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Chapter 6: Problem 8

Draw the state diagram for a finite state machine \(M=(S, g, 0, v, \omega)\), where \(g=0=\\{0,1\\}\), if for any \(x \in \mathcal{g}^{+}, M\) puts out its first 1 when it recognizes the substring 1111 and then puts out its second 1 when it recognizes the substring 0000 , after which its output is constantly 0 .

Short Answer

Expert verified
The corresponding state diagram consists of three states: \(S_0, S_1, S_2\). The FSM begins in state \(S_0\), it stays in \(S_0\) if a 0 appears or if it has seen fewer than four 1s in a row. Once it recognises four consecutive 1s, it moves to \(S_1\). In \(S_1\), if it receives a 1 before seeing four consecutive 0s it resets and waits for consecutive 0s, otherwise, after seeing four consecutive 0s, it moves to \(S_2\). Once in \(S_2\), FSM remains there producing only 0s.

Step by step solution

01

Define States

Starting with a state \(S_0\) as the initial state where the FSM hasn't encountered four consecutive 1s yet. Once it encounters four consecutive 1s, it transitions to state \(S_1\). When it encounters four consecutive 0s, it transitions to state \(S_2\). After that, the FSM stays in state \(S_2\) producing only 0. Therefore, the FSM has 3 states: \(S_0, S_1, S_2\).
02

Define Transitions for Recognizing Substring 1111

In state \(S_0\), if the FSM receives a 1, it remains in state \(S_0\) until it recognizes four consecutive 1s and then moves to state \(S_1\). If the FSM receives a 0 in state \(S_0\), it resets and stays in state \(S_0\), waiting for four consecutive 1s.
03

Define Transition for Recognizing Substring 0000

In state \(S_1\), the FSM stays until it recognizes four consecutive 0s, after which it transitions to state \(S_2\). In state \(S_1\), if a 1 is received before encountering four consecutive 0s, the FSM resets and waits for four consecutive 0s again.
04

Define Transition for Constant Output

Once in state \(S_2\), the FSM stays and its constant output is 0 regardless of the incoming input sequence.

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Key Concepts

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

State Diagram
A state diagram is an essential tool for visualizing how a finite state machine (FSM) operates across different states. It provides a graphical representation that illustrates the states of the FSM and the transitions between these states based on inputs. In the case of our FSM, the state diagram consists of three primary states:
  • S0: The starting state where no significant patterns (like "1111") have been detected yet.
  • S1: The intermediate stage reached after recognizing the specific pattern "1111".
  • S2: The concluding state that remains after spotting the "0000" pattern.
The visual nature of the state diagram makes it easier to understand complex sequences of transitions and can be helpful for debugging and verifying the FSM's logic. Pay close attention to how each state connects with the others and what conditions trigger each transition to get the full picture of how the FSM behaves.
State Transitions
State transitions describe how the FSM moves from one state to another, based on input conditions. These transitions are the crux of any finite state machine's operation. Each state captures a part of the input history that is crucial for future operations. For example, in the given exercise:
  • In S0, the FSM awaits the sequence "1111." During this time, if any zero occurs before this sequence is completed, the FSM stays in S0.
  • Once "1111" is detected, the FSM shifts to S1. Here, it looks for the subsequent sequence "0000." If another sequence interrupts, it resets and waits for "0000" again.
  • Upon recognizing "0000," the FSM transitions to S2. In this state, the output remains 0, indicating a stable state as it endlessly awaits new transitions.
Understanding these transitions equips one with a mechanism to predict the FSM鈥檚 response to different input sequences, highlighting the importance of each state in the overall cycle.
Substring Recognition
Substring recognition is a vital concept in understanding how a finite state machine can efficiently identify patterns within an input stream. It emphasizes the FSM's ability to track sequences, finding particular substrings amidst potentially cluttered input data. In our specific problem:
  • The FSM must detect the "1111" substring to transition from the initial state S0 to the next state S1. This pattern recognition is crucial as it identifies when the FSM should output its first signal change.
  • Afterwards, transitioning from S1 to S2 depends on detecting the "0000" substring. This step prompts the change to a constant, unchanging output of 0, reflecting the FSM has finalized its main detection task.
Through the careful design of state transitions and conditions, the FSM can effectively filter through diverse inputs to recognize specific substrings that lead to significant transitions, providing a robust approach to digital pattern recognition tasks.

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Most popular questions from this chapter

For a given alphabet \(\mathbf{\Sigma}\) and index set \(I\), let \(B_{i} \subseteq \mathbf{\Sigma}^{*}\) for each \(i \in I\). If \(A \subseteq \Sigma^{*}\), prove that (a) \(A\left(\cap_{b e l} B_{i}\right) \subseteq\) \(\bigcap_{i \in i} A B_{i}\); and, (b) \(\left(\cap_{i \in I} B_{t}\right) A \subseteq \bigcap_{i \in I} B_{i} A\). [Here, for example, \(A\left(\bigcap_{i \in I} B_{i}\right)\) denotes the concatenation of the languages \(A\) and \(\bigcap_{i e i} B_{i}\).]

For \(\mathbf{\Sigma}=\\{0,1\\}\) determine all possible languages \(A, B \subseteq \mathbf{\Sigma}^{*}\) where \(A B=\\{01,000,0101\), \(0111,01000,010111\\} .\)

Given a nonempty language \(A \subseteq \Sigma^{*}\), prove that if \(A^{2}=A\), then \(\lambda \in A\).

For languages \(A, B \subset \Sigma^{*}\), does \(A^{*} \subseteq B^{*} \Rightarrow A \subseteq B\) ?

For the alphabet \(\mathbf{\Sigma}=\\{0,1\\}\), let \(A, B, C \subseteq \mathbf{\Sigma}^{*}\) be the following languages: $$ \begin{aligned} &A=\\{0,1,00,11,000,111,0000,1111\\}, \\ &B=\left\\{w \in \mathbf{\Sigma}^{*} \mid 2 \leq\|w\|\right\\}, \\ &C=\\{w \in \mathbf{\Sigma} * \mid 2 \geq\|w\|\\} . \end{aligned} $$ Determine the following subsets (languages) of \(\mathbf{\Sigma}^{*}\). a) \(A \cap B\) b) \(A-B\) c) \(A \triangle B\) d) \(A \cap C\) e) \(B \cap C\) f) \(B \cup C\) g) \(\overline{(A \cap C)}\) h) \(\bar{A} \cap \bar{C}\) i) \(\bar{A} \cap B\)
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