0> java.util.regex.Pattern.compile("a$").matcher("a\nb").find
=> false
irb(main):003:0> "a\nb"[/a$/]
=> "a"

What happened?

The regex /a$/ matches the letter a just before something checked by a dollar sign assertion. The assert criteria is, by default, not the same in Ruby and Java. In Java, by default, the dollar sign matches at the end of the whole text and before any final line breaks. In Ruby, the dollar sign match at the end of every single line.

Here’s what the three lines of code above means:

  1. First, we need to include the Java libraries by writing require 'java'. This might not be necessary, depending on your setup.
  2. We compile a Java regex and test if part of the string of ‘a’ and ‘b’ with newline in-between can be matched. It can’t.
  3. We compile the same regex in Ruby and test if part of the string of ‘a’ and ‘b’ with newline in-between can be matched. It can.

This is just one of many examples. If you are going to use regexes in your program, you need to understand them. It’s as simple as that.

Pomodoro Technique Illustrated -- New book from The Pragmatic Programmers, LLC

Regex slides

Published 2011-03-21 Howto , Programming , Regex , Regexp , Regular Expressions , Technology 2 Comments
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Slides from my regex session at Turku Agile Day are now available below and at Slideshare:

If you can’t see the whole picture due to Slideshare’s embed limits, click ‘zoom’ -> ‘zoom to page’ in the Slideshare tool bar above.

Three Regex presentations in Stockholm upcoming month

Published 2011-02-10 Pomodoro , Pomodoro Technique , Regex , Regular Expressions 1 Comment
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I have three Regex presentations in Stockholm during the upcoming month. Information below is in Swedish.

Tre tillfällen att förstå Regex bättre

Är du rädd för Regex? :-) Eller vill du kanske få en enkel och praktisk beskrivning av den underliggande matematiken? Vill du förstå och kunna använda Regex? Då ska du besöka någon av mina tre sessioner den närmaste månaden:

Fokus är på att förstå och bli bra på – fokus är inte på Regex-syntax.

Obs! Ingen av sessioner förutsätter förkunskaper om Regex.
Pomodorotekniken på svenska

Pomodoro-boken på svenska

Nu finns den svenska boken Pomodorotekniken. Det är alltså en översättning av den amerikanska boken (som även finns på tyska, japanska, kinesiska och koreanska). Du kan t ex köpa den på Adlibris: http://www.adlibris.com/se/product.aspx?isbn=9144070373

Regex at Windows’ command line

Published 2010-12-2 Howto , Regex , Regular Expressions 2 Comments
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Regular Expressions are patterns that match strings in text files or text streams. All strings that can possibly be matched by a particular expression are collectively a regular language. Every regular language has at least one corresponding finite automaton. The finite automaton is a very simple state machine.

In order to match regular languages we need only three operations – Les trois Mousqetaires de Regex:

  1. Kleene star, i.e * – repeat the preceding pattern zero or more times
  2. Concatenation – match two consecutive patterns, e.g. 73 match 7 followed by 3
  3. Alternation – match exactly one out of a list of patterns, i.e. logical OR

The precedence is as the list above. Sometimes, that’s not enough and thus we also need D’Artagnan, the fourth musketeer. I’m talking about parenthesis for grouping – just like in any other programming language.

The unix command-line tool egrep meets the regular expressions litmus test. It supports concatenation, Kleene star, alternation, and grouping.

What about Windows then?

There’s a built-in command-line tool in Windows called findstr. And findstr supports Kleene star and concatenation. That’s 2/4 of the mandatory operations we need to match regular languages. Unlike egrep, it lacks alternation and forced precedence with parenthesis – and thus you can’t use findstr for regular expressions.

What to do then if you want to write regular expressions at the command line in Windows? There are at least two alternatives:

  • UnxUtils are native Win32 ports of common GNU utilities. Native means that the executables only depend on the Microsoft C-runtime (msvcrt.dll) and not an emulation layer. One of the executables is egrep.
  • Cygwin is a Linux API emulation layer and a vast number of tools – among them egrep.

D’Artagnan and his three musketeer friends Athos, Porthos, and Aramis lived by the motto tous pour un, un pour tous. If you don’t have grouping and the three mandatory operations, then you can’t write regular expressions.

Regex is not Regular

Published 2010-09-8 Howto , Programming , Regex , Regular Expressions , Ruby Leave a Comment
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Regular expressions span exactly the same languages as the mathematical construction called Finite atomata. With the Pumping lemma, we can prove that it’s impossible to create regular expressions for many problems, e.g.:

  1. All palindromes
  2. All text strings that doesn’t consist of a prime number of identical characters

Since Regular expressions have become popular as extensions to imperative programming languages, new features have been added. Languages from Context-free grammars and Pushdown automata (essentially a finite automata with an attached stack), can be matched with these features. Regular expressions are Type-3 and Context-free grammars are type-2 in the Chomsky hierarchy.

To not be confused we call these built-in languages that are more than Type-3, Regex. Regexes are more powerful than Regular expressions.

Non regular prime number finder

With the (non regular) feature back-reference, it’s possible to create a regex that only match strings with a prime number of identical characters:

irb(main):001:0> r = /^.?$|^((.)\2+?)\1+$/
=> /^.?$|^((.)\2+?)\1+$/
irb(main):002:0> r.match "22"
=> nil
irb(main):003:0> r.match "333"
=> nil
irb(main):004:0> r.match "4444"
=> #<MatchData "4444" 1:"44" 2:"4">
irb(main):005:0> r.match "55555"
=> nil
irb(main):006:0> r.match "tttttt"
=> #<MatchData "tttttt" 1:"tt" 2:"t">

Non regular palindrome finder

If we also add the (non regular) feature recursion, we can match all palindromes:

irb(main):001:0> p =
/\A(?<palindrome>|.|(?:(?<prefix>.)\g<palindrome>\k<prefix+0>))\z/
=> /\A(?<palindrome>|.|(?:(?<prefix>.)\g<palindrome>\k<prefix+0>))\z/
irb(main):002:0> p.match "otto"
=> #<MatchData "otto" palindrome:"otto" prefix:"t">
irb(main):003:0> p.match "dallas sallad"
=> #<MatchData "dallas sallad" palindrome:"dallas sallad" prefix:"s">
irb(main):004:0> p.match "rais air"
=> nil
irb(main):005:0> p.match "1010"
=> nil

Recursive Regexes

Published 2010-06-22 Howto , Programming , Regex , Regular Expressions , Ruby , Technology Leave a Comment
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Oniguruma supports recursive regular expressions. They can e.g. be used for matching a generative grammar like the following:

  1. expression -> expression + term
  2. expression -> expression - term
  3. expression -> term
  4. term -> ( expression )
  5. term -> digit

I wrote the following regex that corresponds to the grammar above:

  • /^(?<expression>(?<term>\d|\(\g<expression>\))([-+]\g<expression>)?)$/

Note that \g<expression> is invoked recursively – like rule number 4 in the grammar. I chose to only allow single digit numbers to make the expression crisper as example. This could of course easily be changed to general integers, floats or imaginary numbers.

Here goes some testing:

irb(main):001:0> r = /^(?<expression>(?<term>\d|\(\g<expression>\))([-+]\g<expression>)?)$/
=> /^(?<expression>(?<term>\d|\(\g<expression>\))([-+]\g<expression>)?)$/
irb(main):002:0> "2+4"[r]
=> "2+4"
irb(main):003:0> "2+45"[r]
=> nil
irb(main):004:0> "2-3(4+3)"[r]
=> nil
irb(main):005:0> "2-3(+3)"[r]
=> nil
irb(main):006:0> "2-31(+3)"[r]
=> nil
irb(main):007:0> "2-3+(+3)"[r]
=> nil
irb(main):008:0> "2-3+(4+3)"[r]
=> "2-3+(4+3)"

Expressions that return nil are obviously not correct.

Pomodoro Technique Illustrated -- New book from The Pragmatic Programmers, LLC

Regex extension — here’s the Integer Class

Published 2010-05-29 Howto , Programming , Regex , Regular Expressions , Ruby 8 Comments

The IP address problem is well known in the regex community. Here I present an extended regex syntax that would make it possible to match IP addresses in a less chatty way.

Ip-address problem

Let’s say I want to validate an IP address. It is four integers with interleaved dots such as 123.125.126.107. The naïve regex would be:

  • \d*\.\d*\.\d*\.\d*

But it is so imprecise that it matches both 9999.9999.9999.9999 and …. Since the four integers in a IP address must be in the range 0-255, I can be certain that they consist of one, two, or three digits. With the Limiting Repetition operator, I can formulate this requirement:

  • \d{1,3}\.\d{1,3}\.\d{1,3}\.\d{1,3}

Now 0.1.2.3 and 101.0.202.1 matches, while 9999.9999.9999.9999 doesn’t match because there are too many digits in the integers. Unfortunately 999.888.777.666 matches as well. That’s not an acceptable IP address. I told you that the integers can be up to 255. But, don’t give up. In Friedl’s seminal book “Mastering Regular Expressions”, he describes two ways to shrink the match set. Both are, however, very chatty. The first way is to list all authorized integers in a super chubby alternation:

  • (0|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|17|18|19|20
    |21|22|23|24|25|26|27|28|29|30|31|32|33|34|35
    |36|37|38|39|40|41|42|43|44|45|46|47|48|49|50
    |51|52|53|54|55|56|57|58|59|60|61|62|63|64|65
    |66|67|68|69|70|71|72|73|74|75|76|77|78|79|80
    |81|82|83|84|85|86|87|88|89|90|91|92|93|94|95
    |96|97|98|99|100|101|102|103|104|105|106|107
    |108|109|110|111|112|113|114|115|116|117|118
    |119|120|121|122|123|124|125|126|127|128|129
    |130|131|132|133|134|135|136|137|138|139|140
    |141|142|143|144|145|146|147|148|149|150|151
    |152|153|154|155|156|157|158|159|160|161|162
    |163|164|165|166|167|168|169|170|171|172|173
    |174|175|176|177|178|179|180|181|182|183|184
    |185|186|187|188|189|190|191|192|193|194|195
    |196|197|198|199|200|201|202|203|204|205|206
    |207|208|209|210|211|212|213|214|215|216|217
    |218|219|220|221|222|223|224|225|226|227|228
    |229|230|231|232|233|234|235|236|237|238|239
    |240|241|242|243|244|245|246|247|248|249|250
    |251|252|253|254|255)

Note that the above is just one integer in the IP address. I have to write the super chubby alternation four times with interleaved dots to get the correct regex.

The second way to specify an integer in the range 0-255 is to decompose the problem into sub problems based on the initial character. If the first digit is 0 or 1, then all integers consisting of 1-3 digits are acceptable – I also allow leading zeros as in e.g. 054. When the initial digit is 2 and the second number is in the range 0-4, then I accept digits in the ranges 20-24 and 200-249. Finally, if the integer starts with 25, then I only tolerate it if it’s followed by a digit in the range 0-5. Like this:

  • [01]?\d\d?|2[0-4]\d|25[0-5]

(Does this regex really match 25? The third part of this alternation only matches integers in the range 250-255. Yes, it does. The first part matches any integer consisting of one or two digits.)

Again I must rewrite my regex four times with interleaved dots to match an entire IP address.

Integer Class – a new operator suggested

So far I’ve talked about how regex works right now. Let’s imagine now that I can add a new operator. I call the new operator Integer Class.

Both regexes above solve the problem, but they demand so many characters that it reminds me of chatter from a group of monkeys (i.e. not easy to understand). My proposal is to extend the regex syntax with a new operator: Integer Class. It matches integers of any length, if they are in a specific range. For IP numbers — which should be in the range 0-255 — it would look like this:

  • [0..255]

Square brackets surrounds two integers – a lower and an upper limit. There are double dots in-between the integers. Integer Class would work almost as a syntactic sugar for the super chubby alternation above — but not quite. Here are some details:

  • Backwards compability: most regex interpreters permits Character Classes with repeated characters. A regex [0..255] is syntactically correct already today. But now it means something entirely different than what I want. The regex interpreter doesn’t care about the double fives and the double dots. Right now, [0..255] is a redundant way of writing [0.25], i.e. it matches exactly one character and it must be either 0, dot, 2 or 5. With my syntax and semantics, the regex interpreter would notice the double dots and say “Hey, this isn’t a Character Class, because it’s an Integer Class.” How cool is that?
  • Leading Zeroes: An Integer Class matches leading zeros in the candidate, but only if you specify it. How? Well, by writing a zero in front of the lower limit. For example: [00..255] means that even sequences like 012, 0004 and 00255 are appropriate integers.
  • Negative integers: Of course, negative integers are permitted. For example, [-1..1] matches -1, 0 or 1. Sidenote: a leading dash in a Character Class means that dashes are allowed. It may sound trivial, but then you should know that in a Character Class [A-Z], the dash means from/to – the range A to Z. A dash in a Character Class has a different meaning depending of if it’s in the beginning or the middle of an expression. Anyway, if there is a double point, it is an Integer Class and then dash means minus.
  • Greedy: Like many other regex operators – such as repetition – the Integer Class is greedy of type longest-leftmost-match, but charity obedient. This means that [0..255] rather match 255 than 2 in candidate 255. But the regex engine is prepared to release the fives if it means the whole expression match. This differs from the super chubby alternation above. At least a Traditional NFA engine often matches the first one it finds, i.e. 2 rather than 255.
  • Meta Characters inside Integer class: There’s no need for rules to escape characters since only digits, double dots and dashes are permitted in an Integer class. Backslash is not allowed to reside in an Integer Class.
  • Negated Integer Class: A caret ^ after the right square bracket will negate the Integer Class. Any integer except those stated in the Integer Class will be matched. E.g. [^5..7] match any integer except 5, 6 and 7. Note that a negated Integer Class still must match an integer. Match exactly one integer, but not 5, 6 or 7.
  • Not accepting nothing: A Character Class doesn’t match nothing – the empty string. The same is true for an Integer Class. There must be an integer matched in the range. The exception is, of course, when an Integer Class is followed by * or ? — for example: [0..255]*

With the suggested Integer Class, an IP address can be matched with this regex:

  • ([0..255]\.){3}[0..255]

Relevant questions:

  • Is this syntax possible or would it be ambiguous?2
  • There are many possible operators to add to the regex syntax – is Integer Class the most needed?
  • Should the Integer Class be even more capable, e.g. be able to match floats?

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