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Turing machine that halts for any input
computability theory, a decider is a Turing machine that halts for every input. A decider is also called a total Turing machine as it represents a total
Decider_(Turing_machine)
Computation model defining an abstract machine
Church's work intertwined with Turing's to form the basis for the Church–Turing thesis. This thesis states that Turing machines, lambda calculus, and other
Turing_machine
Type of Turing machine
science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper
Universal_Turing_machine
Topics referred to by the same term
Maher: The Decider, a stand-up comedy special Decider (Turing machine), a Turing machine that eventually halts for every input "The Decider", a recurring
Decider
Ability of a computing system to simulate Turing machines
cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician
Turing_completeness
English computer scientist (1912–1954)
encrypted by the German Enigma machine. He also contributed the Turing test to philosophy of artificial intelligence and the Turing pattern concept to mathematical
Alan_Turing
Test of a machine's ability to imitate human intelligence
The Turing test was designed by Alan Turing to assess a machine's ability to exhibit intelligent behaviour equivalent to that of a human by imitating
Turing_test
Thesis on the nature of computability
numbers is called Turing computable if some Turing machine computes the corresponding function on encoded natural numbers. Turing proposed that effectively
Church–Turing_thesis
Problem in computer science
Chapter 7 "Turing Machines." A book centered around the machine-interpretation of "languages", NP-Completeness, etc. Hodges, Andrew (1983). Alan Turing: the
Halting_problem
Formal language in mathematics and computer science
Turing machine that decides the formal language. In theoretical computer science, such always-halting Turing machines are called total Turing machines or
Recursive_language
Finite-state machine
eliminating isomorphic automata. Read-only right-moving Turing machines are a particular type of Turing machine that only moves right; these are almost exactly
Deterministic finite automaton
Deterministic_finite_automaton
Operation in computability theory
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem X a successively
Turing_jump
1950 scientific paper by Alan Turing
Turing test to the general public. Turing's paper considers the question "Can machines think?" Turing says that since the words "think" and "machine"
Computing Machinery and Intelligence
Computing_Machinery_and_Intelligence
Models of computation
super-Turing computation is a set of hypothetical models of computation that can provide outputs that are not Turing-computable. For example, a machine that
Hypercomputation
Hypothetical computational model
Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that
Zeno_machine
Concept in computability theory
theory, a Turing reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyle B} is an oracle machine that decides problem
Turing_reduction
Abstract computation model
Turing machine (or to be more precise, the definition of acceptance for such a machine) alternates between these modes. An alternating Turing machine
Alternating_Turing_machine
2014 film by Morten Tyldum
the 1983 biography Alan Turing: The Enigma by Andrew Hodges. The film's title quotes the name of the game cryptanalyst Alan Turing proposed for answering
The_Imitation_Game
Concept in theoretical computer science
programs used in the game are n-state Turing machines, one of the first mathematical models of computation. Turing machines consist of an infinite tape, and
Busy_beaver
Formal language
recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable
Recursively enumerable language
Recursively_enumerable_language
1965:291) Turing 1937 in (Davis 1967:118) Turing 1937 in (Davis 1967:116) Turing 1937 in (Davis 1967:117) Turing 1937 in (Davis 1967:138) Turing 1937 in
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Mathematical model of abstract computation
2-state 5-symbol Turing machine, and conjectured that a particular 2-state 3-symbol Turing machine (hereinafter (2,3) Turing machine) might be universal
Wolfram's 2-state 3-symbol Turing machine
Wolfram's_2-state_3-symbol_Turing_machine
Measure of unsolvability
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures
Turing_degree
Proof by Alan Turing
Turing's proof is a proof by Alan Turing submitted on 12 November 1936 and first published in 1937 with the title "On Computable Numbers, with an Application
Turing's_proof
Study of computable functions and Turing degrees
(Turing) computable, or recursive function if there is a Turing machine that, on input n, halts and returns output f(n). The use of Turing machines here
Computability_theory
Impossible task in computing
calculability" based on his λ-calculus, and by Alan Turing the next year with his concept of Turing machines. Turing immediately recognized that these are equivalent
Entscheidungsproblem
Academic subfield of computer science
Description was given by Turing Award winner Stephen Cook. Aside from a Turing machine, other equivalent (see Church–Turing thesis) models of computation
Theory_of_computation
Intelligence in machines
8–17), Moravec (1988, p. 3) Turing's original publication of the Turing test in "Computing machinery and intelligence": Turing (1950) Historical influence
Artificial_intelligence
Mathematical function that can be computed by a program
including Turing machines General recursive functions Lambda calculus Post machines (Post–Turing machines and tag machines). Register machines Although
Computable_function
Inherent difficulty of computational problems
deterministic Turing machine, but many complexity classes are based on non-deterministic Turing machines, Boolean circuits, quantum Turing machines, monotone
Computational complexity theory
Computational_complexity_theory
Ability to solve a problem by an effective procedure
computability notions weaker than Turing machines are studied in automata theory, while computability notions stronger than Turing machines are studied in the field
Computability
Set of problems in computational complexity theory
Turing machine so that it is possible for the machine to store the entire input (it can be shown that in terms of computability the two-tape Turing machine
Complexity_class
Mathematical logic concept
computable. According to the Church–Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is computably enumerable
Computably_enumerable_set
Study of abstract machines and automata
different names by different research communities. The earlier concept of Turing machine was also included in the discipline along with new forms of infinite-state
Automata_theory
Whether a decision problem has an effective method to derive the answer
decision problem is decidable if there exists an effective method for deriving the correct answer. Logical systems are decidable if membership in their
Decidability_(logic)
Subset of artificial intelligence
Annotation Game: On Turing (1950) on Computing, Machinery, and Intelligence", in Epstein, Robert; Peters, Grace (eds.), The Turing Test Sourcebook: Philosophical
Machine_learning
Transformation of one computational problem to another
suppose R is a decider for E. We will use this to produce a decider S for H (which we know does not exist). Given input M and w (a Turing machine and some input
Reduction_(complexity)
Complexity class consisting of all recursive languages
of decision problems solvable by a Turing machine, which is the set of all recursive languages (also called decidable languages). R is equivalent to the
R_(complexity)
American computer scientist (born 1964)
the web. In 2005, he founded and became the director of the university's Turing Center. The center investigated problems in data mining, natural language
Oren_Etzioni
1952 puzzle video game
by Alan Turing starting in 1945. The project was delayed for many months, however, due to technical, political, and economic reasons, and Turing abandoned
Checkers_(video_game)
Sequence of operations for a task
size of inputs increase. Any algorithm can be computed by any Turing complete model. Turing completeness only requires four instruction types—conditional
Algorithm
Theorem in computability theory
theorem implies that in dynamically typed programming languages that are Turing-complete, it is impossible to verify the absence of type errors. On the
Rice's_theorem
Yes-or-no question that cannot ever be solved by a computer
input, decide whether the program finishes running or will run forever. Alan Turing proved in 1936 that a general algorithm running on a Turing machine that
Undecidable_problem
Complexity class used to classify decision problems
deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine. NP is the
NP_(complexity)
Branch of philosophy
reducing the complexity of Turing computable tasks and are still restricted to tasks within the scope of Turing machines. [citation needed] [clarification
Philosophy of artificial intelligence
Philosophy_of_artificial_intelligence
String rewriting system
p. 149 Post, following Turing, technically makes use of the undecidability of the printing problem (whether a Turing machine ever prints a particular
Semi-Thue_system
Deterministic model of computation
1, the set of m-tag systems is Turing-complete; i.e., for each m > 1, it is the case that for any given Turing machine T, there is an m-tag system that
Tag_system
Mathematical-logic system
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. Its namesake, the
Lambda_calculus
Unsolved problem in computer science
{\displaystyle \Sigma \cup \{\#\}} is decidable by a deterministic Turing machine in polynomial time. A Turing machine that decides LR is called a verifier for
P_versus_NP_problem
Real number that can be computed within arbitrary precision
between 0 and 1: A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of
Computable_number
Attempts to formalize the concept of algorithms
and Turing [1936]. For example, according to Savage [1987], an algorithm is a computational process defined by a Turing machine. Church and Turing did
Algorithm_characterizations
Class of computational complexity
PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. If we denote by S P A C E ( f (
PSPACE
Complexity class (logarithmic space)
tapes of a logspace Turing machine, the machine also takes a read-only one-way tape filled with random bits. The Turing machine has to halt for every
L_(complexity)
Measure of algorithmic complexity
encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring <M>. If M is a Turing Machine which, on
Kolmogorov_complexity
Algorithmic complexity class
the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2p(n)) time, where p(n) is a polynomial
EXPTIME
Computational input that relies on the length but not content of the input
computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length n of the input, but not on the
Advice_(complexity)
German mathematician (1912–2003)
the subject of mathematical logic. Hans Hermes was a pioneer of the Turing machine as the central concept of predictability. In 1937, Hermes reported under
Hans_Hermes
Memory space for a deterministic Turing machine
resource describing the resource of memory space for a deterministic Turing machine. It represents the total amount of memory space that a "normal" physical
DSPACE
Category of mathematical proof
doi:10.1112/plms/s2-43.6.544). This is the epochal paper where Turing defines Turing machines and shows that it (as well as the Entscheidungsproblem) is unsolvable
Proof_of_impossibility
Complexity class
solution in polynomial time by a deterministic Turing machine (or solvable by a non-deterministic Turing machine in polynomial time). NP-hard Class of problems
NP-hardness
Relation between deterministic and nondeterministic space complexity
if a nondeterministic Turing machine can solve a problem using f ( n ) {\displaystyle f(n)} space, a deterministic Turing machine can solve the same problem
Savitch's_theorem
British video game developer
first-person puzzler The Turing Test". Eurogamer.net. March 10, 2016. Retrieved September 17, 2026. Robertson, John (June 24, 2016). "The Turing Test: A puzzle
Bulkhead_(company)
Type of automaton
about what can be computed by machines. They are more capable than finite-state machines but less capable than Turing machines (see below). Deterministic
Pushdown_automaton
Given more time, a Turing machine can solve more problems
the Turing machine M. Let m be the size of the tuple ([M], x). We know that we can decide membership of Hf by way of a deterministic Turing machine R,
Time_hierarchy_theorem
List of concepts in artificial intelligence
1936 paper, A. M. Turing defined the class of abstract machines that now bear his name. A Turing machine is a finite-state machine associated with a special
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Type of computational algorithm
reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means the Turing machine can keep a constant number of pointers
Log-space_reduction
Lemma that defines a property of regular languages
{\displaystyle q_{s}} for such a state. The transitions that take the machine from the first encounter of state q s {\displaystyle q_{s}} to the second
Pumping lemma for regular languages
Pumping_lemma_for_regular_languages
Determination of whether a given program halts for each input
model of Turing machines as the model of programs implementing computable functions would have the goal of deciding whether a given Turing machine is a total
Termination_analysis
Computational problems no algorithm can solve
configuration). Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states
List_of_undecidable_problems
Method of comparing problems by transforming one into another in computability theory
{\displaystyle B} if A {\displaystyle A} is Turing reducible to B {\displaystyle B} via a single (oracle) Turing machine that produces a total function relative
Reduction (computability theory)
Reduction_(computability_theory)
Hypothesized risk to human existence
August 2025. Turing, Alan (1951). Intelligent machinery, a heretical theory (Speech). Lecture given to '51 Society'. Manchester: The Turing Digital Archive
Existential risk from artificial intelligence
Existential_risk_from_artificial_intelligence
problem. For Turing machines, the halting problem can be stated as follows: Given a Turing machine, and an input, decide whether the machine halts when
Mortality (computability theory)
Mortality_(computability_theory)
all input, is undecidable. Suppose we have a decider for it, D {\displaystyle D} . For any Turing machine M {\displaystyle M} and input w {\displaystyle
Computation_history
System for reasoning about vagueness
J. (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theoretical Computer Science. 317 (1–3):
Fuzzy_logic
Hypothesis in computational complexity theory
machine must be sufficiently powerful to emulate the sequential machine in time polynomially related to the sequential space; compare Turing machine,
Parallel_computation_thesis
Study of mathematical analysis seen through computability theory
Turing machines. The tape configuration and interpretation of mathematical structures are described as follows. A Type 2 Turing machine is a Turing machine
Computable_analysis
Two-dimensional cellular automaton
Conway. Theoretically, the Game of Life has the power of a universal Turing machine: anything that can be computed algorithmically can be computed within
Conway's_Game_of_Life
Type of pumping lemma
context-free Visibly pushdown Regular Star-free Finite Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton restricted Tree
Pumping lemma for context-free languages
Pumping_lemma_for_context-free_languages
British engineer and robotics researcher
Reading, which also featured parallel-paired Turing tests. In 2012, he co-organised with Huma Shah a series of Turing tests held at Bletchley Park. According
Kevin_Warwick
Randomized polynomial time class of computational complexity theory
the complexity class of decision problems for which a probabilistic Turing machine exists with these properties: It always runs in polynomial time in the
RP_(complexity)
Type of Turing reduction
enumerable sets that are neither decidable nor m-complete, and hence that there exist nonuniversal Turing machines whose individual halting problems
Many-one_reduction
1951 chess program
Digital Computing Machines. Sir Isaac Pitman and Sons, Ltd. Copeland, B. Jack; Turing, Alan Mathison (2004). The Essential Turing : Seminal Writings
Chess_(Dietrich_Prinz)
Class of problems solvable in polynomial time
contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's
P_(complexity)
Formal grammar
∩ L2, L1 ∪ L2, and L1 \ L2 are also regular tree languages, and it is decidable whether L1 ⊆ L2, and whether L1 = L2. Regular tree grammars are a generalization
Regular_tree_grammar
Estimate of time taken for running an algorithm
deterministic Turing machine in polynomial time NP: The complexity class of decision problems that can be solved on a non-deterministic Turing machine in polynomial
Time_complexity
Type of computational problem
it is FP, that is, computable in polynomial time by a Turing machine, since otherwise the machine will not have enough time to produce the full output
Counting_problem_(complexity)
Model of computation
single circuit (in contrast to the Turing machine model, in which a language is fully described by a single Turing machine). A language is instead represented
Boolean_circuit
Generative AI chatbot by OpenAI
Nature article that "ChatGPT broke the Turing test". Stanford researchers reported that GPT-4 "passes a rigorous Turing test, diverging from average human
ChatGPT
Computer science concept
within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical
Polynomial_hierarchy
Infinitely many tasks in finite time
hist-ph]. Hamkins, Joel David (November 2002). "Infinite Time Turing Machines". Minds and Machines. 12 (4): 521–539. arXiv:math/0212047. doi:10.1023/A:1021180801870
Supertask
Branch of computational complexity theory
nondeterministic Turing machine. The machine may be specified by one of any of the standard formulations. One usually considers the one-tape Turing machine, but the
Parameterized_complexity
Yes/no problem in computer science
field of recursion theory categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution
Decision_problem
Computational model used in machine learning
universal Turing machine, using a finite number of neurons and linear connections. Further, the use of irrational values for weights results in a machine with
Neural network (machine learning)
Neural_network_(machine_learning)
Discrete model of computation
thought to be computationally universal, or capable of simulating a Turing machine. Special types of cellular automata are reversible, where only a single
Cellular_automaton
Complexity class
polynomial p ( n ) {\displaystyle p(n)} and a polynomial-time bounded Turing machine M such that for every instance x, x is a no-instance if and only if:
Co-NP
Complexity class
2EXPTIME) is the set of all decision problems solvable by a deterministic Turing machine in O(22p(n)) time, where p(n) is a polynomial function of n. In terms
2-EXPTIME
British philosopher and chess player (born 1958)
Philosophical Significance of the Turing Machine and the Turing Test", in S. Barry Cooper and Jan van Leeuwen (eds) Alan Turing: His Work and Impact (2013)
Peter_Millican
Type of a context-free grammar
It is decidable whether a given grammar G is LL(k), but it is not decidable whether an arbitrary grammar is LL(k) for some k. It is also decidable if a
LL_grammar
class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages
NL-complete
Sequence of rational numbers
can be given using an enumeration of Turing machines. Let {n}(n) denote the action of the n-th Turing machine on the number n. Then let the n-th decimal
Specker_sequence
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DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
DECIDER TURING-MACHINE
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