Automata theory

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science with close connections to cognitive science and mathematical logic. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically.

An automaton with a finite number of states is called a finite automaton (FA) or finite-state machine (FSM). These machines help us understand how computers and other systems can process information step by step. Automata theory is closely related to formal language theory. In this area, automata help us represent and understand languages that can have infinitely many words, using models that have a limited number of states.

Automata play an important role in many areas, including the theory of computation, building compilers for programming languages, artificial intelligence, parsing information, and checking that systems work correctly through formal verification. They help scientists and engineers build smarter and more reliable technology.

History

Automata theory started in the mid-20th century, focusing on simple machines called finite automata. It was first part of a bigger area called systems theory, but it used math called abstract algebra to study information, not physical things.

In 1956, a book called Automata Studies brought together work by many scientists. That same year, Noam Chomsky created the Chomsky hierarchy, which connects machines to language rules. By the 1960s, automata theory became a main part of computer science.

Automata

Automata theory is the study of abstract machines and how they can solve problems. The term "automata" comes from the Greek word for "self-acting" or "self-moving." An automaton processes inputs step by step, changing its state based on each input. It has a set of possible states and rules for moving from one state to another.

An automaton can be described using a few key parts: a set of input symbols, a set of output symbols, a set of states, a rule for moving between states, and a rule for producing outputs. One common example is an electronic lock, which checks a code and either opens or stays closed based on whether the code matches. Automata can recognize different types of languages, such as regular languages, which are sets of strings that follow certain patterns.

Hierarchy in terms of powers

Automata theory looks at different types of virtual machines and what they can do. These machines can solve certain problems, and some are more powerful than others. The hierarchy shows how these machines are organized, with more powerful ones able to handle more complex tasks. This helps us understand the limits of what computers and other machines can compute.

Main articles: Turing machine, Finite-state machine, Regular expression

Automaton

Deterministic Finite Automaton (DFA) -- Lowest Power (same power)    | | {\displaystyle ||}   (same power) Nondeterministic Finite Automaton (NFA) (above is weaker)    ∩ {\displaystyle \cap }    (below is stronger) Deterministic Push Down Automaton (DPDA-I) with 1 push-down store ∩ {\displaystyle \cap } Nondeterministic Push Down Automaton (NPDA-I) with 1 push-down store ∩ {\displaystyle \cap } Linear Bounded Automaton (LBA) ∩ {\displaystyle \cap } Deterministic Push Down Automaton (DPDA-II) with 2 push-down stores | | {\displaystyle ||} Nondeterministic Push Down Automaton (NPDA-II) with 2 push-down stores | | {\displaystyle ||} Deterministic Turing Machine (DTM) | | {\displaystyle ||} Nondeterministic Turing Machine (NTM) | | {\displaystyle ||} Probabilistic Turing Machine (PTM) | | {\displaystyle ||} Multitape Turing Machine (MTM) | | {\displaystyle ||} Multidimensional Turing Machine

Applications

Automata theory has many useful applications in different areas. For example, finite automata help with processing text, creating programs, and designing computer hardware. Context-free grammar is important for building programming languages and studying how people speak.

Cellular automata are used to mimic life processes, like the growth patterns of shells and plants. One famous example is John Conway's Game of Life. Some scientists even think the whole universe might work like a giant computer following simple rules.

Automata simulators

Automata simulators are tools that help teach and study automata theory. These simulators let you describe an automaton in different ways, such as using a special language, filling out a form, or drawing its diagram by clicking and dragging. Popular automata simulators include Turing's World, JFLAP, VAS, TAGS, and SimStudio. They let you test how the automaton works with any input you choose.

Main article: symbolic language

Category-theoretic models

Automata theory looks at different types of machines and how they work. In math, we can group these machines into categories. For example, deterministic automata and Turing machines form a special kind of category called a Cartesian closed category, which has special properties.

We can also study variable automata, which change over time. These changing automata can form groups or groupoids, which are special kinds of mathematical structures. Reversible automata fit into even more complex structures called 2-categories.