Regular Expression Module-2

Transcription

1 Regular Expression Module-2 Harivinod N, Dept of CSE, VCET Puttur 1 Introduction Let's now take a different approach to categorizing problems. Instead of focusing on the power of a computing device, let's look at the task that we need to perform. In particular, let's consider problems in which our goal is to match finite or repeating patterns. 1

2 Applications using Patterns The first step of compiling a program This step is called lexical analysis. Its job is to break the source code into meaningful units such as keywords, variables, and numbers. Filtering for spam Searching a complex directory structure by specifying patterns that are known to occur in the file we want. Regular Languages Generates Regular Language Regular Expression Recognizes or Accepts Finite State Machine Harivinod N, Dept of CSE, VCET Puttur 4 2

3 Stephen Cole Kleene , mathematical logician One of many distinguished students (e.g., Alan Turing) of Alonzo Church (lambda calculus) at Princeton. Best known as a founder of the branch of mathematical logic known as recursion theory. Also invented regular expressions. Kleene pronounced his last name KLAY-nee. `kli:ni and `kli:n are common mispronunciations. His son, Ken Kleene, wrote: "As far as I am aware this pronunciation is incorrect in all known languages. I believe that this novel pronunciation was invented by my father. " Kleeneness is next to Godelness Cleanliness is next to Godliness Regular expression Regular expression (RE) is defined in recursive way. The regular expressions over an alphabet Σ are all and only the strings that can be obtained as follows: is a regular expression. ε is a regular expression. Every element of Σ is a regular expression. If α, β are regular expressions, then so is αβ. If α, β are regular expressions, then so is α β. If α is a regular expression, then so is α*. If α is a regular expression, then so is α +. If α is a regular expression, then so is (α). 3

4 Semantic interpretation L( ) =. The language that contains no strings L(ε) = {ε}. The language that contains just the empty string. L(c), where c Σ = {c}. The language that contains the single one-character string c L(αβ) = L(α) L(β). Concatenation of RE is same as concatenation of Languages L(α β) = L(α) L(β). Union of RE s is same as union of the two constituent languages. Semantic interpretation L(α*) = (L(α))*. * is a Klene star operation. Defines the language that is formed by concatenating together zero or more strings drawn from L(α). L(α + ) = L(αα*) = L(α) (L(α))*. If L(α) is equal to, then L(α + ) is also equal to. Otherwise L(α + ) is language formed by concatenating together one or more strings drawn from L(α). L((α)) = L(α). 4

5 Semantic interpretation If the meaning of a regular expression α is the language L, then we say that α defines or describes L. Example-1: A simple RE The meaning of the regular expression (a U b)*b is the set of all strings over the alphabet {a, b} that end in b. 5

6 Example 2: Another simple RE So the meaning of the regular expression ((a b) (a b))a(a b)* is: {xay : x and y are strings of a's and b's and lxl = 2}. Harivinod N, Dept of CSE, VCET Puttur 11 Example 3 Harivinod N, Dept of CSE, VCET Puttur 12 6

7 Example 4 Harivinod N, Dept of CSE, VCET Puttur 13 Operators on RE Ordered as there priority Harivinod N, Dept of CSE, VCET Puttur 14 7

8 Ex: No more than one b Ex: No Two Consecutive letters are the Same Harivinod N, Dept of CSE, VCET Puttur 15 Ex: Floating point Numbers Harivinod N, Dept of CSE, VCET Puttur 16 8

9 Harivinod N, Dept of CSE, VCET Puttur 17 Kleene s Theorem Harivinod N, Dept of CSE, VCET Puttur 18 9

10 Kleene s Theorem Theorem - Part 1 - Any language that can be defined with a regular expression can be accepted by some FSM and so is regular. Theorem - Part 2 - Every regular language (i.e., every language that can be accepted by some DFSM) can be defined with a regular expression. RE to FSM Harivinod N, Dept of CSE, VCET Puttur 20 10

11 To each regular expression there is a corresponding FSM. Harivinod N, Dept of CSE, VCET Puttur 21 To each regular expression there is a corresponding FSM. Harivinod N, Dept of CSE, VCET Puttur 22 11

12 To each regular expression there is a corresponding FSM. Harivinod N, Dept of CSE, VCET Puttur 23 Harivinod N, Dept of CSE, VCET Puttur 24 12

13 Algorithm: RE to FSM Harivinod N, Dept of CSE, VCET Puttur 25 Example-1 Harivinod N, Dept of CSE, VCET Puttur 26 13

14 Example-1 Error Harivinod N, Dept of CSE, VCET Puttur 27 FSM to RE Harivinod N, Dept of CSE, VCET Puttur 28 14