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Problem in computer science
In computability theory, the halting problem is the decision problem of, given an arbitrary computer program and an input, determining whether said program
Halting_problem
Halting probability of a random computer program
one could calculate the halting problem for all programs of a size up to N. Let the program p for which the halting problem is to be solved be N bits
Chaitin's_constant
Concept in theoretical computer science
noncomputable. This has implications in computability theory, the halting problem, and complexity theory. The concept of a busy beaver was first introduced
Busy_beaver
Theorem in computability theory
for every program. The theorem generalizes the undecidability of the halting problem. It has far-reaching implications on the feasibility of static analysis
Rice's_theorem
Complexity class
that the halting problem is NP-hard but not NP-complete. For example, the Boolean satisfiability problem can be reduced to the halting problem by transforming
NP-hardness
Yes-or-no question that cannot ever be solved by a computer
an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly
Undecidable_problem
Abstract machine used to study decision problems
complexity class, or it can even be an undecidable problem such as the halting problem. If another problem R ′ {\displaystyle R'} is reducible to R
Oracle_machine
Philosophical idea of things impossible to know
include the limits of knowledge, ignorabimus, unknown unknowns, the halting problem, and chaos theory. Nicholas Rescher provides the most recent focused
Unknowability
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
Ability to solve a problem by an effective procedure
is not recursive. The halting problem is therefore called non-computable or undecidable. An extension of the halting problem is called Rice's theorem
Computability
computational limitations does Turing's Halting Problem imply? What are the philosophical consequences of the P vs NP problem? What is information? How do ethics
Philosophy of computer science
Philosophy_of_computer_science
Study of computable functions and Turing degrees
the terminology. Not every set of natural numbers is computable. The halting problem, which is the set of (descriptions of) Turing machines that halt on
Computability_theory
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
Measure of algorithmic complexity
Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a lower bound for each text's
Kolmogorov_complexity
Deterministic model of computation
t)} time. This version of the halting problem is among the simplest, most-easily described undecidable decision problems: Given an arbitrary positive integer
Tag_system
Models of computation
not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that could correctly evaluate
Hypercomputation
Computational problems no algorithm can solve
undecidable in ZFC. The halting problem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it
List_of_undecidable_problems
Impossible task in computing
method' that decides whether any given Turing machine halts or not (the halting problem). If 'algorithm' is understood as meaning a method that can be represented
Entscheidungsproblem
Problem a computer might be able to solve
factors of n. An example of a computational problem without a solution is the Halting problem. Computational problems are one of the main objects of study in
Computational_problem
termination analysis utilizes this principle in order to solve the universal halting problem for a certain class of programs. When applied to general programs,
Size-change termination principle
Size-change_termination_principle
Computation model defining an abstract machine
whether M will eventually produce s. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits
Turing_machine
Complexity class
the halting problem. "NP-complete problems are difficult because there are so many different solutions." On the one hand, there are many problems that
NP-completeness
types. The halting problem is not in ExpGenP for any model of Turing machine, The Post correspondence problem is in ExpGenP. The decision problem for Presburger
Generic-case_complexity
Topics referred to by the same term
Gödel's first incompleteness theorem Tarski's undefinability theorem Halting problem Kleene's recursion theorem Lawvere's fixed-point theorem (categorical
Diagonal_argument
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
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
Problem that can be possibly solved via mathematics
are so-called undecidable problems, such as the halting problem for Turing machines. Some well-known difficult abstract problems that have been solved relatively
Mathematical_problem
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)
Limitative results in mathematical logic
unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem. The incompleteness theorems apply to formal systems that are of sufficient
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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 on
Decider_(Turing_machine)
Quality of an algorithm being correct with respect to a specification
proof (termination proof) can never be fully automated, since the halting problem is undecidable. For example, successively searching through the positive
Correctness (computer science)
Correctness_(computer_science)
Theorem implying that no algorithm can optimally perform a task done by humans
provably perfect size-optimizing compiler would imply a solution to the halting problem, which cannot exist. This also implies that there may always be a better
Full-employment_theorem
String rewriting system
decision problem is undecidable. However, that there is some Turing machine with undecidable halting problem means that the halting problem for a universal
Semi-Thue_system
Any type of calculation
well-defined characterisation under this definition. This includes the halting problem and the busy beaver game. It remains an open question as to whether
Computation
Programming idiom
whether a computer program contains an infinite loop or not; this is the halting problem. An infinite loop is a sequence of instructions in a computer program
Infinite_loop
Determination of whether a given program halts for each input
input program computes a total function. It is closely related to the halting problem, which is to determine whether a given program halts for a given input
Termination_analysis
Ability of a computing system to simulate Turing machines
instance, 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
Turing_completeness
System with multiple networked computers
solves a given problem. A complementary research problem is studying the properties of a given distributed system. The halting problem is an analogous
Distributed_computing
Hypothetical computational model
than classical Turing machines, based on their ability to solve the halting problem for classical Turing machines. Cristian Calude and Ludwig Staiger present
Zeno_machine
Proposition in mathematical logic
studied problems in set theory, and establishing its truth or falsehood was the first of Hilbert's problems presented in 1900. The answer to this problem is
Continuum_hypothesis
Technique used in computer science
locking. A lot of confusion revolves around the halting problem. But this logic does not solve the halting problem because the conditions in which locking occurs
Deadlock prevention algorithms
Deadlock_prevention_algorithms
Academic subfield of computer science
concrete problem that is both easy to formulate and impossible to solve using a Turing machine. Much of computability theory builds on the halting problem result
Theory_of_computation
Analysis of computer programs without executing them
and abstract interpretation. By a straightforward reduction to the halting problem, it is possible to prove that (for any Turing complete language), finding
Static_program_analysis
Types of populated places used by Irish Travellers
halting site (or a halting bay site) is purpose-built residential accommodation for Travellers provided by a local municipal authority. The halting site
Halting_site
The emptiness problem is undecidable for context-sensitive grammars, a fact that follows from the undecidability of the halting problem. It is, however
Emptiness_problem
computability theory, the mortality problem is a decision problem related to the halting problem. For Turing machines, the halting problem can be stated as follows:
Mortality (computability theory)
Mortality_(computability_theory)
Control flow construct for executing code repeatedly
called infinite loops. The problem of determining whether a program contains an infinite loop is known as the halting problem. A conditional loop (also
Loop_(statement)
Post's problem. Post had to prove two things in order to obtain his result: that the simple set A is not computable, and that the K, the halting problem, does
Simple_set
Type of Turing reduction
enumerable problems. Thus the halting problem is r.e. complete. Note that it is not the only r.e. complete problem. The specialized halting problem for an
Many-one_reduction
Square tiles with a color on each edge
the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem. Combining
Wang_tile
American mathematician and computer scientist (1903–1995)
result preceded Alan Turing's work on the halting problem, which also demonstrated the existence of a problem unsolvable by mechanical means. Upon hearing
Alonzo_Church
computation than simply using primitive recursive functions. Since the halting problem cannot be solved in general, there must still be programs that terminate
Walther_recursion
Category of mathematical proof
that there are problems that cannot be solved in general by any algorithm, with one of the more prominent ones being the halting problem. Gödel's incompleteness
Proof_of_impossibility
Contractual transaction on a decentralized platform
executed by the Ethereum Virtual Machine. Due to the halting problem and other security problems, Turing-completeness is considered to be a risk and is
Smart_contract
Basic framework of mathematics
theorem. 1936: Alan Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. 1938: Gödel proved
Foundations_of_mathematics
Hierarchy of complexity classes for formulas defining sets
machine is capable of solving its own halting problem (a variation of Turing's proof applies). The halting problem for a Δ n 0 , Y {\displaystyle \Delta
Arithmetical_hierarchy
Gödel's incompleteness theorem Group (mathematics) Halting problem insolubility of the halting problem Harmonic series (mathematics) divergence of the (standard)
List_of_mathematical_proofs
Transformation of one computational problem to another
function. In particular, we often show that a problem A is undecidable by showing that the halting problem reduces to A. The complexity classes P, NP and
Reduction_(complexity)
Algorithmic complexity class
problems is the halting problem: deciding whether a deterministic Turing machine (DTM) halts. One of the most fundamental EXPTIME-complete problems is
EXPTIME
Class of mathematical problems
discrete, tree based, calculation of the optimal time to exercise. Halting problem Markov decision process Optional stopping theorem Prophet inequality
Optimal_stopping
Section of code in a program that can never be executed
Oxbow code Halting problem – the general problem of determining whether a piece of code is unreachable is at least as hard as the halting problem and hence
Unreachable_code
Computational input that relies on the length but not content of the input
contains some undecidable problems, such as the unary version of every undecidable problem, including the halting problem. Because of that, it is not
Advice_(complexity)
Logical paradox
to discuss the crocodile dilemma. List of paradoxes Self-reference Halting problem – the usual proof of its undecidability uses a similar contradiction
Crocodile_dilemma
Process of repeating items in a self-similar way
optimization problem in recursive form. The key result in dynamic programming is the Bellman equation, which writes the value of the optimization problem at an
Recursion
Binary sequence
\Delta _{2}^{0}} , that is, computable relative to an oracle for the Halting problem. (Schnorr 1971) Chaitin's Ω is an example of such a sequence. No random
Algorithmically random sequence
Algorithmically_random_sequence
Argument that leads to a logical absurdity
the condition is not acceptable, as it would allow us to solve the Halting problem. To see how, consider the statement H(M) stating "Turing machine M
Reductio_ad_absurdum
Two-dimensional cellular automaton
Turing-complete and may therefore execute arbitrary programs. As per the halting problem it is undecidable whether an arbitrary program executed in the Game
Conway's_Game_of_Life
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 of
Outline_of_algorithms
Theorem for proving more complex theorems
Lemma. Merriam-Webster. Loewen, Nathan R. B. (12 March 2018), Beyond the Problem of Evil, Lexington Books, p. 47, ISBN 9781498555739 Higham, Nicholas J
Lemma_(mathematics)
Interpreted programming language first released in 1987
need to decide the halting problem in order to complete parsing in every case. It is a longstanding result that the halting problem is undecidable, and
Perl
Colloquial version of Russell's paradox
{\displaystyle \bot } Cantor's theorem Gödel's incompleteness theorems Halting problem List of paradoxes Self-reference List of self–referential paradoxes
Barber_paradox
Operation in computability theory
jump of X {\displaystyle X} can be thought of as an oracle to the halting problem for oracle machines with an oracle for X {\displaystyle X} . Formally
Turing_jump
Subfield of mathematics
unsolvable. Turing proved this by establishing the unsolvability of the halting problem, a result with far-ranging implications in both recursion theory and
Mathematical_logic
Mathematical logic concept
computably enumerable (cf. picture for a fixed x). This set encodes the halting problem as it describes the input parameters for which each Turing machine
Computably_enumerable_set
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
Computer hardware technology that uses quantum mechanics
computers. Thus, quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove
Quantum_computing
English mathematician, mathematical physicist (born 1931)
transcends formal logic because factors such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based
Roger_Penrose
Approach to static program analysis
computable within finite time and memory (see Rice's theorem and the halting problem). Abstraction is used to allow for generalized answers to questions
Abstract_interpretation
Mathematical function that can be computed by a program
Similarly, most subsets of the natural numbers are not computable. The halting problem was the first such set to be constructed. The Entscheidungsproblem
Computable_function
Covering by shapes without overlaps or gaps
only if, the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane
Tessellation
Proof in set theory
objects. For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally
Cantor's_diagonal_argument
Branch of mathematics
classification of finitely presented groups. By the word problem for groups, which is equivalent to the halting problem, it is impossible to classify such groups, so
Differential_topology
Proof by Alan Turing
lead to his final proof. His first theorem is most relevant to the halting problem, the second is more relevant to Rice's theorem. First proof: that no
Turing's_proof
Logical incompatibility between two or more propositions
of postulates] a new definition must be given. Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute
Contradiction
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
Computable_number
In logic, a statement which is always true
period. The problem of determining whether there is any valuation that makes a formula true is the Boolean satisfiability problem; the problem of checking
Tautology_(logic)
Mathematical problem
In mathematical logic, Tarski's high school algebra problem is a question posed by Alfred Tarski. It asks whether there are identities involving addition
Tarski's high school algebra problem
Tarski's_high_school_algebra_problem
Numbers that arise in the theory of Turing machines
play a key role in Alan Turing's proof of the undecidability of the halting problem, and are very useful in reasoning about Turing machines as well. Say
Description_number
English computer scientist (1912–1954)
that there was no solution to the decision problem by first showing that some variants of the halting problem for Turing machines are undecidable: in particular
Alan_Turing
Soviet mathematician (1903–1979)
classifying all four-manifolds would imply a solution to Turing's halting problem. Embedding implies failure to create a correspondence between algorithms
Andrey_Markov_Jr.
Software that can produce new versions of itself
proofs in theoretical computer science, such as the proof that the halting problem is undecidable. Ken Thompson started development on Unix in 1968 by
Self-hosting_(compilers)
Symbolic description of a mathematical object
well-defined characterisation under this definition. This includes the halting problem and the busy beaver game. It remains an open question as to whether
Expression_(mathematics)
Function, homomorphism, or morphism
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Map_(mathematics)
Inherent flaw in computer instructions
potential problems. Although in general the problem of finding all programming errors given a specification is not solvable (see halting problem), these
Software_bug
Statistical technique for smoothing categorical data
no value may be computable or observable in a finite time (see the halting problem). But at least one possibility must have a non-zero pseudocount, otherwise
Additive_smoothing
Collection of mathematical objects
characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate
Set_(mathematics)
3-volume treatise on mathematics, 1910–1913
established for Principia's axioms of set theory. (See Hilbert's second problem.) Russell and Whitehead suspected that the system in PM is incomplete:
Principia_Mathematica
Mathematical operation with two operands
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Binary_operation
Mathematical set formed from two given sets
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Cartesian_product
Process of writing a self-compiling compiler
theoretical computer science, such as the variation of the proof that the halting problem is undecidable that uses Rice's Theorem. Due to security concerns regarding
Bootstrapping_(compilers)
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