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Chapter 12: Problem 27

Draw the transition diagram of a finite-state automaton that accepts the given set of strings over \(\\{a, b\\}\). Starts with baa

Short Answer

Expert verified
The transition diagram for strings over the alphabet set {a, b} that start with 'baa' comprises 5 states. It transitions from the initial state to new states upon sequentially receiving inputs 'b', 'a', and 'a', respectively. All other sequences lead to a catch-all non-accepting state, and any further inputs after the sequence 'baa' keeps the automaton in the accepting state.

Step by step solution

01

Define the Alphabet Set and Identify the Initial State

First, define the alphabet set as \(\{a, b\}\). Identify an initial state, often denoted as \(q_0\). This will be the state where the finite-state automaton starts.
02

Create Transition States for baa

The automaton should accept strings that start with 'baa'. Thus, from the initial state \(q_0\), create a transition for 'b' that leads to a new state \(q_1\). Then from state \(q_1\), create another transition for 'a' that leads to state \(q_2\). Finally, create one more transition for 'a' from state \(q_2\) leading to state \(q_3\). Essentially, for the input sequence 'baa', our automaton should reach the state \(q_3\).
03

Create Catch-all Transition State

To account for all other possible input sequences from the alphabet set, create a catch-all state \(q_4\) which will be a non-accepting state. From every state, if an unanticipated symbol arrives, the automaton should go to this catch-all state \(q_4\).
04

Mark the Final State

The final state of the automaton will be \(q_3\), because we want the automaton to accept the strings that start with 'baa'. From state \(q_3\), for each 'a' or 'b', the automaton will remain in the same state \(q_3\), indicating that all the strings that begin with 'baa' are accepted.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Diagram
A transition diagram is a graphical representation of a finite-state automaton.It showcases how the automaton changes states in response to input symbols from the alphabet set. In essence, it acts like a map, detailing the journey from one state to another. In our case, the transition diagram is crafted to detect strings beginning with 'baa'.
  • Each circle represents a state. These are denoted by letters like \(q_0, q_1, q_2,\) etc.
  • Arrows show how you transition from state to state based on the input symbol.
  • The starting state, usually denoted by an arrow pointing to it, helps us understand where we begin in processing a string.
  • Final states are typically indicated by double circles, showing that once reached, the input sequence is accepted.
Creating a transition diagram requires a clear understanding of the input sequence rules we've defined and ensured each possible input route is represented.
Alphabet Set
The alphabet set in a finite-state automaton is the collection of symbols the automaton can process.It serves as the building blocks for forming strings that are analyzed by the automaton. In our problem, the alphabet set is \(\{a, b\}\).
  • This means only the letters 'a' and 'b' are valid inputs for our automaton.
  • Defining the alphabet set helps us establish which inputs should prompt transitions between states in our automaton.
  • It's crucial to limit the alphabet set so the automaton appropriately manages and responds to predefined patterns or sequences.
Understanding the alphabet set is fundamental as it constrains the possible strings and transitions we have to consider in the automaton design.
Initial State
The initial state is the starting point for any state machine or finite-state automaton.It signifies the place where the automaton begins processing input symbols.
  • In our exercise, the initial state is named \(q_0\).
  • This state is crucial as it lays the foundation for how the string input is analyzed, allowing the automaton to react appropriately to the first symbol from the alphabet set.
  • An arrow pointing to the initial state distinguishes it from other states in a transition diagram, marking its importance in the sequence of processing.
Correctly identifying the initial state ensures that the transitions are correctly managed from the start.
Final State
The final state in a finite-state automaton signifies a successful parsing of a string.It indicates that the input string meets the criteria set by the transition rules and is accepted by the automaton.
  • In our scenario, the final state is \(q_3\).
  • A string that leads the automaton through all the transitions to this state signals acceptance by the automaton, particularly when the string begins with 'baa'.
  • The final state is often depicted as a double circle, visually setting it apart from other states in the transition diagram.
  • Once in the final state, further inputs either keep the automaton in this state or trigger transitions depending on predefined rules.
Recognizing the final state is essential because it defines when a sequence of input symbols has been fully and correctly parsed according to specified conditions.

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Most popular questions from this chapter

Each grammar in Exercises is proposed as generating the set \(L\) of strings over \(\\{a, b\\}\) that contain equal numbers of a's and b's. If the grammar generates \(L\), prove that it does so. If the grammar does not generate \(L\), give a counterexample and prove that your counterexample is correct. In each grammar, \(S\) is the starting symbol. $$ S \rightarrow a S b|b S a| \lambda $$

Draw the transition diagram of a finite-state automaton that accepts the given set of strings over \(\\{a, b\\}\). Exactly two \(a\) 's

Each grammar in Exercises is proposed as generating the set \(L\) of strings over \(\\{a, b\\}\) that contain equal numbers of a's and b's. If the grammar generates \(L\), prove that it does so. If the grammar does not generate \(L\), give a counterexample and prove that your counterexample is correct. In each grammar, \(S\) is the starting symbol. $$ S \rightarrow a B|b A, A \rightarrow a| S A, B \rightarrow b \mid S B $$

Each grammar in Exercises is proposed as generating the set \(L\) of strings over \(\\{a, b\\}\) that contain equal numbers of a's and b's. If the grammar generates \(L\), prove that it does so. If the grammar does not generate \(L\), give a counterexample and prove that your counterexample is correct. In each grammar, \(S\) is the starting symbol. $$ S \rightarrow a S b|b S a| a b S|b a S| S a b|S b a| \lambda $$

Design a finite-state machine having the given properties. The input is always a bit string. Outputs 1 when it sees the first 0 and until it sees another 0 ; thereafter, outputs \(0 ;\) in all other cases, outputs 0
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