In Exercises 15-17 construct a regular grammar G= (V, T, S, P) that generates the language recog

In Exercises 15-17 construct a regular grammar G= (V, T, S,...

Key Concepts

Regular Grammar

A regular grammar is a formal grammar that consists of a finite set of production rules which are either right-linear or left-linear. These grammars generate regular languages and are characterized by having rules that involve a nonterminal on the left-hand side and either a terminal followed optionally by a nonterminal, or just a terminal, on the right-hand side.

Finite-State Machine

A finite-state machine (FSM) is an abstract computational model used to represent and recognize regular languages. It consists of a finite number of states, a set of input symbols (alphabet), transition functions that describe state changes upon reading symbols, a start state, and one or more accept (final) states.

Regular Languages

Regular languages are a class of languages that can be described by regular expressions, recognized by finite-state machines, and generated by regular grammars. Their defining feature is that they are closed under operations such as union, concatenation, and Kleene star, which makes them relatively simple in terms of expressive power.

Conversion Process from FSM to Regular Grammar

There exists a systematic procedure to convert a finite-state machine into an equivalent regular grammar. This involves mapping the states of the FSM to nonterminal symbols in the grammar, converting each transition in the FSM into a production rule in the grammar, and incorporating the accept states by adding productions that terminate the derivation. The resulting grammar generates exactly the language recognized by the FSM.

Transcript

00:01 We are given a finite state machine and we're asked to construct a regular grammar that generates the language recognized by this machine.

00:12 The machine can be found in exercise 17 of this section.

00:18 Looking at the machine, we see it has four states, s0, s1, s2, and s3.

00:32 We see that, of course, s0 is the start state, and we see that s1 and s2 are final states.

00:43 To find a grammar that generates the same language recognized by this machine, first let's find the terminal symbols.

01:08 These are the labels of the arrows in the figure.

01:11 You see that the only possible labels are zero or one, so that our set of terminal symbols, t, is simply going to be set 0, 1.

01:21 Now, because this is a grammar, we know that there exists some start symbol s.

01:36 And of course we know that s represents the start state s zero.

01:47 And we see that there are three other states.

02:02 And so we should have three other non -terminal states.

02:08 And so our vocabulary v will consist of s, a, b, and c, along with our terminal symbols 0 and 1.

02:37 Now we'll let a represent state s1, b will represent state s2, and c will represent state s3.

03:13 Now, we see that if there is an arrow from a state u to a state v with a label of x, then we'll add a product.

03:24 In the production, symbol u can be replaced by the symbol xv, with u and v, the string xv, with u and v being the symbols representing the states u and v.

03:39 Also, if we see that u is a final state, state u is a final state, and if there's an arrow with an input of x from state v to state u, then we'll add a production that the symbol of v can be replaced by symbol x, terminal symbol x.

04:04 Also, we notice that the start state is an accepting state...

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