Pattern Matching and Regular Expressions

Abstract

Here are some interesting and important questions:

• The goal of this course is to understand the foundations of computation. We will ask some very basic questions, such as

• How hard is it to determine whether a given string x matches a given pattern α? This is an important practical question. There are very efficient algorithms, as we will see.

• Is every set represented by some pattern? Answer: no. For example, the set

  $$\left\{ {{a^n}{b^n}|n \geqslant 0} \right\}$$

  is not represented by any pattern. We’ll prove this later.

• Patterns α and γ are equivalent if L(α) = L(β). How do you tell whether α and β are equivalent? Sometimes it is obvious and sometimes not.

• Which operators are jedundant? For example, we can get rid of ∊ since it is equivalent to ~(#@) and also to ∅*. We can get rid of @ since it is equivalent to #*. We can get rid of unary ⁺ since α⁺ is equivalent to αα*. We can get rid of #, since if S = {a₁,...,a_n} then # is equivalent to the pattern

  $${a_1} + {a_2} + ... + {a_n}$$