COMS W3261
Computer Science Theory
Lecture 2: September 14, 2009
Regular Expressions

1. Outline

• Review
• Two inductive proofs
• Regular expressions
• Examples of regular expressions
• Practice problems

2. Review

• Operations on languages
• Union
• L ∪ M = { w | w is in either L or M }
• Concatenation
• LM = { xy | x is in L and y is in M }
• Exponentiation
• L⁰ is { ε }, the set containing the empty string.
• Lⁱ = L^i-1L, for i > 0
• Example: {0, 1}* is the set of all strings of 0's and 1's (including ε).
• Kleene closure
• L* = L⁰ ∪ L¹ ∪ L² ...
• Note that ∅* = ∅⁰ = { ε }.

3. Two inductive proofs

• Let x be a string of left and right parentheses. The profile of x is the number of left parentheses in x minus the number of right parenthesis in x.
• Definition P. A string w of left and right parentheses is profile-balanced if the profile of w is zero and the profile of any prefix of w is nonnegative.
• Definition B. Balanced strings.
B1: The empty string is balanced.
B2: If x and y are balanced strings, then (x)y is a balanced string.
• We want to show L(B) = L(P).


• We first show L(B) is contained in L(P) by proving the following statement S(n) is true by induction on n for n ≥ 0.
S(n): If w is a string generated by n applications of rule B2 in the definition B, then w is profile-balanced.
• Basis: n = 0. The only string that can be generated by zero applications of rule B2 is the empty string, which is clearly profile-balanced.
• Induction: Suppose S(i) is true for i = 0, 1, 2,..., n. Consider an instance of S(n + 1) which generates a string w using n + 1 applications of rule B2.
• The last application of rule B2 that generates w says w is of the form (x)y where x and y are balanced.
• Both x and y were generated by n or fewer applications of rule B2, so by the inductive hypothesis both x and y are profile-balanced. It now follows, (x)y is profile-balanced.
• We therefore conclude S(n) is true for all n ≥ 0.


We now show L(P) is contained in L(B) by proving the following statement T(n) is true by induction on n for n ≥ 0.
T(n): If w is a profile-balanced string of length n, then w balanced.
• Basis: n = 0. Here, w is the empty string, which is balanced by rule B1.
• Induction: Suppose all profile-balanced strings of length up to n are balanced. Consider a profile-balanced string w of length n + 1.
• We can write w as (x)y where (x) is the shortest prefix of w that is profile-balanced. It now follows that y must be profile-balanced.
• By the inductive hypothesis both x and y are balanced since each of their lengths is less than n. By rule B2, (x)y is balanced.
• We therefore conclude T(n) is true for all n ≥ 0.


• We have now shown L(B) = L(P). Thus, we can use either P or B to define what it means for a string of parentheses to have balanced parentheses.

4. Regular Expressions

• A regular expression E is an algebraic expression that denotes a language L(E).
• Programming languages such as awk, java, javascript, perl, python use regular expressions to match patterns in strings.
• Regular expressions come in many different forms but almost all have the operations of union, concatenation, and Kleene closure.


• Inductive definition of regular expressions over an alphabet Σ:
• Basis
• The constants ε and ∅ are regular expressions that denote the languages { ε } and { }, respectively.
• A symbol c in Σ by itself is a regular expression that denotes the language { c }.
• Induction: Let E and F be regular expressions.
• E + F is a regular expression that denotes L(E) ∪ L(F).
• EF is a regular expression that denotes L(E)L(F), the concatenation of L(E) and L(F).
• E* is a regular expression that denotes (L(E))*.
• (E) is a regular expression that denotes L(E).


• Precedence and associativity of the regular-expression operators
• The regular-expression operator star has the highest precedence and is left associative.
• The regular-expression operator concatenation has the next highest precedence and is left associative.
• The regular-expression operator + has the lowest precedence and is left associative.
• Thus the regular expression a + b*c would be grouped a + ((b*)c.

5. Examples of Regular Expressions and the Languages They Denote

• 0*10* denotes the set of all strings of 0's and 1's containing a single 1.
• (0+1)*1(0+1)* denotes the set of all strings of 0's and 1's containing at least one 1.
• (a+b)*abba(a+b)* denotes the set of all strings of a's and b's containing the substring abba.
• Let R? be a shorthand for the regular expression (R + ε).
We can interpret the operator ? as meaning "zero or one of".
The regular expression a?b?c? denotes the language {ε, a, b, c, ab, ac, bc, abc}.
• The Unix command
egrep '^a?b?c?d?e?$' file
would print all lines in file consisting of the letters a, b, c, d, e in increasing alphabetic order.
The metacharacters ^ and $ match the empty string at the beginning and end of a line, respectively.
aegilops is the longest English word whose letters are in increasing alphabetic order.

6. Practice Problems

1. Do the two regular expressions (a+b)* and (a*b*)* denote the same language?
2. Write a regular expression for all strings of a's and b's with an even number of a's.
3. Write a regular expression for all strings of a's and b's with an even number of a's and an odd number of b's.
4. Write a regular expression for all strings of a's and b's that do not contain aba as a substring.
5. Write a regular expression for all strings of a's, b's, and c's that do not contain two identical adjacent characters.
6. Write a Unix regular expression for all English words ending in dous.
7. Write a Unix regular expression for all English words with the five vowels a,e,i,o,u in order. (The vowels do not have to be next to one another.)

7. Reading Assignment

• HMU: Sect. 3.1