8 ε. Figure 1: An NFA-ǫ

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Alan Morris
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1 a LECTURE 27 Figure 1: An FA-ǫ 12.1 ǫ Transitions In all automata that we have seen so far, every time that it has to change from one state to another, it must use one input symol. An FA-ǫ is an FA that allows transitions laeled ǫ called ǫ-transitions. These transitions are non-deterministic transitions in the sense it the machine might follow one (or several) of those transitions without reading (consuming) any input symol. Look carefully at the FA-ǫ of figure 12.1 Is a in the language accepted y?. Is in the language accepted y?. The following computation traces show that oth a, and are in L(). [0,a] [1,a] [2,a] [3,a] [4,a] [6,a] [5,ǫ] [7,ǫ] [9,ǫ] [8,ǫ] [10,ǫ] Since there is a computation such that [0,a] [10,ǫ] where 10 is a final state, then a L(). As we can see ǫ transitions allow the machine to change state without having to consume any input symols. Definition 12.1 The ǫ-closure of a state q is the set of all those states that can e reached from q using only ǫ transitions. Formally, ǫ-closure(q) = {r Q [q,x] [r,x], x Σ } 1

2 The definition of ǫ-closure can e generalized to a set of states. Definition 12.2 If R Q then ǫ-closure(r) = ǫ-closure(q) q R Example. Let us compute the ǫ-closure(0) in the FA given aove. ǫ-closure(0) = {0,1,2,3,4,6} Every state is in the ǫ-closure of itself, since it is not necessary to consume any input symol to remain in the same state Closure of Regular Languages and ǫ-closure We were trying to show that regular languages are closed under concatenation when we decided to explore the idea of FAs. ow it is time to put the ideas that we developed so far to good use and show that Regular Languages are closed under concatenation. Formally, what we need to show is that if L 1 and L 2 are regular languages then L 1 L 2 is regular too. SinceL 1 andl 2 areregular,thentherearetwodfas M 1 andm 2 thataccept the languages L 1 and L 2 respectively. We will show that there is an FA-ǫ that accepts L 1 L 2. We know that FAs accept regular languages, since we were ale to show that given an FA it can e transformed into an equivalent DFA. We will show in the next section that an FA-ǫ can also e transformed into an equivalent DFA. For now, assume that FA-ǫ accept regular languages. Assume that the following pictures represent M 1 and M 2, each having one start state and several accepting states. M1 M2 To uild an FA-ǫ that accepts L 1 L 2, we just have to add an ǫ transition from the accepting states of M 1 to the start state of M 2 as shown elow 2

3 Following a similar idea we can show that regular languages are closed under union and Kleene star too. In the case of the Union, we add one new start state and ǫ transitions from this new state to each one of the original start states of the machine. In the case of the Kleene star operation we add one new start state and one new final state. We then add ǫ transitions from the new start state to the original start state, from the original final states to the new final state, from the original final states to the original start state, and from the new start state to the new final state. This last transition makes sure that ǫ is accepted, as it must, since ǫ L. 3

4 13 Regular Expressions Regular expressions provide us with a simple and readale way to represent languages. Suppose that we want do represent the language otained y the following operations: 1. Concatenate the elements of {a,a} to the elements of {aa,} 2. Apply the kleene star operator to the result 3. Concatenate the answer to the set otained from the union of {a,a} and {aa,}. If we wanted to write it down using set notation and regular operations, we would need to write lots of { and }. So, instead of writing: ({a,a}{aa,}) ({a,a} {aa,}) We will write the regular expression: ((a a)(aa )) ((a a) (aa )). Definition 13.1 Assume that x and y are regular expressions that descrie the languages X and Y respectively. 1. is a regular expression that represents the language. 2. ǫ is a regular expression that represents the language {ǫ} 3. a is a regular expression that represents the language {a}, where a Σ. 4. x y is a regular expression that represents the language X Y. 5. xy is a regular expression that represents the language XY 6. x is a regular expression that represents the language X The type of definition given aove is usually called a recursive definition, since the regular expressions are used to define regular expressions. However, the definition can e applied to generate longer regular expressions ased on shorter ones, with the ase cases eing the and a. As a matter of notation, the regular expression xx is areviated as x +. This means that x appears at least once. Operator precedence is similar to that of regular algeraic expressions, parentheses are computed first, then kleene star, then concatenation, and finally union. For example, the regular expression a c, represents the language {a,c}, not {a,ac}. Example. Some examples of regular expressions: 4

5 1. Language of those strings that have exactly one a a 2. Language of those strings that have at least one a (a ) a(a ) It can also e represented y the regular expression Σ aσ 3. Language of those strings that have aa as a sustring Σ aaσ 4. Language of those strings that start and end with the same symol aσ a Σ a There are many regular expression identities. We will give only a few of them: x = x = x xǫ = x = ǫ We must e very careful to distinguish etween the empty set, which has O elements, and the regular expression ǫ, which denotes a set that has exactly one element, which is the empty string, i.e. {ǫ}. Example. There are many applications where regular expressions are very useful. One of them is in the area of compilers, where we can descrie tokens y using regular expressions. Assume that the following regular expressions represent the given sets d 0 represents {0,1,2,3,4,5,6,7,8,9} d represents {1,2,3,4,5,6,7,8,9} x represents {a,...,z,a,...,z, } 5

6 1. A regular expression for integers (assuming that they don t egin with a 0) is dd 0 2. A regular expression for floats (without scientific notation) is dd 0.d Regular Expressions and Regular Languages In this section we will show that the languages represented y regular expressions are regular languages. To show this we will show the following theorem (in two parts): Theorem Given a regular expression x that represents language L, there is an FA-ǫ such that L() = L. 2. Given an FA-ǫ, there is a regular expression x that represents L(M). Proof. (Part 1) The proof is done y induction, using the definition of regular expressions. In each one of the cases show that there is an FA-ǫ that accepts the same language. Assume that M x is an FA-ǫ that accepts the language represented y x and that M y is an FA-ǫ that accepts the language represented y y. 1. is accepted y the following machine: 2. ǫ is accepted y the following machine: 3. a is accepted y the following machine: a 6

7 4. x y is accepted y the following machine: Mx My 5. xy is accepted y the following machine: Mx My 6. x is accepted y the following machine: Mx 7

Languages and Finite Automata

Languages and Finite Automata or how to talk to machines... Costas Busch - RPI 1 Languages A language is a set of strings String: A sequence of letters (a word) Examples: cat, dog, house, Defined over