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Yes-or-no question that cannot ever be solved by a computer
computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an
Undecidable_problem
Computational problems no algorithm can solve
In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not
List_of_undecidable_problems
Problem in computer science
in 1937 that the halting problem is undecidable, meaning that no general algorithm exists that can correctly solve the problem for all possible program–input
Halting_problem
Undecidable decision problem introduced by Emil Post
correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the
Post_correspondence_problem
Impossible task in computing
decidabilities. On the top are the undecidable problems. Below it are the decidable problems. Furthermore, the decidable problems can be divided into a complexity
Entscheidungsproblem
Square tiles with a color on each edge
halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's
Wang_tile
problems in mathematics List of undecidable problems List of NP-complete problems List of PSPACE-complete problems List of problems in loop theory and quasigroup
Lists_of_problems
On solvability of Diophantine equations
answering that question. In modern terms, Hilbert's tenth problem is an undecidable problem. In a Diophantine equation, there are two kinds of variables:
Hilbert's_tenth_problem
Yes/no problem in computer science
accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational complexity theory
Decision_problem
Unsolved problem in computer science
Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. They cannot
P_versus_NP_problem
Limitative results in mathematical logic
answers every question in the problem set (see undecidable problem). Because of the two meanings of the word undecidable, the term independent is sometimes
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
from this set. The matrix mortality problem is known to be undecidable when n ≥ 3. In fact, it is already undecidable for sets of 6 matrices (or more) when
Matrix_mortality_problem
Complexity class
as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult
NP-hardness
Topics referred to by the same term
Look up undecidable or undecidability in Wiktionary, the free dictionary. Undecidable may refer to: Undecidable problem in computer science and mathematical
Undecidable
Abstract machine used to study decision problems
any complexity class, or it can even be an undecidable problem such as the halting problem. If another problem R ′ {\displaystyle R'} is reducible to
Oracle_machine
Problem of deciding whether an expression equals zero
proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x1, ..., xn are real numbers, then there is an
Constant_problem
Whether a decision problem has an effective method to derive the answer
arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining
Decidability_(logic)
Problem in finite group theory
of G {\displaystyle G} . The word problems for certain groups provide well-known examples of undecidable problems. If A {\displaystyle A} is a finite
Word_problem_for_groups
Theorem in computability theory
nor false for every program. The theorem generalizes the undecidability of the halting problem. It has far-reaching implications on the feasibility of
Rice's_theorem
Computational problem in algebraic topology
whether it is homeomorphic to another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more. An abstract simplicial complex
Simplicial complex recognition problem
Simplicial_complex_recognition_problem
Form of source code, without regard to meaning
distinction between parsing and execution, and makes syntax analysis an undecidable problem in these languages, meaning that the parsing phase may not finish
Syntax (programming languages)
Syntax_(programming_languages)
Rule system for formal languages
to this problem from the well-known undecidable problem of determining whether a Turing machine accepts a particular input (the halting problem). The reduction
Context-free_grammar
which is undecidable for certain alternating multi-head finite automata over single-letter alphabets. The emptiness problem is undecidable for context-sensitive
Emptiness_problem
Problem that can be possibly solved via mathematics
unsolvable are so-called undecidable problems, such as the halting problem for Turing machines. Some well-known difficult abstract problems that have been solved
Mathematical_problem
Decision problem
ranging over all finitely presented groups, are undecidable. In the case of the isomorphism problem, this means that there does not exist a computer
Group_isomorphism_problem
written on it. Philip K. Hooper proved in 1966 that the mortality problem is undecidable. This is true both for a machine with a tape infinite in both directions
Mortality (computability theory)
Mortality_(computability_theory)
semantic properties of programs are undecidable. The theorem has the immediate consequence that it is an undecidable problem to determine if two lambda terms
Scott–Curry_theorem
Energy difference between ground and first excited states
correlations. In 2015, it was shown that the problem of determining the existence of a spectral gap is undecidable in two or more dimensions. The authors used
Spectral_gap_(physics)
Philosophical idea of things impossible to know
remain unknowable. Modern inquiry encompasses undecidable problems and questions such as the halting problem, which in their very nature cannot be possibly
Unknowability
Question in abstract algebra
the first purely algebraic problem to be proved undecidable. Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum
Whitehead_problem
Open problem on 3x+1 and x/2 functions
proved that the problem Given g and n, does the sequence of iterates gk(n) reach 1? is undecidable, by representing the halting problem in this way. Closer
Collatz_conjecture
Ability of a computing system to simulate Turing machines
the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle
Turing_completeness
Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2
Richardson's_theorem
Topics referred to by the same term
mathematical logic Decidable problem and Undecidable problem Gödel's incompleteness theorem, a theorem on the undecidability of languages consisting of
Decidability
Transformation of one computational problem to another
Likewise, a reduction computing a noncomputable function can reduce an undecidable problem to a decidable one. As Michael Sipser points out in Introduction
Reduction_(complexity)
Computational input that relies on the length but not content of the input
It also contains some undecidable problems, such as the unary version of every undecidable problem, including the halting problem. Because of that, it
Advice_(complexity)
Number representing a continuous quantity
computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that
Real_number
Theory in fundamental physics
case of superinformation. Calculating Space Computability theory Undecidable problem Quantum circuit Generalized probabilistic theory Heaven, Douglas
Constructor_theory
Sequence of operations for a task
his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result). Kleene
Algorithm
Pattern that has no predecessors
by an efficient algorithm, but for higher dimensions this is an undecidable problem. Nevertheless, computer searches have succeeded in finding these
Garden of Eden (cellular automaton)
Garden_of_Eden_(cellular_automaton)
run continuously or terminate. It is widely accepted as an undecidable problem (a problem with a solution that cannot be found through an algorithm)
Philosophy of computer science
Philosophy_of_computer_science
Complexity class
problems: The domino problem for Wang tiles. The satisfiability problem for first-order logic. Knuth–Bendix completion algorithm List of undecidable problems
RE_(complexity)
Real number that can be computed within arbitrary precision
including: any number that encodes the solution of the halting problem (or any other undecidable problem) according to a chosen encoding scheme. Chaitin's constant
Computable_number
Computer science concept
correctness at the cost of making the type checking itself an undecidable problem (as in the halting problem). In a type system with automated type checking, a program
Type_system
Ability to solve a problem by an effective procedure
not recursive. The halting problem is therefore called non-computable or undecidable. An extension of the halting problem is called Rice's theorem, which
Computability
Class of problems solvable in polynomial time
contains some impractical problems, including some undecidable problems such as the unary version of any undecidable problem. In 1999, Jin-Yi Cai and D
P_(complexity)
Language for controlling a computer
programmer to alter the behavior of the parser make syntax analysis an undecidable problem, and generally blur the distinction between parsing and execution
Programming_language
1931 paper by Kurt Gödel
Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Question in theoretical computer science
Subsequently, the problem was shown to lie in TOWER, the least non-elementary complexity class. It becomes an undecidable problem for pushdown automata
Equivalence_problem
Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences
Trakhtenbrot's_theorem
Decision problem pertaining to equivalence of expressions
the word problem for groups, but there are many other instances as well. Some deep results of computational theory concern the undecidability of this question
Word_problem_(mathematics)
Problem-solving procedures with certain characteristics
(logic) Decision problem Effective results in number theory Function problem Model of computation Recursive set Undecidable problem Hunter, Geoffrey (1996)
Effective_method
Formal language in mathematics and computer science
decision problem is decidable by exhibiting a Turing machine running an algorithm that terminates on all inputs. An undecidable problem is a problem that
Recursive_language
Set of problems in computational complexity theory
{P}}\subsetneq {\textsf {P/poly}}} (for example, there are some undecidable problems that are in P/poly). P/poly has a number of properties that make
Complexity_class
Study of computable functions and Turing degrees
mathematical propositions are true or false. Many problems in mathematics have been shown to be undecidable after these initial examples were established
Computability_theory
American mathematician (1919–1985)
"Definability and Decision Problems in Arithmetic". Her dissertation showed that the theory of the rational numbers was an undecidable problem, by demonstrating
Julia_Robinson
Sequence of words formed by specific rules
an early example of an undecidable problem. Post would later use this paper as the basis for a 1947 proof "that the word problem for semigroups was recursively
Formal_language
String rewriting system
notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was obtained independently
Semi-Thue_system
Type of mathematical set
recognition problem is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is undecidable for any
Simplicial_complex
2002 phrase from Donald Rumsfeld
Johari window Knightian uncertainty Outside Context Problem Russell's teapot Undecidable problem Wild card (foresight) "Defense.gov News Transcript: DoD
There_are_unknown_unknowns
Moreover, determining if ρ ≤ 1 {\displaystyle \rho \leq 1} is an undecidable problem. Nevertheless, in recent years much progress has been done on its
Joint_spectral_radius
Problem a computer might be able to solve
solving a given problem will require, and explain why some problems are intractable or undecidable. Solvable computational problems belong to complexity
Computational_problem
Problem on words in group theory
and genus 1 cases being trivial). It is known that the conjugacy problem is undecidable for many classes of groups. Classes of group presentations for which
Conjugacy_problem
Turing machine that halts for any input
determining whether it is a decider is an undecidable problem. This is a variant of the halting problem, which asks for whether a Turing machine halts
Decider_(Turing_machine)
Mathematical logic concept
poset is a lattice. The theory of this lattice is known to be an undecidable problem. Similarly, the set of all computably enumerable vector spaces also
Computably_enumerable_set
Structure of a formal language
Press, p. 233, ISBN 9781466513457. For more on this subject, see undecidable problem. Chomsky, Noam (Sep 1956). "Three models for the description of language"
Formal_grammar
Subfield of mathematics
problem, a result with far-ranging implications in both recursion theory and computer science. There are many known examples of undecidable problems from
Mathematical_logic
Logical problem studied in computer science
satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Researchers study which theories or subsets
Satisfiability modulo theories
Satisfiability_modulo_theories
Undecidability theorem in group theory
"reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adyan (1955) and, independently, Michael
Adian–Rabin_theorem
Overview of and topical guide to algorithms
NP-completeness NP-hardness EXPTIME PSPACE BPP (complexity) BQP Undecidable problem Halting problem Rice's theorem No free lunch theorem List of algorithms List
Outline_of_algorithms
Algorithmic complexity class
hierarchy theorem. In computability theory, one of the basic undecidable problems is the halting problem: deciding whether a deterministic Turing machine (DTM)
EXPTIME
Computer hardware technology that uses quantum mechanics
computability. This means that quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove
Quantum_computing
theory, ALL is the class of all decision problems. ALL contains all of the complex classes of decision problems, including RE and co-RE, and uncountably
ALL_(complexity)
Computation model defining an abstract machine
Davis, Martin, ed. (2004) [1965]. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Mineola, NY:
Turing_machine
Product Puzzle", which is not impossible -gry, a word puzzle List of undecidable problems, no algorithm can exist to answer a yes–no question about the input
List_of_impossible_puzzles
Structure in computing
every possible run of the program. The exact static call graph is an undecidable problem, so static call graph algorithms are generally overapproximations
Call_graph
Consistency of the axioms of arithmetic
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent
Hilbert's_second_problem
Concept in computability theory
undecidable problem for which no algorithm exists. M. Davis, ed., 1965. The Undecidable—Basic Papers on Undecidable Propositions, Unsolvable Problems
Turing_reduction
Millennium Prize Problem
The general problem of determining the presence of a mass gap (a special case of a spectral gap) in a system is known to be undecidable, meaning no computer
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Value approached by a mathematical object
modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. There are
Limit_(mathematics)
Conformance of AI to intended objectives
Nei Y.; Castro, Paulo A. L. (2025). "Machines that halt resolve the undecidability of artificial intelligence alignment". Scientific Reports. 15 (1): 15591
AI_alignment
Problem in number theory
this problem was used by Bjorn Poonen as the opening example in a survey on undecidable problems in number theory, of which Hilbert's tenth problem is the
Sums_of_three_cubes
2009 book by Robert Hearn and Erik Demaine
discovery that optimal play in certain multiplayer games can be an undecidable problem. A third part of the book provides a compendium of known hardness
Games, Puzzles, and Computation
Games,_Puzzles,_and_Computation
German-American mathematician (1878–1952)
Non-Archimedean ordered field Scissors congruence Two ears theorem Undecidable problem The story of his travel in 1940 from Norway via Stockholm, Moscow
Max_Dehn
theorem Computable model theory Tarski's exponential function problem Undecidable problem Institutional model theory Institution (computer science) Non-standard
List of mathematical logic topics
List_of_mathematical_logic_topics
Notation techniques for grammars in computer science
enumerable languages, which makes parsing impossible in general: it is an undecidable problem to decide whether a given string can be generated by a given W-grammar
Van_Wijngaarden_grammar
it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties
Hopfian_group
Overview of and topical guide to logic
of the Church–Turing thesis Lambda calculus List of undecidable problems Post correspondence problem Post's theorem Primitive recursive function Recursion
Outline_of_logic
Model of computation
(i.e. P ⊊ {\displaystyle \subsetneq } P/poly) because there are undecidable problems that are in P/poly. P/poly turns out to have a number of properties
Boolean_circuit
Algorithm to be run on quantum computers
superposition or quantum entanglement. Problems that are undecidable using classical computers remain undecidable using quantum computers. What makes quantum
Quantum_algorithm
Study of abstract machines and automata
29–122. Section 4.1: Decidable Languages, pp. 152–159. Section 5.1: Undecidable Problems from Language Theory, pp. 172–183. Elaine Rich (2008). Automata,
Automata_theory
Models of computation
that encode the solution to the halting problem) as an input can solve a large number of useful undecidable problems; a system granted an uncomputable random-number
Hypercomputation
is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Logical formalism using combinators instead of variables
shown in a similar way as for the corresponding problems for lambda terms. The undecidable problems above (equivalence, existence of normal form, etc
Combinatory_logic
Model of computation
FIFO overflows. This is however not possible for all KPNs. It is an undecidable problem to test whether a KPN is strictly bounded by b {\displaystyle b}
Kahn_process_networks
Certain kind of modal formula
print. with corr.). Cambridge Univ. Press. L. A. Chagrova, 1991. An undecidable problem in correspondence theory. Journal of Symbolic Logic 56:1261–1272
Sahlqvist_formula
Complexity class used to classify decision problems
Unsolved problem in computer science P = ? N P {\displaystyle {\mathsf {P\ {\overset {?}{=}}\ NP}}} More unsolved problems in computer science In
NP_(complexity)
Category of mathematical proof
found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm
Proof_of_impossibility
Finite-state machine with fifo buffers for memory
{\displaystyle (c,w_{1},\dots ,w_{m})\in R(S)} . Most problem related to perfect channel system are undecidable. This is due to the fact that such a machine may
Channel system (computer science)
Channel_system_(computer_science)
Concept in theoretical computer science
that Σ is not a computable function. Moreover, this implies that it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy
Busy_beaver
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
Boy/Male
Arabic, Indian, Muslim
Problem Solver
Girl/Female
Indian, Telugu
Destroyer of Problems
Boy/Male
Muslim
Problem solver
Boy/Male
Hindu, Indian
Problem
Boy/Male
Indian, Tamil
People with this Name are Preferably Intelligent and Very Generous; Highly Knowledgeable in Problem Solving Skills
Boy/Male
Hindu, Indian, Sanskrit
And the Undesirable; The Unpleasant
Girl/Female
Muslim/Islamic
Away from all Problems
Girl/Female
Bengali, Indian
Eternity; Problem Solver
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
Boy/Male
Indian, Punjabi, Sikh
Religious Holy Virtues
Girl/Female
Hindu, Indian, Tamil, Telugu
Goddess Durga; Who Wears Sindhoor
Boy/Male
Indian
Old Arabic name
Girl/Female
Hindu, Indian
Goddess Laxmi
Boy/Male
Tamil
Lord Shiva, Auspicious, Lucky, Always pure
Girl/Female
British, English
Noble Maiden
Male
English
English form of French Eustache, EUSTACE means "fruitful."
Boy/Male
English
He who holds Christ in his heart. Famous Bearers: actors Christopher Plummer and Christopher...
Girl/Female
Muslim
The Arabic letter m, Mim (1)
Boy/Male
Tamil
Embellishment, To be charming
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
UNDECIDABLE PROBLEM
v. t.
To have just and adequate ideas of; to apprehended the meaning or intention of; to have knowledge of; to comprehend; to know; as, to understand a problem in Euclid; to understand a proposition or a declaration; the court understands the advocate or his argument; to understand the sacred oracles; to understand a nod or a wink.
a.
Undesirable.
n.
One who proposes problems.
a.
To rescue from something undesirable or hurtful; to prevent from doing something; to spare.
a.
Unobjectionable; unquestionably excellent; as, a person of undeniable connections.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
a.
Not declinable; not varied by inflective terminations; as, nihil (nothing), in Latin, is an indeclinable noun.
adv.
Those which have the value of independent words, inasmuch as the simple words are either not used at all, or are rarely, or at least much less frequently, used; as, unavoidable, unconscionable, undeniable, unspeakable, unprecedented, unruly, and the like; or inasmuch as they are used in a different sense from the usual meaning of the primitive, or especially in one of the significations of the latter; as, unaccountable, unalloyed, unbelieving, unpretending, unreserved, and the like; or inasmuch as they are so frequently and familiarly used that they are hardly felt to be of negative origin; as, uncertain, uneven, and the like.
a.
Not refragable; not to be gainsaid or denied; not to be refuted or overthrown; unanswerable; incontestable; undeniable; as, an irrefragable argument; irrefragable evidence.
a.
Not deniable; incapable of denial; palpably true; indisputable; obvious; as, undeniable evidence.
a.
Capable of being decided; determinable.
a.
Not decimable, or liable to be decimated; not liable to the payment of tithes.
a.
Alt. of Problematical
n.
A problem of more than usual difficulty added to another on an examination paper.
n.
An indeclinable word.
a.
Having the nature of a problem; not shown in fact; questionable; uncertain; unsettled; doubtful.
v. t.
To propose problems.
adv.
In an undeniable manner.
a.
Questionable; equivocal; indefinite; problematical.