How can a string be accepted by a nondeterministic finite state machine?
A string can be accepted by a nondeterministic finite state machine (NFSM) if there exists at least one computation path that leads to an accepting state when the machine processes the string. In order to understand how this is achieved, it is important to have a clear understanding of the components and behavior of an NFSM.
An NFSM is a mathematical model used to describe the behavior of a system that can be in a finite number of states and transitions between these states based on input symbols. It consists of a set of states, a set of input symbols, a transition function, an initial state, and a set of accepting states.
The transition function of an NFSM determines the next state of the machine based on the current state and the input symbol. However, unlike a deterministic finite state machine (DFSM), an NFSM can have multiple possible next states for a given input symbol and current state. This non-determinism allows the NFSM to explore multiple computation paths simultaneously.
To accept a string, an NFSM starts in the initial state and reads the input symbols one by one. At each step, the machine can make non-deterministic choices by transitioning to multiple possible next states. This means that the machine can be in multiple states at the same time during its computation.
If, at any point, there exists at least one computation path that leads to an accepting state, the NFSM accepts the string. This means that even if some computation paths lead to non-accepting states, as long as there is at least one path to an accepting state, the string is considered accepted.
Let's consider an example to illustrate this concept. Suppose we have an NFSM with three states: A, B, and C. The initial state is A, and the accepting state is C. The input alphabet consists of two symbols: 0 and 1.
The transition function of the NFSM is as follows:
– From state A, if the input symbol is 0, the machine can transition to either state A or B.
– From state A, if the input symbol is 1, the machine can transition to state B.
– From state B, if the input symbol is 0, the machine can transition to state C.
– From state B, if the input symbol is 1, the machine can transition to either state B or C.
– From state C, regardless of the input symbol, the machine remains in state C.
Now, let's consider the string "010". The NFSM would start in state A and read the symbols one by one. Here is the computation path for this string:
1. Start in state A.
2. Read the first symbol, 0. Transition to state A or B.
3. Read the second symbol, 1. Transition to state B.
4. Read the third symbol, 0. Transition to state C.
Since there exists at least one computation path that leads to the accepting state C, the string "010" is accepted by the NFSM.
A string can be accepted by a nondeterministic finite state machine if there exists at least one computation path that leads to an accepting state. The NFSM can explore multiple computation paths simultaneously due to its non-determinism, allowing it to accept strings that would not be accepted by a deterministic finite state machine.
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