Transition Table in Automata Theory

In the very basic discussion of automata theory, we talk about state diagrams or state-transition diagrams to visually represent a machine or automata. Another representation of the same is transition tables which are nothing but the same representation in a tabular form.

In this chapter, we will cover the transition tables in detail including how they are formed, their properties, and applications.

A Little Background on Automata

Before discussing the concept of transitions in automata, we must understand the background. In computer science, automata is a fundamental branch which deals with abstract machines and the problems they can solve.

We know the automaton is a mathematical model of a computing device that can be used to recognize patterns within input data. There are mainly two things involved in automata.

• States − These represent the different configurations or conditions of an automaton, or what actions the automaton can do.
• Transitions − These define how the automaton moves from one state to another based on the input that state consumed.

If we talk about the finite state automaton, we mostly focus −

• Deterministic Finite Automata (DFA) − In a DFA, for each state and input symbol, there is exactly one transition to a next state.
• Non-deterministic Finite Automata (NFA) − An NFA allows multiple transitions from a state for a given input symbol, including the possibility of no transition.

A little variation of FSM is the Pushdown Automata (PDA), which is nothing but an FSM with additional stack data-structure.

Transition and Transition Table

Consider the following state diagram along with the transition table −

In this finite state machine, there are four states. The initial state is q1 and the final state is qf. There are several transitions from one state to another. However, not all state has all possible transitions, and for a single input it is moving more than one states. So this is an example of non-deterministic FSM. Let us see the corresponding transition table for this same state diagram.

The table shows the states and for which symbol it will jump to which state. This transition tables are equivalent to the state transition diagrams to represent an automaton.

Let us see another example with inputs and outputs −

Here we have 4 states but there is no final state since this automaton is making output while transitioning. On the arc, we have input and output separated by the "/" symbol. The corresponding transition table will be look like the following −

Advantages of Transition Tables

Transition Tables provide a clear and concise way to visualize the state transitions based on different input symbols.

• Purpose − Transition tables are crucial for comprehending, analyzing, and designing automata by visualizing their behavior, determining language acceptance, and converting between representations like state diagrams.
• Components − A transition table typically consists of −
  • States − A list of all possible states the automaton can be in.
  • Input symbols − The set of symbols the automaton can process.
  • Next states − The state the automaton transitions to based on the current state and input symbol.
• Transition tables are versatile tools for analyzing automata behavior. Some possible advantages of using them are −
  • Simulating automata − By starting at the initial state and following the transitions based on the input string, one can simulate the automaton's execution.
  • Determining language acceptance − An automaton accepts a string if it reaches an accepting state after processing the entire input. Transition tables help in tracing this process.
  • Converting to other representations − Transition tables can be easily converted to state diagrams, which provide a visual representation of the automaton.

Conclusion

Transition tables are crucial in automata theory, providing a structured representation of automaton behavior. They are essential for understanding, designing, and analyzing automata.

In this chapter, we explained the specific types of automata (DFA, NFA, PDA) and covered the concept of transition diagram and transition tables.