Automata Theory Study Cards

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Automata Theory

A branch of computer science that deals with the study of abstract machines and computational models, including finite automata, regular expressions, context-free grammars, and Turing machines.

Finite Automaton

A mathematical model of computation that consists of a finite set of states, an input alphabet, a transition function, a start state, and a set of accepting states.

Regular Expression

A sequence of characters that defines a search pattern, used for matching strings in formal language theory and text processing.

Context-Free Grammar

A formal grammar in which production rules are of the form A -> α, where A is a nonterminal symbol and α is a string of terminals and/or nonterminals.

Pushdown Automaton

A type of automaton that extends the capabilities of a finite automaton by adding a stack, allowing it to recognize context-free languages.

Turing Machine

A theoretical computing device that can simulate the logic of any computer algorithm, using an infinite tape divided into cells and a read-write head.

Decidability

The property of a language or problem being solvable by an algorithm, where the algorithm always terminates and produces the correct output.

Undecidability

The property of a language or problem being unsolvable by an algorithm, where no algorithm can determine whether a given input belongs to the language or problem.

Computability Theory

The study of what can and cannot be computed by various computational models, including Turing machines and other abstract machines.

Formal Language

A set of strings over a given alphabet, defined by a formal grammar or automaton, used in the study of formal languages and automata theory.

Regular Language

A formal language that can be described by a regular expression or recognized by a finite automaton.

Context-Free Language

A formal language that can be described by a context-free grammar or recognized by a pushdown automaton.

Chomsky Hierarchy

A classification of formal grammars and languages into four levels, based on their generative power and the types of formal languages they can describe.

Pumping Lemma

A tool used in the theory of formal languages to prove that certain languages are not context-free or not regular.

Halting Problem

The problem of determining, given a description of a program and an input, whether the program will eventually halt or run forever.

Universal Turing Machine

A Turing machine that can simulate the logic of any other Turing machine, used to define the concept of computability.

Church-Turing Thesis

The hypothesis that any function that can be computed by an algorithm can be computed by a Turing machine, or equivalently, by any other computational model.

Regular Operation

An operation that can be performed on regular languages, such as union, concatenation, and Kleene star.

Context-Free Operation

An operation that can be performed on context-free languages, such as union, concatenation, and Kleene star.

Pumping Lemma for Regular Languages

A lemma used to prove that a language is not regular by showing that it cannot satisfy the pumping property.

Pumping Lemma for Context-Free Languages

A lemma used to prove that a language is not context-free by showing that it cannot satisfy the pumping property.

Non-deterministic Finite Automaton

A type of finite automaton where multiple transitions may be possible for a given input symbol and current state.

Deterministic Finite Automaton

A type of finite automaton where there is exactly one transition for each input symbol and current state.

Regular Grammar

A type of formal grammar where all production rules are of the form A -> aB or A -> ε, where A and B are nonterminal symbols and a is a terminal symbol.

Context-Sensitive Grammar

A type of formal grammar where production rules are of the form α -> β, where α and β are strings of terminals and/or nonterminals, and the length of α is less than or equal to the length of β.

Unrestricted Grammar

A type of formal grammar where production rules are of the form α -> β, where α and β are strings of terminals and/or nonterminals, and there are no restrictions on the lengths of α and β.

Regular Set

A set of strings that can be described by a regular expression or recognized by a finite automaton.

Context-Free Set

A set of strings that can be described by a context-free grammar or recognized by a pushdown automaton.

Context-Sensitive Set

A set of strings that can be described by a context-sensitive grammar or recognized by a linear-bounded automaton.

Recursively Enumerable Set

A set of strings that can be recognized by a Turing machine, where the machine may not halt for strings not in the set.

Decidable Language

A language for which there exists an algorithm that can determine, for any given input, whether the input belongs to the language or not.

Undecidable Language

A language for which no algorithm can determine, for any given input, whether the input belongs to the language or not.

Halting Language

The set of descriptions of Turing machines that halt on a given input.

Universal Language

The set of descriptions of Turing machines that accept a given input.

Formal System

A set of symbols, axioms, and rules for deriving new symbols, used in the study of formal languages and formal proofs.

Formal Proof

A sequence of statements, each of which is either an axiom or follows from previous statements using the rules of inference, used to establish the truth of a statement in a formal system.

Formal Proof System

A set of rules for deriving formal proofs in a formal system, including axioms, rules of inference, and rules for manipulating symbols.

Formal Proof Theory

The study of formal systems, formal proofs, and their properties, including consistency, completeness, and soundness.

Formal Logic

A branch of mathematics and philosophy that studies formal systems, formal proofs, and their applications in reasoning and computation.

Formal Language Theory

A branch of computer science and linguistics that studies formal languages, formal grammars, and their applications in computation and communication.

Formal Semantics

A branch of linguistics and computer science that studies the meaning of programming languages and natural languages using formal methods.

Formal Verification

A process of proving or disproving the correctness of a system or program using formal methods, such as formal proofs or model checking.

Formal Methods

A set of techniques and tools for specifying, designing, and verifying software and hardware systems using formal languages, formal proofs, and formal models.

Formal Specification

A precise description of the behavior and properties of a system or program using a formal language, used in formal methods and formal verification.

Formal Model

A mathematical or logical representation of a system or program, used in formal methods and formal verification to reason about its behavior and properties.

Formal Language Processor

A software tool or system that processes formal languages, such as compilers, interpreters, and parsers.

Formal Language Recognition

The process of determining whether a given string belongs to a formal language, using automata, grammars, or other formal models.

Formal Language Generation

The process of generating strings that belong to a formal language, using automata, grammars, or other formal models.

Formal Language Complexity

The study of the computational complexity of formal languages, including time complexity, space complexity, and other measures of complexity.

Formal Language Equivalence

The study of the equivalence of formal languages, including language containment, language equality, and other notions of language equivalence.

Formal Language Operations

The study of operations that can be performed on formal languages, such as union, intersection, concatenation, and Kleene star.

Formal Language Closure

The study of closure properties of formal languages, including closure under union, intersection, concatenation, Kleene star, and other operations.

Formal Language Pumping Lemma

A lemma used to prove that a language is not regular or not context-free by showing that it cannot satisfy the pumping property.

Formal Language Decision Problem

A problem of determining, for a given formal language, whether a given string belongs to the language or not.

Formal Language Membership Problem

A problem of determining, for a given formal language and string, whether the string belongs to the language or not.

Formal Language Equivalence Problem

A problem of determining, for two given formal languages, whether the languages are equivalent or not.

Formal Language Containment Problem

A problem of determining, for two given formal languages, whether one language is contained in the other or not.

Formal Language Intersection Problem

A problem of determining, for two given formal languages, whether the intersection of the languages is empty or not.

Formal Language Concatenation Problem

A problem of determining, for two given formal languages, whether the concatenation of the languages contains a given string or not.

Formal Language Kleene Star Problem

A problem of determining, for a given formal language and string, whether the string can be obtained by concatenating zero or more strings from the language.