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A finite state machine has a limited or finite no. of possible states.It is mainly used as a development tool for solving problems. For example in DIgital electronics a sequence detector problem- which has to detect a particular pattern only and provide output only for the required pattern. A infinite state machine can be conceived but it is not practical one.
A finite state machine is a machine which has a finite no of states. In computer science finite state machines are used to develop lexical analyzers for the compilers and also when program for a standard device is to be written in embedded systems we use finite state machines to keep track of all the actions to be performed by the machine and also to perform time sharing between the functions to be performed by the device. A finite state machine takes finite no of input and according to the inputs it changes its state. This change of state may even occur after performing some specific action. In electronics mealy an moore machines are used. I had not heard of an infinite state machine up till now but if it exists then I confidently say that it is not realizable.