INFO 2950 Prof Carla Gomes gomescs cornell edu

INFO 2950 Prof. Carla Gomes gomes@cs. cornell. edu Module Modeling Computation: Language Recognition Rosen, Chapter 12. 4 1

What sets can be recognized by a Finite State Automata? Regular Sets 2

Regular Sets Definition: A regular set is a set that can be generated starting from the empty set, empty string, and single elements from the alphabet, using concatenations, unions, and Kleene closures in arbitrary order. We will give a more precise definition after we define a regular expression.

Regular Expressions Definition: The regular expressions over a set I are defined recursively by: – Æ (the empty set) is a regular expression, – (the set containing the empty string) is a regular expression, – x is a regular expression for all xÎI, – (AB) , (A B) , and A* are regular expressions if A and B are regular expressions Definition: A regular set is a set represented by a regular expression.

Regular Expression Examples: 001*, 1(0 1)0, (0 1)*11, and AB*C are regular expressions The regular set defined by the regular expression 01* is the set of strings starting with a 0 followed by 0 or more 1 s. The regular set defined by (10)* is the set of strings containing 0 or more copies of 10. The regular set defined by 0(0 1)*1 is the set of all binary strings beginning with 0 and ending with 1. The regular set defined by (0 1)1(0 1) is the set of strings {010, 011, 110, 111}.

What are the strings represented by 10* A 1 followed by any mnumber of 0 s (including no zeros) (10)* Any number of copies of 10 (including null string) 6

0 01 the string 0 or the string 01 0 (0 1)* Any string beginning with 0 (0*1)* Any string not ending with a 0 (including null string) 7

Find a regular expression The set of bit strings with even length (00 01 10 11)* Set of bit strings ending with a 0 not containing 11 Concatenations of 0 or 10 ; not the null string (0 10)*(0 10) 8

The set of bit strings containing and odd number of 0 s At least one 0 Zero or more 1 s, followed by a 0, followed by zero or more 1 1*01*(01*01*)* 9

Regular Expression Applications Regular expressions are actually used quite often in computer science. For instance, if you are editing a file with vi, and want to see if it contains the string blah followed by a number followed by any character followed by the letter Q, you can use the regular expression blah[0 -9]*. Q This works because vi uses regular expressions for searching.

Regular Expression Regular Grammar a* (a+b)* a* + b* S | a. S | b. S S |A|B A a | a. A B b | b. B S b | a. S S b. A A | a. A S | ab. S a*b ba* (ab)*

EXAMPLE 1 Consider the language { ambn | m, n N}, which is represented by the regular expression a*b*. A regular grammar for this language can be written as follows: S | a. S | B B b | b. B.

Grammars, Expressions, and Automata • Consider the set A={binary strings which start with 0 and end with 1} We saw that A is recognized by a finite-state automata. A is generated by the grammar with V={S, A, 0, 1}, T={0, 1}, and P={S 0 A, A 1 A, A 1} We also saw that A is defined by the regular expression 0(0 1)*1 • This is no coincidence, as we will see next.

Three Equivalent Representations Regular expressions Finite automata Each can describe the others Regular languages Kleene’s Theorem: For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.

Grammars, Expressions, and Automata • Theorem: Let L be a language. The following three statements are equivalent L is regular set (that is, L generated by a regular expression) L is a regular language (that is, L generated by a regular grammar) L is recognized by a finite-state automaton • Put another way, L is a regular set if and only if L is a regular language if and only if L is recognized by a finitestate automaton. • In other words, regular sets, regular languages, and languages recognized by finite-state automata are all the same thing.

Example • Example: What language does the following finite-state automaton recognize?

Complex Example Continued • If start by going to state S 1 can recognize 000, 011100, 011111100, 0010100, 01110110, 01110100, … • It is not easy to see the pattern right away, but notice that they Start with 0 Can have any number of instances of 111 or 01 interleaved Can then have either 00 or 110 Can end with any number of 1 s. • These are all of the form 0(111 01)*(00 110)1* • But we can also start by going to S 6

Complex Example Continued • If we start by going to S 6 we notice that the strings Start with 1 Have any number of occurrences of 01 Have a 1 End with as many 0 s as we want • These are of the form 1(01)*10* • Thus, we can recognize (0(111 01)*(00 110)1*) (1(01)*10*)

Limitations • Problem: Find a finite-state automaton that recognizes the following language L={0 n 1 n | n=0, 1, 2, …} • Solution: It cannot be done. • Proof: Take an advanced course. • Can you describe L with a regular expression? • Can you give a regular grammar that generates L? • Can you give any grammar that generates L?

Models of computing DFA Push down automata Bounded Turing M’s Turing machines - regular languages Context-free Context sensitive Phrase-structure 20

Summary • Hopefully it is clear that although finite-state machines and finite-state automata are useful models of computation, they have serious limitations. • Are there more powerful ways to model computation? • The answer is: Yes. • Some more powerful models include Pushdown automaton Linear bounded automaton Turing machines Quantum computation models