Define the language recognized by a finite state machine and provide an example.
A finite state machine (FSM) is a mathematical model used in computer science and cybersecurity to describe the behavior of a system that can be in a finite number of states and transitions between those states based on input. It consists of a set of states, a set of input symbols, a set of transitions, and an initial state. The language recognized by a finite state machine is the set of all possible input sequences that cause the machine to reach an accepting state.
To define the language recognized by a finite state machine, we need to consider the structure of the machine itself. Let's assume we have a finite state machine M = (Q, Σ, δ, q0, F), where:
– Q is the set of states, which is a finite non-empty set.
– Σ is the input alphabet, which is a finite non-empty set of symbols.
– δ is the transition function, which maps a state and an input symbol to a new state.
– q0 is the initial state, which is a member of Q.
– F is the set of accepting states, which is a subset of Q.
The language recognized by M, denoted as L(M), is the set of all possible input sequences that cause the machine to reach an accepting state. In other words, L(M) is the set of all strings over the input alphabet Σ that, when fed into the machine M, result in the machine reaching an accepting state.
To illustrate this concept, let's consider a simple example of a finite state machine. Suppose we have a machine M with three states: q0, q1, and q2. The input alphabet Σ consists of two symbols: 0 and 1. The transition function δ is defined as follows:
– δ(q0, 0) = q1
– δ(q0, 1) = q0
– δ(q1, 0) = q2
– δ(q1, 1) = q1
– δ(q2, 0) = q2
– δ(q2, 1) = q2
In this example, the initial state q0 is also the only accepting state. The language recognized by this machine, L(M), is the set of all input sequences that lead to the machine reaching the accepting state q0. This includes strings such as "10", "100", "110", "1000", and so on.
The language recognized by a finite state machine is the set of all possible input sequences that cause the machine to reach an accepting state. It is determined by the structure of the machine, including the set of states, the input alphabet, the transition function, the initial state, and the set of accepting states.
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