A Finite State Machine (FSM) is a mathematical model used to describe the behavior of a system that can be in a finite number of states. It consists of a set of states, a set of input symbols, a set of output symbols, and a transition function that determines the next state based on the current state and the input symbol. In the context of cybersecurity, FSMs are often used to model the behavior of protocols, systems, or algorithms.

To determine whether a string is accepted or rejected by an FSM, we need to understand how the FSM processes the input string and transitions between states. The FSM starts in an initial state and reads the input string symbol by symbol. At each step, the FSM consults its transition function to determine the next state based on the current state and the input symbol.

The transition function of an FSM is typically defined as a table or a graph. In the table representation, each row corresponds to a state, and each column corresponds to an input symbol. The entries in the table specify the next state that the FSM transitions to when it is in a particular state and receives a specific input symbol. In the graph representation, each state is represented by a node, and the edges between nodes represent the transitions. The labels on the edges indicate the input symbol that triggers the transition.

To determine whether a string is accepted or rejected, we start with the FSM in the initial state and feed it the input string symbol by symbol. After each symbol is processed, the FSM transitions to the next state based on the transition function. If the FSM reaches the end of the input string and ends up in a designated final state, the string is accepted. Otherwise, if the FSM reaches the end of the input string but does not end up in a final state, the string is rejected.

Let's consider an example to illustrate this process. Suppose we have an FSM with three states: A, B, and C. The initial state is A, and the final state is C. The transition function is as follows:

State | Input Symbol | Next State

——|————–|———–

A | 0 | A

A | 1 | B

B | 0 | C

B | 1 | A

C | 0 | B

C | 1 | C

Now, let's say we want to determine whether the string "101" is accepted or rejected by this FSM. We start with the FSM in state A. We read the first symbol, which is 1. Consulting the transition function, we see that when the FSM is in state A and receives a 1, it transitions to state B. We update the FSM's current state to B. Next, we read the second symbol, which is 0. According to the transition function, when the FSM is in state B and receives a 0, it transitions to state C. We update the current state to C. Finally, we read the third symbol, which is 1. The transition function tells us that when the FSM is in state C and receives a 1, it remains in state C. We update the current state to C.

At this point, we have reached the end of the input string "101". Since the FSM is in the final state C, we can conclude that the string "101" is accepted by this FSM.

An FSM determines whether a string is accepted or rejected by processing the input string symbol by symbol and transitioning between states based on a transition function. If the FSM reaches the end of the input string and ends up in a final state, the string is accepted. Otherwise, if the FSM reaches the end of the input string but does not end up in a final state, the string is rejected.

#### Other recent questions and answers regarding EITC/IS/CCTF Computational Complexity Theory Fundamentals:

- Can PDA detect a language of palindrome strings?
- Is Chomsky’s grammar normal form always decidible?
- Can a regular expression be defined using recursion?
- How to represent OR as FSM?
- Is there a contradiction between the definition of NP as a class of decision problems with polynomial-time verifiers and the fact that problems in the class P also have polynomial-time verifiers?
- Is verifier for class P polynomial?
- Can a Nondeterministic Finite Automaton (NFA) be used to represent the state transitions and actions in a firewall configuration?
- Is using three tapes in a multitape TN equivalent to single tape time t2(square) or t3(cube)? In other words is the time complexity directly related to number of tapes?
- If the value in the fixed point definition is the lim of the repeated application of the function can we call it still a fixed point? In the example shown if instead of 4->4 we have 4->3.9, 3.9->3.99, 3.99->3.999, … is 4 still the fixed point?
- If we have two TMs that describe a decidable language is the equivalence question still undecidable?

View more questions and answers in EITC/IS/CCTF Computational Complexity Theory Fundamentals