Archive for the 'Python' Category

Regular Expressions – a brief history

Regular Expressions is a programming language with which we can specify a set of strings. With the help of two operators and one function, we can be more concise than if we would have to list all the strings that are included in the set. Where does Regular Expressions come from? Why is it called Regular and how does it differ from Regex?

The story begins with a neuroscientist and a logician who together tried to understand how the human brain could produce complex patterns using simple cells that are bound together. In 1943, Warren McCulloch and Walter Pitts published “A logical calculus of the ideas immanent in nervous activity”, in the Bulletin of Mathematical Biophysics 5:115-133. Although it was not the objective, it turned out this paper had a great influence on computer science in our time. In 1956, mathematician Stephen Kleene developed this model one step further. In the paper “Representation of events in nerve nets and finite automata” he presents a simple algebra. Somewhere at this point the terms Regular Sets and Regular Expressions were born. As mentioned above, Kleene’s algebra had only two operations and one function.

In 1968, the Unix pioneer Ken Thompson published the article “Regular Expression Search Algorithm” in Communications of the ACM (CACM), Volume 11. With code and prose he described a Regular Expression compiler that creates IBM 7094 object code. Thompson’s efforts did not end there. He also implemented Kleene’s notation in the editor QED. The aim was that the user could do advanced pattern matching in text files. The same feature appeared later on in the editor ed.

To search for a Regular Expression in ed you wrote g/<regular expression>/p The letter g meant global search and p meant print the result. The command — g/re/p — resulted in the standalone program grep, released in the fourth edition of Unix 1973. However, grep didn’t have a complete implementation of regular expressions, and it was not until 1979, in the seventh edition of Unix that we were blessed with Alfred Aho’s egrepextended grep. Now the circle was closed. The program egrep translated any regular expressions to a corresponding DFA.

Larry Wall’s Perl programming language from the late 80’s helped Regular Expressions to become mainstream. Regular Expressions are seamlessly integrated in Perl, even with its own literals. New features were also added to Regular Expressions. The language was extended with abstractions and syntactic sugar, and also brand new features that may not even be possible to implement in Finite Automata. This raises the question if modern Regular Expressions can be called Regular Expressions. I don’t think so. The term Regex denotes not only Kleene’s algebra but also the additions made by Perl, Java, Ruby, and other implementations.

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The Scent of Regex Requisite

Did you know that the famous quote “Some people, when confronted with a problem, think: I know, I’ll use regular expressions. Now they have two problems.” dates all the way back to 1997? However, most programmers agree that regex has its time and place. But, how can we know when to use regex? It’s really simple. We must use our nose and feel the scent of regex requisite. Below is a list of five scents that puts the R-word in our working memory.

Text to type

A text sequence is also a kind of data type. You may have read it from a file or perhaps a user entered it into your system. But you don’t confine yourself to text. You want to transform it into a bunch of structured data records. You read record after record from the text hank. Each record consists of a series of comma delimited fields and each record is terminated with a semicolon. Regex loves to parse text hanks.

Non recursive

A recursively-defined data type may be instantiated with values that contains values of the very same type. Think of fir branches. Each branch consists of one stem, zero or more sub branches, and many needles. The sub branches are fir branches as well. A branch may have a sub branch, which may have a sub branch, which may have a sub branch etc. In theory there is no limit to how many levels we can have. As soon as you want to translate text into recursive data — then regex is usually not the best tool. To parse an entire HTML document with nested div tags is an example of recursive data.

Not lucid

If the input is small, regex often doesn’t add anything. But, when you do search-and-replace in 2000 files and what you want to replace has a variable appearance — then a neat little regex is the generalized solution. You can capture different versions and replace them with something that actually depends on the input data. It is quality — no mistakes — and quantity — no misses. In a small input, you can modify by hand. You can easily see what should be changed and to what.


Suddenly it happens: you have input from a user, from the network, from another system or from a file. You can not predict what will come, more than that it’s text. It may be a lot and it may be a tiny little piece. Yes, it can even be an empty string. This very uncertainty makes the generalized description of the input data characteristics, useful. You describe a pattern, not a specific entity. Regex is a superhero when it comes to describing generalized patterns.

Complex logic

I’ve described before how 20+ lines of Java code could be transformed into one small regex. This is not a general law. Regex is a limited programming language suited to solve a very specific class of problems. However, in this case the imperative Java code had a lot of nested as well as consecutive conditional statements; if-else-if-if-else — i.e. complex logic. Regex is a declarative language. You describe what you want, and not how to get there. Thus, you don’t have to state all these scrubby paths.

Some of these five scents partly overlap, but each of them are well worth to remember. Facing a programming problem, if you can’t feel any of them, then you can be pretty sure that there are better tools than regex.

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From Regular Expression to Finite Automaton

For each regular expression — and I mean the three operators and the six recursive rules style — there is a finite automaton that accepts exactly the same strings. Since this is not a university book in mathematics, I’ll show you an inductive reasoning about this and not a formal proof.

The hypothesis is thus that for an arbitrary regular expression p, we can create a finite automaton that has exactly one start state, no paths into the start state, no paths out from the acceptance state and that accepts exactly the same strings that are matched by p.

  • The empty string ε is a regular expression corresponding to a finite automaton with a start state, a path that accepts the empty string ε and leads from the start state to an acceptance state. We’ll call this an //ε-path//.
  • The empty set Ø is the set equivalent to a regular expression that can’t match any single string — not even the empty string ε. It is the same as a two-state automaton, with no single path. One state is start and the other one acceptance. But, they are not linked.
  • A regular expression that only matches the symbol b corresponds to a finite automaton with two states: start and acceptance. There’s a path from start t acceptance, and it only accepts the symbol b.

All three finite automata above have two states. One is start and the other one is acceptance. The difference is that the first one has an ε-path from start to acceptance, the second one has no path, and the third one has a b-path. Now we’ll continue. Imagine that we have two regular expressions p and q corresponding to finite automata s and t respectively.

  • Concatenation of two regular expressions p and q means that we first match a string with p, directly followed by a string that’s matched by q. To create this finite automaton we first add ε-paths from every acceptance state in s to the start state of t. Then we deprive all acceptance states in s their acceptance status and we’ll also withdraw the start status of the start state in t.
  • Alternation of two regular expressions p and q, i.e. p|q is like a finite automaton with a new start state that has ε-paths to all start staes of s and t. The new finite automaton also has a new acceptance state that is reached with ε-paths from all acceptance states of s and t. The start and acceptance states of s and t are thus not start and acceptance state in our new automaton.
  • Kleene star is the concatenation closure. Assume that p = q*. Then s is the finite automaton we get if we take t and add two states and four paths as follows: One new state is the start state and the other one is an acceptance state. All acceptance states of ´s´ loses that status in s, but instead gets an ε-path to the new acceptance state. We add two ε-paths from the new initial state — one to the old start state and one to the new acceptance state. In addition to that, we insert one ε-path from each of the old acceptance states to the old start state.

Look at the pictures above. Then take a deep breath and feel if you can translate an arbitrary regular expression to a finite automaton. Finally assess the last picture where the regular expression (w|bb)* is depicted as a graph using the method described above. Does it feel reasonable?

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Simple Regular Expression Examples

Now, we have three operators and a small framework. After all this theory, you might wonder if it’s possible for us to solve any problems. Yes, of course we can. Here are some examples:

All binary strings with no more than one zero:

'01101'.match /1*(0|)1*/ #=> #<MatchData "011">
'0111'.match /1*(0|)1*/ #=> #<MatchData "0111">
'1101'.match /1*(0|)1*/ #=> #<MatchData "1101">
'11010'.match /1*(0|)1*/ #=> #<MatchData "1101">

All binary strings with at least one pair of consecutive zeroes:

'101001'.match /(1|0)*00(1|0)*/ #=> #<MatchData "101001">
'10101'.match /(1|0)*00(1|0)*/ #=> nil
'1010100'.match /(1|0)*00(1|0)*/ #=> #<MatchData "1010100">

All binary strings that have no pair of consecutive zeros:

'1010100'.match /1*(011*)*(0|)/ #=> #<MatchData "101010">
'101001'.match /1*(011*)*(0|)/ #=> #<MatchData "1010">
'0010101'.match /1*(011*)*(0|)/ #=> #<MatchData "0">
'0110101'.match /1*(011*)*(0|)/ #=> #<MatchData "0110101">

All binary strings ending in 01:

'110101'.match /(0|1)*01/ #=> #<MatchData "110101">
'11010'.match /(0|1)*01/ #=> #<MatchData "1101">
'1'.match /(0|1)*01/ #=> nil
'01'.match /(0|1)*01/ #=> #<MatchData "01">

All binary strings not ending in 01:

'010'.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "010">
'011'.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "011">
''.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "">
'1'.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "1">
'01'.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "0">
'101'.match /(0|1)*(0|11)|1|0|/ #=> #<MatchData "10">

All binary strings that have every pair of consecutive zeroes before every pair of consecutive ones:

'0110101'.match /0*(100*)*1*(011*)*(0|)/ #=> #<MatchData "0110101">
'00101100'.match /0*(100*)*1*(011*)*(0|)/ #=> #<MatchData "0010110">
'11001011'.match /0*(100*)*1*(011*)*(0|)/ #=> #<MatchData "110">
'1100'.match /0*(100*)*1*(011*)*(0|)/ #=> #<MatchData "110">
'0011'.match /0*(100*)*1*(011*)*(0|)/ #=> #<MatchData "0011">

See if you can find even better regular expressions that solve these problems. Remember that there’re an infinite number of synonyms to each regular expression.

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Regular Expression Precedence

You might be tempted to read the following regular expression as third or fifth row:

'fifth row'.match /third|fifth row/ #=> #<MatchData "fifth row">
'third row'.match /third|fifth row/ #=> #<MatchData "third">

But unfortunately, as you can see, it’s more like either third (only) or else fifth row. This is due to something called order of operations or operator precedence. The invisible operator for concatenation has higher precedence than the alternation operator |.

To oil these wheels, we now add parentheses to our three operators. In a regular expression, the sub expression enclosed in parentheses get the highest priority:

'fifth row'.match /(third|fifth) row/ #=> #<MatchData "fifth row">
'third row'.match /(third|fifth) row/ #=> #<MatchData "third row">

Note that the parentheses are meta-characters, not literals. They won’t match anything in the subject string. And of course it’s possible to nest parentheses:

'third row'.match /(third|(four|fif)th) row/ #=> #<MatchData "third row">
'fourth row'.match /(third|(four|fif)th) row/ #=> #<MatchData "fourth row">
'fifth row'.match /(third|(four|fif)th) row/ #=> #<MatchData "fifth row">

There are three things we need to remember, to know in what order and with what operands the regular expression engine will execute the operators:

  • Operator precedence is an ordered list that tells you if one operator should be executed before another operator in a regular expression. Several operators can have the same priority. In mathematics, the terms inside the parentheses have the highest priority. Multiplication and division have a lower priority. Addition and subtraction have the lowest. This is why 6+6/(2+1) = 8.
  • Operator position indicates where the operands are located in relation to the operator. The position can be prefix, infix, or postfix. If the operator is prefix, then the operand resides to the right of the operator, as the unary minus sign e.g. -3. An infix operator has an operand on each side, as in addition 1+2. A postfix operator stands to the right of its operand, as the exclamation point that represents the faculty operator in 5!.
  • Operator associativity tells us how to group two operators on the same precedence level. An infix operator can be right-associative, left-associative or non-associative. In mathematics, the infix operations addition and subtraction have the same precedence. Since both are left-associative the following equation holds: 1-2+3 = (1-2)+3 = 2. Prefix or postfix operators are either associative or non-associative. If they are associative, we start with the operator that is closest to the operand. An operator that is non-associative can’t compete with operators of same precedence.

Here goes the table for the operators we have studied so far. Later on, there’s a complete table of all regex operators.

Operator Symbol Precedence Position Associativity
Kleene star * 1 Postfix Associative
Concatenation N/A 2 Infix Left-associative
Alternation | 3 Infix Left-associative

If you think this is hard to remember, then try to memorize the mnemonic SCA. It stands for Star-Concat-Alter, i.e. the order of precedence in regular expressions.

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The Kleene Star operator

All finite languages ​​can be described by regular expressions. You can simply list the strings as an alternation string1|string2|string3 etc. But there are also some languages with an infinite number of strings that can be described by regular expressions. To achieve that, we use an operator we call Kleene star after the American mathematician Stephen Cole Kleene. If p is a regular expression, then p* is the smallest superset of p that contains ε (the empty string) and is closed under the concatenation operation. It is the set of finite length strings that can be created by concatenating strings that match the expression p. If p can match any other string than ε, then p* will match an infinite number of possible strings.

The real name of the typographic symbol (glyph) used to denote the Kleene Star is asterisk. It’s a Greek (not geek) word and it logically enough means little star. Normally, it has five or six arms, but in its original purpose — to describe birth dates in a family tree — it had seven arms. This is a very popular symbol. In Unicode, there are a score of different variants and many communities have chosen their own meaning of the asterisk. In typography it means a footnote. In musical notation, it may mean that the sustain pedal of the piano should be lifted. On our cell phones, we use the asterisk to navigate menus in touch-tone systems. So there is no world-wide interpretation of *. However, in this book it’ll always mean the operator Kleene star.

Do you want to see a simple example? The concatenation closure of one single symbol, such as a, is a* = { ε, a, aa, aaa,... }. Want to see a more academic example? The concatenation closure of the set consisting solely of the empty string ε is — well, the set consisting solely of the empty string ε* = ε. Want to see a more complicated example? (1|0)* = { ε, 0, 1, 01, 10, 001, 010, 011,... }. It may thus be different matches of the expression that concatenates. Can you write a regular expression that matches all binary strings that contain at least one zero? Or all binary strings with an even number of ones?

The operator Kleene star — pronounced /ˈkleɪniː stɑ:r/ klay-nee star — is unary, i.e. it only takes one operand. The operand is the regular expression to the left, which allows us to say that it’s a postfix operator. It has the highest priority of all operators and it is associative. The latter means that if two operators of the same priority are competing, then the operator closest to the operand wins. Since p** = p*, we say that the Kleene star is idempotent. And I want to emphasize again that p* = (p|)*, i.e. the empty string ε is always present in a closure. We’ll later on see that there is a very common — and none the less disastrous — mistake to forget this very fact.

Here are some possible answers to two the questions above:

'110'.match /(0|1)*0(0|1)*/ #=> #<MatchData "110"> - all strings with at least oe zero
'1111'.match /(0|1)*0(0|1)*/ #=> nil
'1001'.match /((10*1)|0*)*/ #=> #<MatchData "1001"> - all strings with an even number of ones
'11001'.match /((10*1)|0*)*/ #=> #<MatchData "1100">
''.match /((10*1)|0*)*/ #=> #<MatchData ""> - even the empty string has an even number of ones
'1001'.match /((10*1)|0)|/ #=> #<MatchData "1001"> - another way to express an even number of ones
'11001'.match /((10*1)|0)|/ #=> #<MatchData "11">
''.match /((10*1)|0)|/ #=> #<MatchData "">
'1'.match /((10*1)|0)|/ #=> #<MatchData "">
'010'.match /((10*1)|0)|/ #=> #<MatchData "0">
'01'.match /((10*1)|0)|/ #=> #<MatchData "0">

The positive closure operator + and the at least n operator {n,} are abstractions for expressing infinite concatenation. We’ll soon explore them in more detail.

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Regular Expression Alternation

From rule number 2 and rule number 3 we can define paradigms — a number of possible patterns. This means that we add two or more languages by applying the set operator union to them. The union of the sets {a, b} and {c, d} is {a, b, c, d}. Hence, it’s all the elements that are either in one or more of the sets. In boolean logic, we call this the inclusively or. In regular expressions, it is called alternation and is written with a vertical bar |. Here are some examples:

'a'.match /a|b/ #=> #<MatchData "a"> - a is either a or b
'ab'.match /a|b/ #=> #<MatchData "a"> - leftmost chosen
'ba'.match /a|b/ #=> #<MatchData "b"> - leftmost chosen
'c'.match /a|b/ #=> nil - here we found neither a nor b

Note that most regex engines selects the leftmost alternative. There are exceptions to this rule. A regex engine based on DFA or POSIX NFA selects the longest alternative. Most regex engines are basic NFA and select the leftmost.

Can you write a regular expression that matches all binary strings of length one? The binary alphabet is { 0, 1 }. Since there aren’t a huge number of binary strings of length one, you can pretty quickly list them: { 0, 1 }. The regular expression with alternation then becomes 0|1:

'0'.match /0|1/ #=> #<MatchData "0">
'1'.match /0|1/ #=> #<MatchData "1">
'2'.match /0|1/ #=> nil
'10'.match /0|1/ #=> #<MatchData "1">

There are four binary strings of length two — {00, 01, 10, 11}. We can capture them with 00|10|01|11:

'10'.match /00|10|01|11/ #=> #<MatchData "10">
'01'.match /00|10|01|11/ #=> #<MatchData "01">
'12'.match /00|10|01|11/ #=> nil
'11'.match /00|10|01|11/ #=> #<MatchData "11">
'1210'.match /00|10|01|11/ #=> #<MatchData "10">

Maybe you didn’t notice, but we used concatenation in the regular expression above (can you see the invisible concatenation symbol between the two binary digits in the regular expression; if not, maybe you should make an appointment with an optometrist; or maybe not; not even an optometrist can help you see invisible symbols). Each of the binary strings of length two are made up of two concatenated binary strings of length one. Since concatenation has higher precedence than alternation, we didn’t need any parentheses.

Alternation is commutative: for two regular expressions p and q it holds that p|q = q|p. It is also associative: p|(q|r) = (p|q)|r. An interesting and very useful fact is that concatenation distributes over alternation. This means that for all regular expressions p, q and r it’s true that p(q|r) = pq|pr and (p|q)r = pq|pr. The consequence of that is that (0|1)(0|1) = (0|1)0|(0|1)1 = 00|10|01|11. So another way to match any binary strings of length two is:

'10'.match /(0|1)(0|1)/ #=> #<MatchData "10">
'01'.match /(0|1)(0|1)/ #=> #<MatchData "01">
'12'.match /(0|1)(0|1)/ #=> nil
'11'.match /(0|1)(0|1)/ #=> #<MatchData "11">
'1210'.match /(0|1)(0|1)/ #=> #<MatchData "10">

The brackets were needed, of course, because concatenation has higher precedence than alternation. We can also have the empty string ε as one of our alternatives:

'moda'.match /moda|/ #=> #<MatchData "moda"> - either moda or nothing is moda
'moda'.match /mado|/ #=> #<MatchData ""> - either mado or nothing is nothing

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