What is Queue Automata?

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In automata theory, Push-Down automata is quite common where stacks are used, but there is another type of automata where queue are used. These are the queue automata. In this chapter, we are going to discuss the fundamentals of queue automata.

Foundations of Queue Automata

A queue automaton is a finite automaton with a queue storage following the first-in-first-out principle, which is a fundamental starting point in formal language theory. These models have desirable properties like equivalence, minimization algorithms, closure under operations, and decidable questions, but their computational capacity is limited to regular languages.

• A classical queue automaton consists of a finite state control, one-way input tape, and a queue. It can remove or keep a symbol at the front and enter a symbol at the end based on current state, input symbol, and front queue symbol.
• A nondeterministic queue automaton (NQA) is a system with finite states, input symbols, queue symbols, initial state, empty-queue symbol, accepting states, and partial transition function.

An example could be a deterministic queue automaton that accepts non-semilinear language {aⁿ | n is a Fibonacci number}, which operates in quasi-realtime by revolving the queue to double the number of symbols in each phase.

Comparison with Other Automata

We can differentiate the idea of queue automata with other automata. Follow the following table for a better understanding.

Some Variations of Queue Automata

Let us see the types for queue automata as Quasi-Realtime and Realtime, next will see the input driven queue automata with push down automata.

Closure Properties and Decidability

Let us check the closure properties for queue automata and then the famous decidability problem for queue automata. We will discuss them one by one in simple terms.

Closure Properties

Realtime Nondeterministic Queue Automata (NQA) −

• Closed under − Union, concatenation, Kleene star, reversal, and λ-free homomorphism
• Not closed under − Complementation, intersection, arbitrary homomorphisms

Realtime Deterministic Queue Automata (DQA) −

• Closed under − Intersection with regular sets
• Not closed under − Union, concatenation, Kleene star, complementation, intersection, homomorphism, reversal

Input-Driven Queue Automata (IDQA) −

• Closed under − Union, intersection with compatible signatures, intersection with regular languages
• Not closed under − Complementation, general intersection, concatenation, Kleene star, homomorphism, inverse homomorphism

Decidability

• k-Turn Nondeterministic Queue Automata (NQA): Emptiness and finiteness are decidable.
• k-Turn Deterministic Queue Automata (DQA): Equivalence with regular languages and universality are decidable
• k-Turn Input-Driven Queue Automata (IDQA): Inclusion and equivalence are decidable for compatible signatures

Practical Implications and Applications

Queue automata, despite their versatility, have high computational power and low manageability, limiting their practical use. They are often studied in restricted versions to regain manageable properties.

Applications in theoretical computer science include the characterization of languages accepted by multi-reset machines and the generation of certain language classes through breadth-first grammars.

The study of reversible computations in queue automata is motivated by the need to reduce information loss and heat dissipation in computations.

Conclusion

Queue automata, with their high computational power and versatility, are comparable to Turing machines. In this chapter, we explored the concept of queue automata along with their types and how they are different from other automata. We described the practical implications and the closure properties as well.