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 consist of states and transitions between them. As the automaton processes input, it moves from one state to another based on a specific transition function. Automata theory helps us understand how machines can recognize and process languages, playing important roles in areas like compiler construction, artificial intelligence, and formal verification.

History

Automata theory started in the middle of the 20th century, linked to the study of finite automata. Early work used abstract algebra to study information, instead of using math that describes physical systems. Important books and ideas appeared in 1956, including work by Claude Shannon, Noam Chomsky, and others. These contributions helped automata theory become its own area of study.

Later developments included important theorems and the growth of computational complexity theory. By the late 1960s, automata theory was seen as a key part of the mathematics behind computer science.

Automata

Automata theory is the study of abstract machines and how they can solve problems. The word "automata" comes from the Greek word meaning "self-acting" or "self-moving." An automaton processes inputs step by step, changing its state based on each input. It can either accept or reject a sequence of inputs, depending on whether it ends in a specific state.

Automata can be described using a set of symbols for inputs and outputs, a set of states, and rules for moving between states. Different types of automata, like finite-state machines and Turing machines, can recognize different kinds of languages or sets of input sequences.

Hierarchy in terms of powers

This section shows a hierarchy of different types of virtual machines based on their abilities. It explains which machines can handle more complex tasks than others. The hierarchy helps us understand how these machines are related and which ones are more powerful in terms of the problems they can solve.

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 fields. For example, finite automata help with tasks like processing text, building compilers, and designing hardware. Context-free grammar (CFGs) are important for programming languages and artificial intelligence, and they were originally studied to understand human languages.

Cellular automata are used in artificial life, with John Conway's famous Game of Life as a well-known example. Automata theory can also explain patterns in nature, like how some shells and pine cones grow and get their colors. Some scientists even suggest that the entire universe might work like a giant computer, an idea started by Konrad Zuse and later shared in America by Edward Fredkin. Automata are also useful in mathematics, such as studying finite fields and solving problems about irreducible polynomials and the induction of regular languages.

Automata simulators

Automata simulators are tools that help teach and learn about automata theory. These simulators let you input a description of an automaton, such as using a symbolic language, a special form, or by drawing its diagram with a mouse. Popular simulators include Turing's World, JFLAP, VAS, TAGS, and SimStudio. They show how the automaton works with different inputs.

Main article: symbolic language

Category-theoretic models

Automata theory includes ways to organize different types of machines into categories. These categories help us understand how machines can be transformed from one to another. For example, certain types of machines called deterministic automata can be organized into a special kind of category that allows for both combining and breaking down these machines in structured ways.

Variable automata, explored by mathematician Norbert Wiener, can also be grouped into categories. These categories help us study how these machines change and interact, forming groups or more complex structures. This approach provides a deeper look at the relationships between different kinds of automata.

Main article: The Human Use of Human Beings