How are finite state machines combined to represent the union of languages recognized by two machines?
In the field of computational complexity theory, finite state machines (FSMs) are widely used to model and analyze the behavior of systems. FSMs are mathematical models that consist of a finite number of states and transitions between these states based on input symbols. They are commonly used to represent regular languages, which are a subset of formal languages that can be described by regular expressions or generated by FSMs.
To represent the union of languages recognized by two FSMs, we need to combine the two machines in a way that allows us to recognize strings that belong to either of the languages. This can be achieved through a process called the union operation.
The union of two FSMs, M1 and M2, involves creating a new FSM, M, that recognizes the language formed by the union of the languages recognized by M1 and M2. This can be done by introducing a new start state and connecting it to the start states of M1 and M2 using ε-transitions (transitions that do not consume any input symbol). The ε-transitions allow the machine to choose between the two starting states and proceed with the recognition process accordingly.
The union operation also requires some modifications to the original machines. First, we need to ensure that the final states of M1 and M2 remain final states in the new machine M. This can be achieved by introducing ε-transitions from the final states of M1 and M2 to a new final state in M. These ε-transitions allow the machine to accept a string if it is accepted by either M1 or M2.
Furthermore, we need to ensure that the transitions of M1 and M2 are preserved in the new machine M. This can be done by simply copying the transitions of M1 and M2 to M. If there are any common transitions between M1 and M2, they can be merged into a single transition in M.
Let's consider a simple example to illustrate the process. Suppose we have two FSMs, M1 and M2, as shown below:
M1:
Start state: q0
Final state: q2
Transitions: (q0, a) -> q1, (q1, b) -> q2
M2:
Start state: p0
Final state: p2
Transitions: (p0, c) -> p1, (p1, d) -> p2
To represent the union of the languages recognized by M1 and M2, we create a new FSM, M:
M:
Start state: s0 (new start state)
Final state: f2 (new final state)
Transitions: (s0, ε) -> q0, (s0, ε) -> p0, (q2, ε) -> f2, (p2, ε) -> f2
(q0, a) -> q1, (q1, b) -> q2, (p0, c) -> p1, (p1, d) -> p2
In this example, the new FSM M recognizes the union of the languages recognized by M1 and M2. It starts in the new start state s0 and can transition to either q0 or p0 using ε-transitions. From there, it follows the transitions of M1 and M2 based on the input symbols. If it reaches the final state of either M1 or M2, it can transition to the new final state f2 using ε-transitions.
To summarize, the union of languages recognized by two FSMs can be represented by combining the machines and introducing ε-transitions to allow for choice between the starting states. Additionally, ε-transitions can be used to connect the final states of the original machines to a new final state in the combined machine. By preserving the original transitions, the new machine can recognize strings that belong to either of the languages recognized by the original machines.
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