How can we design a finite state machine that recognizes strings that do not contain a specific sequence, such as "0011"?

/ / Published in Cybersecurity, EITC/IS/CCTF Computational Complexity Theory Fundamentals, Finite State Machines, Examples of Finite State Machines, Examination review

A finite state machine (FSM) is a mathematical model used to represent and analyze systems which exhibit a finite number of states. In the field of computational complexity theory, FSMs are widely used to study the complexity of problems and algorithms. In this context, designing an FSM that recognizes strings not containing a specific sequence, such as "0011", requires careful consideration of the problem at hand.

To design an FSM that recognizes strings not containing the sequence "0011", we need to define the states, inputs, and transitions of the machine. Let's denote the states as Q, the inputs as Σ, the initial state as q0, and the transition function as δ.

First, we define the states of the FSM. In this case, we can define two states: "NoMatch" and "Match". The "NoMatch" state represents the condition where the input string does not contain the sequence "0011", while the "Match" state represents the condition where the input string contains the sequence "0011".

Next, we define the inputs of the FSM. Since we are dealing with binary strings, the inputs can be either 0 or 1. Thus, Σ = {0, 1}.

Now, let's define the transitions of the FSM. We need to determine the transition function δ, which maps a state and an input to a new state. In this case, we can define the following transitions:

1. If the current state is "NoMatch" and the input is 0, the FSM remains in the "NoMatch" state.
2. If the current state is "NoMatch" and the input is 1, the FSM remains in the "NoMatch" state.
3. If the current state is "Match" and the input is 0, the FSM remains in the "Match" state.
4. If the current state is "Match" and the input is 1, the FSM transitions to the "NoMatch" state.

These transitions ensure that the FSM stays in the "NoMatch" state as long as the input string does not contain the sequence "0011". Once the sequence "0011" is encountered, the FSM transitions to the "Match" state and remains there.

To complete the FSM, we need to define the initial state. In this case, the initial state can be set to "NoMatch".

To illustrate the design, let's represent the FSM using a state transition diagram:

         0
NoMatch ----> NoMatch
  |            |
  | 1          | 1
  v            v
Match <----- NoMatch

In this diagram, the arrows represent the transitions between states, and the labels on the arrows represent the inputs that trigger the transitions.

To test the FSM, we can consider some example strings:

1. "010101": The FSM starts in the "NoMatch" state. Following the transitions, it remains in the "NoMatch" state for each input. Thus, the FSM recognizes this string as not containing the sequence "0011".

2. "001100110": The FSM starts in the "NoMatch" state. Following the transitions, it transitions to the "Match" state after encountering the first "0011" sequence. Thus, the FSM recognizes this string as containing the sequence "0011".

By designing an FSM with appropriate states, inputs, and transitions, we can effectively recognize strings that do not contain a specific sequence, such as "0011". This approach can be extended to handle more complex patterns and sequences, providing a valuable tool in the field of computational complexity theory.

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