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Concept in computability theory
In computability theory, a Turing reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyle B} is an oracle machine
Turing_reduction
Method for solving one problem using another
polynomial-time reduction, from the most to the least restrictive, are polynomial-time many-one reductions, truth-table reductions, and Turing reductions. The most
Polynomial-time_reduction
Type of Turing reduction
{\displaystyle L_{1}} . Many-one reductions are a special case and stronger form of Turing reductions. With many-one reductions, the oracle (that is, our solution
Many-one_reduction
English computer scientist (1912–1954)
algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father
Alan_Turing
Method of comparing problems by transforming one into another in computability theory
reducibility relation. For example, the Turing degrees are the equivalence classes of sets of naturals induced by Turing reducibility. The degrees of any reducibility
Reduction (computability theory)
Reduction_(computability_theory)
Concept from evolutionary biology
The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis", which describes
Turing_pattern
Biography by Andrew Hodges
Alan Turing: The Enigma (1983) is a biography of the British mathematician, codebreaker, and early computer scientist, Alan Turing (1912–1954) by Andrew
Alan_Turing:_The_Enigma
Concept in theoretical computer science
performed by a truth-table reduction, but every truth-table reduction can be performed by a Turing reduction.) A Turing reduction from a set B to a set A
Truth-table_reduction
Transformation of one computational problem to another
main types of reductions used in computational complexity theory, the many-one reduction and the Turing reduction. Many-one reductions map instances of
Reduction_(complexity)
Type of computational algorithm
log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means the Turing machine can
Log-space_reduction
Theorem about Turing reductions
mathematical logic, the Friedberg–Muchnik theorem is a theorem about Turing reductions that was proven independently by Albert Muchnik and Richard Friedberg
Friedberg–Muchnik_theorem
List
Turing OS Turing pattern Turing Pharmaceuticals Turing (programming language) Turing reduction Turing Robot, China Turing scheme Turing table Turing tarpit
List of things named after Alan Turing
List_of_things_named_after_Alan_Turing
Philosophical view explaining systems in terms of smaller parts
computability (or recursive) theory, where it assumes the form of e.g. Turing reduction, but also in the realm of real-world computation in time (or space)
Reductionism
Complexity class
term reduction was used in the technical meaning of a polynomial-time many-one reduction. Another type of reduction is polynomial-time Turing reduction. A
NP-completeness
Complexity class
non-deterministic Turing machine. The problem is #P-hard, meaning that every other problem in #P has a polynomial-time Turing reduction or polynomial-time
♯P-complete
Thesis on the nature of computability
computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise
Church–Turing_thesis
1950 scientific paper by Alan Turing
what is now known as the Turing test to the general public. Turing's paper considers the question "Can machines think?" Turing says that since the words
Computing Machinery and Intelligence
Computing_Machinery_and_Intelligence
Abstract machine used to study decision problems
of oracle Turing machines, as discussed below. The one presented here is from van Melkebeek (2003, p. 43). An oracle machine, like a Turing machine, includes:
Oracle_machine
Abstract calculator
Post machine or Post–Turing machine is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of
Post–Turing_machine
for an example. The definition of NP-easy uses a Turing reduction rather than a many-one reduction because the answers to problem Y are only TRUE or
NP-easy
American annual computer science prize
Staff (2014). "ACM's Turing Award prize raised to $1 million". Communications of the ACM. 57 (12): 20. doi:10.1145/2685372. "A. M. Turing Award". Association
Turing_Award
1952 scholarly article by Alan Turing
patterns have come to be known as Turing patterns. For example, it has been postulated that the protein VEGFC can form Turing patterns to govern the formation
The Chemical Basis of Morphogenesis
The_Chemical_Basis_of_Morphogenesis
Impact of English computer scientist
Institute Turing Lecture Turing machine Turing patterns Turing reduction Turing test Turing Award Various institutions have paid tribute to Turing by naming
Legacy_of_Alan_Turing
Study of computable functions and Turing degrees
Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post. The fundamental results the researchers obtained established Turing computability as the correct
Computability_theory
Unsolved problem in computational complexity theory
defining a new class GI, the set of problems with a polynomial-time Turing reduction to the graph isomorphism problem. If in fact the graph isomorphism
Graph_isomorphism_problem
Limit of a uniformly computable sequence of functions
therefore suffices to show that if limit computability is preserved by Turing reduction, as this will show that all sets computable from 0 ′ {\displaystyle
Computation_in_the_limit
Type of set in mathematics
i.e. sets whose Turing jump is computable from the Halting problem, and form a Turing ideal, i.e. class of sets closed under Turing join and closed downward
K-trivial_set
Turing machine Deterministic Turing machine Non-deterministic Turing machine Alternating automaton Alternating Turing machine Turing-complete Turing tarpit
List of computability and complexity topics
List_of_computability_and_complexity_topics
Inherent difficulty of computational problems
deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and
Computational complexity theory
Computational_complexity_theory
1938 doctoral thesis by Alan Turing
id.tue.nl/lecturenotes/DDM110%20CAS/Turing/Turing-1939%20Sysyems%20of%20logic%20based%20on%20ordinals.pdf "Turing's Princeton Dissertation". Princeton
Systems of Logic Based on Ordinals
Systems_of_Logic_Based_on_Ordinals
and of the complement is the set of all natural numbers. There is a Turing reduction from every problem to its complement problem. The complement operation
Complement_(complexity)
genetics". Turing's 1948 paper has been re-printed as Turing AM. Intelligent Machinery. In: Ince DC, editor. Collected works of AM Turing — Mechanical
Unorganized_machine
Mathematical-logic system based on functions
N]. 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,
Lambda_calculus
Mathematical proof about the permanent of matrices
completeness of 01-permanent, both used polynomial-time Turing reductions. In this kind of reduction, a single hard instance of some other problem in #P is
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
logspace (many-one) reduction, then L = P. Although not all languages in P/poly are sparse, there is a polynomial-time Turing reduction from any language
Sparse_language
The polynomial hierarchy is contained in probabilistic Turing machine in polynomial time
polynomial-time Turing reduction to a counting problem. An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real Turing machines)
Toda's_theorem
Japanese computer scientist
that every problem in the polynomial hierarchy has a polynomial-time Turing reduction to a counting problem. S. Toda Archived 2007-08-18 at the Wayback Machine
Seinosuke_Toda
Type of decision problem in computer science
different type. However, it is also possible to define completeness using Turing reductions, in which one problem can be solved in a polynomial number of calls
PSPACE-complete
Problem in computer science
problem considered in Turing's 1936 paper ("does a Turing machine starting from a blank tape ever print a given symbol?"). However, Turing equivalence is rather
Halting_problem
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 function
Decider_(Turing_machine)
Binary sequence
allowing laws of randomness that are Turing-computable. In other words, a sequence is random iff it passes all Turing-computable tests of randomness. The
Algorithmically random sequence
Algorithmically_random_sequence
Thought experiment on artificial intelligence
technology that could pass the Turing test. If Searle's room could pass the Turing test, but still does not have a mind, then the Turing test is not sufficient
Chinese_room
Problem transformation for counting solutions
non-deterministic Turing machine that runs in polynomial time, for which the output to the problem is the number of accepting paths of the Turing machine. Intuitively
Polynomial-time counting reduction
Polynomial-time_counting_reduction
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
Set of problems solved by small circuits
complexity, P/poly is defined in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend
P/poly
Combinatorial optimization problem
is precisely the difficult core of the NP-hard problems. Although a Turing reduction can get around this issue by trying all values of k. A simple greedy
Metric_k-center
Unsolved problem in computer science
deterministic polynomial-time Turing machine. Meaning, P = { L : L = L ( M ) for some deterministic polynomial-time Turing machine M } {\displaystyle
P_versus_NP_problem
Concept in computability theory
from Q, that compute functions from P. Mučnik reducibility Turing reducibility Reduction (computability) Hinman, Peter G. (2012). "A survey of Mučnik
Medvedev_reducibility
as the set of all Turing degrees [ Y ] {\displaystyle [Y]} such that X ≤ T Y {\displaystyle X\leq _{T}Y} ; that is, the set of Turing degrees that are
Martin_measure
Reflexive and transitive binary relation
called asymptotic equivalence. Polynomial-time, many-one (mapping) and Turing reductions are preorders on complexity classes. Subtyping relations are usually
Preorder
Theory of a quantum origin of consciousness
Orchestrated objective reduction (Orch OR) is a controversial theory postulating that consciousness originates at the quantum level inside neurons (rather
Orchestrated objective reduction
Orchestrated_objective_reduction
Set of problems in computational complexity theory
"Other models of computation"), the Turing machine is used to define most basic complexity classes. With the Turing machine, instead of using standard
Complexity_class
Infinite sequence of numbers satisfying a linear equation
{\displaystyle -s_{n}} (because the answer must be negated, this is a Turing reduction). The Skolem-Mahler-Lech theorem would provide answers to some of these
Constant-recursive_sequence
Academic subfield of computer science
Several models exist for this purpose, such as the Turing machine. Computer scientists study the Turing machine because it is simple to formulate, can be
Theory_of_computation
Complexity class
and randomized black-box Turing reductions between the associated search problems, this rules out the corresponding reduction from the t {\displaystyle
PPP_(complexity)
Subroutine for testing independence
elements of the matroid; in complexity-theoretic terms, this is a Turing reduction. Two oracles are said to be polynomially equivalent if they are polynomially
Matroid_oracle
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
NP-hardness
in the phonology of consonant clusters. The H-cluster reductions are various consonant reductions that have occurred in the history of English, involving
Phonological history of English consonant clusters
Phonological_history_of_English_consonant_clusters
Class in computational complexity theory
a general simulation of a sequential computer (i.e. the Turing machine simulation of Turing machines), then we will be able to parallelize any program
P-complete
in the same way that Turing reducibility relates to μ-recursiveness. Turing reduction Many-one reduction Truth-table reduction Arithmetical hierarchy
Enumeration_reducibility
Measure of algorithmic complexity
the Turing-machine model and the arithmetic model: In the arithmetic model, every real number requires a single memory cell, whereas in the Turing model
Strongly-polynomial_time
American computer scientist (born 1943)
For his contributions to database research, Stonebraker received the 2014 Turing Award, often described as "the Nobel Prize for computing." Stonebraker's
Michael_Stonebraker
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
Framework for studying interactive computational tasks through logic
⊓x(p(x)⊔¬p(x))⟜⊓x(q(x)⊔¬q(x)) expresses the problem of Turing-reducing q to p (in the sense that q is Turing reducible to p if and only if the interactive problem
Computability_logic
1989 book by Roger Penrose
to The Emperor's New Mind. Alan Turing Anathem Church–Turing thesis Mind–body dualism Orchestrated objective reduction Quantum mind Raymond Smullyan Shadows
The_Emperor's_New_Mind
1984 science fiction novel by William Gibson
Neuromancer, but because of the severe restrictions placed on AI programs by the Turing Registry, it cannot achieve this on its own. It directed Armitage to recruit
Neuromancer
Algorithmic complexity class
solved by an alternating Turing machine in polynomial space. This is one way to see that PSPACE ⊆ EXPTIME, since an alternating Turing machine is at least
EXPTIME
Unsolved problem in structural complexity theory
Mahaney's theorem. Even for a relaxed definition of NP-completeness using Turing reductions, the existence of a sparse NP-complete language would imply an unexpected
Berman–Hartmanis_conjecture
Mathematical model of computation
computation such as the Turing machine. The computational power distinction means there are computational tasks that a Turing machine can do but an FSM
Finite-state_machine
Book by Stephen Wolfram
new technical result in describing the Turing completeness of the Rule 110 cellular automaton. Very small Turing machines can simulate Rule 110, which
A_New_Kind_of_Science
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
polynomial time by a deterministic Turing machine. (The statements verifiable in polynomial time by a deterministic Turing machine and solvable in polynomial
Cook–Levin_theorem
Contractual transaction on a decentralized platform
and proposed a stronger version based on the Solidity language, which is Turing complete. Since then, various cryptocurrencies have supported programming
Smart_contract
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
Computability
Argument that leads to a logical absurdity
In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical
Reductio_ad_absurdum
Thesis in the philosophy of mind
Machines 19(2): 181-197. Turing, A. (1950). Computing machinery and intelligence. Mind 59. In J. Copeland (Ed.), The essential Turing (pp. 433–460). Oxford:
Multiple_realizability
Claim that human mathematicians are not describable as formal proof systems
is not a computation of a Turing machine, and thus not an effective procedure; or it is a product of an inconsistent Turing Machine that could be reasoning
Penrose–Lucas_argument
Intelligence of 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
1997 book by David Deutsch
falsified. Alan Turing's theory of computation, especially as developed in Deutsch's "Turing principle", where Turing's Universal Turing machine is replaced
The_Fabric_of_Reality
Quantum measurement phenomenon
quantum mechanics, frequent measurements cause the quantum Zeno effect, a reduction in transitions away from the system's initial state, slowing a system's
Quantum_Zeno_effect
If there is a polynomial time algorithm for unambiguous-SAT, then NP equals RP
The promise problem Unambiguous-SAT can be decided by a nondeterministic Turing machine that has at most one accepting computation path, thus it belongs
Valiant–Vazirani_theorem
log-space Turing machine M that accepts a language in NL. Since there is only logarithmic space on the work tape, all possible states of the Turing machine
St-connectivity
British theoretical physicist (born 1953)
falsification. Alan Turing's theory of computation, especially as developed in Deutsch's Turing principle, in which the Universal Turing machine is replaced
David_Deutsch
Computer architecture where code and data share a common bus
to Turing—in so far as not anticipated by Babbage.... Both Turing and von Neumann, of course, also made substantial contributions to the "reduction to
Von_Neumann_architecture
147 (2007), pages 187-227. G. Japaridze, "The logic of interactive Turing reduction". Journal of Symbolic Logic 72 (2007), pages 243-276. G. Japaridze
Giorgi_Japaridze
Complexity class (logarithmic space)
can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space. Formally, the Turing machine has two tapes, one of which
L_(complexity)
Natural language processing computer program
recent Turing test study". Ars Technica. Retrieved December 3, 2023. Jones, Cameron R.; Bergen, Benjamin K. (April 20, 2024), Does GPT-4 pass the Turing test
ELIZA
Class of computational complexity
{\displaystyle {\mathsf {PSPACE}}} , because a deterministic Turing machine can simulate a nondeterministic Turing machine while roughly squaring the amount of space
PSPACE
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
Co-NP
Programmable machine that processes data
of the modern computer was proposed by Alan Turing in his seminal 1936 paper, On Computable Numbers. Turing proposed a simple device that he called "Universal
Computer
String rewriting system
process exactly follows the run of the Turing machine encoded. This proves that string rewrite systems are Turing complete. The reason for having two halted
Semi-Thue_system
Parallel computing platform and programming model
"Dissecting the NVidia Turing T4 GPU via Microbenchmarking". arXiv:1903.07486 [cs.DC]. Burgess, John (2019). "RTX ON – The NVIDIA TURING GPU". 2019 IEEE Hot
CUDA
Conversational software
1950, Alan Turing published an article entitled "Computing Machinery and Intelligence" in which he proposed what is now called the Turing test as a criterion
Chatbot
Book by Roger Penrose
"warm and wet" quantum processes have been discovered. Alan Turing, creator of the Turing test Quantum mind "Minds, Machines and Gödel". Archived from
Shadows_of_the_Mind
Computer programming language
0). The operational semantics of BCL, apart from eta-reduction (which is not required for Turing completeness), may be very compactly specified by the
Binary_combinatory_logic
Type of computational problem
of an nondeterministic Turing machine running in polynomial time. Just as NP has NP-complete problems via many-one reductions, #P has #P-complete problems
Counting_problem_(complexity)
the 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
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
Kolmogorov_complexity
Theoretical model of computation
is an abstract machine. As an abstract machine, it shares features with Turing machines and the SECD machine. The Krivine machine explains how to compute
Krivine_machine
Class of problems in computer science
using only deterministic Turing machines. A language L is in PP if and only if there exists a polynomial p and deterministic Turing machine M, such that M
PP_(complexity)
Replacing subterm in a formula with another term
rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects,
Rewriting
TURING REDUCTION
TURING REDUCTION
Boy/Male
Hindu, Indian
A Thought
Male
Welsh
Welsh name derived from the element aur, EURIG means "gold."
Surname or Lastname
English
English : habitational name from places in Oxfordshire and West Sussex named Goring, from Old English GÄringas ‘people of GÄra’, a short form of the various compound names with the first element gÄr ‘spear’.German (Göring) : see Goering.
Surname or Lastname
English and French
English and French : from an Anglo-Norman French form of the Old Norse personal name þórfinnr, composed of the elements þórr, the name of the god of thunder in Scandinavian mythology (see Thor) + the ethnic name Finnr ‘Finn’. This may have absorbed another name, Turpius, Turpinus (from Latin turpis ‘ugly’, ‘base’), one of the self-abasing names adopted as a mark of humility by the early Christians. It was borne by the archbishop of Rheims in the Charlemagne legend.A Turpin of unknown geographic origin is documented in Montreal in 1681.
Surname or Lastname
English
English : patronymic from Dear 1.German (Döring) : see Doering.
Surname or Lastname
English
English : variant of Darling.
Girl/Female
Indian
Lively, Entertainer, From a stream or a Spring, The Spring season, The Spring season
Surname or Lastname
English
English : patronymic from Dear 1.German : probably a variant of Döring (see Doering).
Surname or Lastname
German and Jewish (Ashkenazic)
German and Jewish (Ashkenazic) : from Middle High German hærinc ‘herring’, German Hering, a nickname for someone supposedly resembling a herring or a metonymic occupational name for a fish seller. In some cases the Jewish surname is ornamental.English : variant spelling of Herring.
Female
English
English name derived from the season name, "spring," (Mar. 21 thru Jun. 21), derived from the verb spring, "to burst forth," from Proto-Indo-European *sprengh-, SPRING means "rapid movement."Â
Surname or Lastname
English
English : of uncertain origin. Early examples, as for example William Spring (Yorkshire 1280), all point to a personal name or nickname, perhaps going back to an Old English byname derived from the verb springan ‘to jump or leap’ (see Springer 1). Alternatively, it could be a topographic name from Middle English spring ‘young wood’, ‘spring’. Compare Springer. Reaney derives the surname from the word denoting the season, although the word is not attested in this sense until the 16th century, the usual Middle English word being lenten. Compare Lenz. The surname has also been established in Ireland (County Kerry) for several centuries.German : from Middle High German sprinc, Middle Low German sprink ‘spring’, ‘well’, hence a topographic name for someone who lived by a spring or well, or habitational name from Springe near Hannover.Jewish (Ashkenazic) : variant of Springer.John Spring emigrated from England and settled in Watertown, MA, in 1634.
Female
English
Elaborated form of English Tara, TARINA means "hill."
Boy/Male
Indian
Loving, Caring, Daring
Surname or Lastname
English
English : perhaps be a nickname from Middle English daring ‘trembling’, ‘crouching or transfixed with fear’.
Surname or Lastname
English
English : from a pet form of the personal name Hugh.
Surname or Lastname
English
English : ethnic name from Old French Lohereng ‘man from Lorraine’ (see Lorraine).
Boy/Male
Muslim/Islamic
Loving Caring, Daring
Boy/Male
Muslim
Loving, Caring, Daring
Surname or Lastname
English (Kent)
English (Kent) : unexplained.Possibly an altered spelling of the German surname Dulling, which is likewise unexplained.
Girl/Female
American, Australian, Bengali, British, Christian, English, Indian
Springtime; Spring Season; Rapid Movement
TURING REDUCTION
TURING REDUCTION
Surname or Lastname
English (Derbyshire)
English (Derbyshire) : unexplained; possibly a habitational name from a lost or unidentified place.
Female
Babylonian
, a sea-goddess.
Boy/Male
Hindu
Lord of victory, Brilliant
Girl/Female
Arabic, Muslim
Kind
Female
English
Feminine form of English unisex Kimberley, KIMBERLY means "King's City Meadow."
Girl/Female
Latin
Goddess of Rome.
Biblical
the seat, alteration, or captivity of Jehovah
Boy/Male
Hindu
Supporting
Boy/Male
Bengali, Hindu, Indian, Malayalam, Marathi
Solitary
Female
Egyptian
, the wife of Toti.
TURING REDUCTION
TURING REDUCTION
TURING REDUCTION
TURING REDUCTION
TURING REDUCTION
n.
The chips or fragments made by boring.
n.
An instrument turning on a center, for boring holes. See Bit, n., 3.
n.
A tiring-room.
n.
A hole made by boring.
n.
Boldness; fearlessness; adventurousness; also, a daring act.
n.
The pieces, or chips, detached in the process of turning from the material turned.
n.
A series of tubes; tubes, collectively; a length or piece of a tube; material for tubes; as, leather tubing.
a.
Consuming; intense; inflaming; exciting; vehement; powerful; as, burning zeal.
n.
A line for hauling the reef cringle to the yard; -- also called reef earing.
n.
A variety of the domestic pigeon remarkable for its habit of tumbling, or turning somersaults, during its flight.
n.
Alt. of Goring cloth
n.
An exposure to air, or to a fire, for warming, drying, etc.; as, the airing of linen, or of a room.
prep.
In the time of; as long as the action or existence of; as, during life; during the space of a year.
n.
The act or state of that which curls; as, the curling of smoke when it rises; the curling of a ringlet; also, the act or process of one who curls something, as hair, or the brim of hats.
a.
Bold; fearless; adventurous; as, daring spirits.
n.
A line used to fasten the upper corners of a sail to the yard or gaff; -- also called head earing.
n.
The act or process of one who, or that which, bores; as, the boring of cannon; the boring of piles and ship timbers by certain marine mollusks.
n.
An obscure road; a way turning from the main road.