1 INFO 2950 Prof. Carla Gomes Module Modeling Computation: Language Recognition Rosen, Chapter 12.4

What sets can be recognized by a Finite State Automata? 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: –  (the empty set) is a regular expression, –  (the set containing the empty string) is a regular expression, –x is a regular expression for all x  I, –(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 Example Examples: 001 *, 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 1s. 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 0s (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 0s At least one 0 Zero or more 1s, 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][0-9]*.Q This works because vi uses regular expressions for searching.

Regular ExpressionRegular Grammar a* S   | aS (a+b)* S   | aS | bS a* + b* S   | A | B A  a | aA B  b | bB a*b S  b | aS ba* S  bA A   | aA (ab)* S   | abS

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

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  0A, A  0A, A  1A, 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 Finite automata Regular expressions Regular languages Each can describe the others 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 finite- state 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, 0110, , , , 00100, , , , … 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 1s. 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 0s 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?

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

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