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called computably isomorphic if there exists a computable bijection f {\displaystyle f} so that ν = μ ∘ f {\displaystyle \nu =\mu \circ f} . Computably isomorphic
Computable_isomorphism
Unsolved problem in computational complexity theory
science Can the graph isomorphism problem be solved in polynomial time? More unsolved problems in computer science The graph isomorphism problem is the computational
Graph_isomorphism_problem
Study of computable functions and Turing degrees
Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function. Although initially skeptical, by 1946 Gödel
Computability_theory
Bijection between the vertex set of two graphs
in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs
Graph_isomorphism
Mathematical function that can be computed by a program
of computability that can be imagined can compute only functions that are computable in the above sense. Before the precise definition of computable functions
Computable_function
Computation model defining an abstract machine
It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with the tape on the beginning
Turing_machine
reduction is an injective reduction, and a computable isomorphism is a bijective reduction. Myhill's isomorphism theorem: Two sets A , B ⊆ N {\displaystyle
Myhill_isomorphism_theorem
Relationship between programs and proofs
first formulation of the isomorphism was referred to (a variant of) Gentzen's sequent calculus. The observation that the isomorphism is best understood with
Curry–Howard_correspondence
Uniqueness of countable dense linear orders
strengthened result that when two computably enumerable linear orders have a computable comparison predicate, and computable functions representing their density
Cantor's_isomorphism_theorem
Problem in computer science
verification that g is computable relies on the following constructs (or their equivalents): computable subprograms (the program that computes f is a subprogram
Halting_problem
Topological space associated to a vector bundle
B} be a real vector bundle of rank n. Then there is an isomorphism called a Thom isomorphism Φ : H k ( B ; Z 2 ) → H ~ k + n ( T ( E ) ; Z 2 ) , {\displaystyle
Thom_space
Axiomatic set theories based on the principles of mathematical constructivism
are computable trees K {\displaystyle K} for which no computable such path through it exists. To prove this, one enumerates the partial computable sequences
Constructive_set_theory
Thesis on the nature of computability
definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil
Church–Turing_thesis
Mathematical logic concept
Enumerability: The set S is the range of a partial computable function. The set S is the range of a total computable function, or empty. If S is infinite, the
Computably_enumerable_set
Set with algorithmic membership test
undecidable) if it is not computable. A subset S {\displaystyle S} of the natural numbers is computable if there exists a total computable function f {\displaystyle
Computable_set
Proof by Alan Turing
to practical computation... (Hodges p. 124) 1 computable number — a number whose decimal is computable by a machine (i.e., by finite means such as an
Turing's_proof
Decision problem
isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups. The isomorphism problem
Group_isomorphism_problem
Very general problem in computer science
logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor's algorithms
Hidden_subgroup_problem
Measure of algorithmic complexity
2^{*}} be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable f : 2 ∗ →
Kolmogorov_complexity
Limitative results in mathematical logic
or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Correspondence between quantum channels and quantum states
kind of correspondence is called Choi-Jamiołkowski isomorphism. The Choi-Jamiołkowski isomorphism is a mathematical concept that connects quantum gates
Choi–Jamiołkowski_isomorphism
Mathematical-logic system based on functions
usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via
Lambda_calculus
Inherent difficulty of computational problems
"Graph isomorphism is in SPP", Information and Computation, 204 (5): 835–852, doi:10.1016/j.ic.2006.02.002. Schöning, Uwe (1988), "Graph Isomorphism is in
Computational complexity theory
Computational_complexity_theory
Hungarian-American mathematician and computer scientist
2016-01-21. Theory of Computing editors, retrieved 2010-07-30. A Big Result On Graph Isomorphism // November 4, 2015, A Fast Graph Isomorphism Algorithm // November
László_Babai
Area of mathematical logic
an isomorphism of A {\displaystyle {\mathcal {A}}} with a substructure of B {\displaystyle {\mathcal {B}}} . If it can be written as an isomorphism with
Model_theory
Problem in theoretical computer science
that subgraph isomorphism remains NP-complete even in the planar case. Subgraph isomorphism is a generalization of the graph isomorphism problem, which
Subgraph_isomorphism_problem
Property of graphs that depends only on abstract structure
of a graph. Easily computable graph invariants are instrumental for fast recognition of graph isomorphism, or rather non-isomorphism, since for any invariant
Graph_property
Unsolved problem in computer science
"Graph isomorphism is in SPP". Information and Computation. 204 (5): 835–852. doi:10.1016/j.ic.2006.02.002. Schöning, Uwe (1988). "Graph isomorphism is in
P_versus_NP_problem
Concept in theoretical computer science
\to \mathbb {N} } is any computable function, then Σ(n) > f(n) for all sufficiently large n, and hence that Σ is not a computable function. Moreover, this
Busy_beaver
Complexity class used to classify decision problems
decision version repeatedly (a polynomial number of times). The subgraph isomorphism problem of determining whether graph G contains a subgraph that is isomorphic
NP_(complexity)
Function computable with bounded loops
closely with our intuition of what a computable function must be. Certainly the initial functions are intuitively computable (in their very simplicity), and
Primitive_recursive_function
Number representing a continuous quantity
number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists
Real_number
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Mathematical_object
Logical principle
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Law_of_excluded_middle
Mathematical function, in linear algebra
way described in § Matrices (below) is a linear map, and even a linear isomorphism. The expected value of a random variable is a linear function of the
Linear_map
Infinite cardinal number
sense), the set of all algebraic numbers, the set of all computable numbers, the set of all computable functions, the set of all binary strings of finite length
Aleph_number
Type of computational problem
divisible by k?". For all k≥2, ModkP contains the graph isomorphism problem. Further, the graph isomorphism problem is low in ModkP. When k is prime, the set
Counting_problem_(complexity)
Branch of mathematical logic
where "recursive" means "computable", as in computable function. This name is used because RCA0 corresponds informally to "computable mathematics". In particular
Reverse_mathematics
Process of repeating items in a self-similar way
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Recursion
Impossible task in computing
intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda
Entscheidungsproblem
Theorem in order theory
Rosenstein proved) there exist computable linear orders with no computable non-identity self-embedding. Cantor's isomorphism theorem Laver's theorem Downey
Dushnik–Miller_theorem
Proving validity without revealing other data
questions to ask Peggy. He can either ask her to show the isomorphism between H and G (see graph isomorphism problem), or he can ask her to show a Hamiltonian
Zero-knowledge_proof
Form of mathematical proof
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Mathematical_induction
General theory of mathematical structures
morphisms g1, g2 : b → x. a bimorphism if f is both epic and monic. an isomorphism if there exists a morphism g : b → a such that f ∘ g = 1b and g ∘ f =
Category_theory
Typed lambda calculus
(without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to second-order propositional intuitionistic logic
System_F
Yes-or-no question that cannot ever be solved by a computer
or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The
Undecidable_problem
Subfield of mathematics
been established. Recursion theory, also called computability theory, studies the properties of computable functions and the Turing degrees, which divide
Mathematical_logic
Statement that is taken to be true
domain of real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that
Axiom
Complexity class
Graph Isomorphism: Is graph G1 isomorphic to graph G2? Subgraph Isomorphism: Is graph G1 isomorphic to a subgraph of graph G2? The Subgraph Isomorphism problem
NP-completeness
Mathematical set containing no elements
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Empty_set
Axioms for the natural numbers
model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing
Peano_axioms
Ordered listing of items in collection
domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration is the complement
Enumeration
Yes/no problem in computer science
time is computed as a function of the pair (x,y)) when the function is not computable in polynomial time (in which case running time is computed as a function
Decision_problem
Concept from category theory
mathematics, specifically in the field of category theory, the associativity isomorphism implements the notion of associativity with respect to monoidal products
Associativity_isomorphism
Branch of mathematics
space is associated with exactly one in the first) is an isomorphism. Because an isomorphism preserves linear structure, two isomorphic vector spaces
Linear_algebra
Standard system of axiomatic set theory
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Zermelo–Fraenkel_set_theory
Connects homology and cohomology groups for oriented closed manifolds
such an isomorphism, one chooses a fixed fundamental class [M] of M, which will exist if M {\displaystyle M} is oriented. Then the isomorphism is defined
Poincaré_duality
Distance-preserving mathematical transformation
diffeomorphism, such a map is called an isometry (or isometric isomorphism), and provides a notion of isomorphism ("sameness") in the category Rm of Riemannian manifolds
Isometry
Mapping a graph onto itself without changing edge-vertex connectivity
"Graph isomorphisms in quasi-polynomial time". arXiv:1710.04574 [math.GR]. Lubiw, Anna (1981), "Some NP-complete problems similar to graph isomorphism", SIAM
Graph_automorphism
Relationship where one statement follows from another
Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281. Papers include those
Logical_consequence
Collection of mathematical objects
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Set_(mathematics)
Model of (first-order) Peano arithmetic that contains non-standard numbers
Peano axioms; for the original second-order formulation, there is, up to isomorphism, only one model: the natural numbers themselves. There are several methods
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Basic notion of sameness in mathematics
assistants. The concept of isomorphism extends to numerous branches of mathematics, including graph theory (graph isomorphism), topology (homeomorphism)
Equality_(mathematics)
Reasoning for mathematical statements
centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical
Mathematical_proof
Set of all things that may be the input of a mathematical function
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Domain_of_a_function
Vector space equipped with a bilinear product
. {\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).} A K-algebra isomorphism is a bijective K-algebra homomorphism. A subalgebra of an algebra over
Algebra_over_a_field
Set of elements in any of some sets
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Union_(set_theory)
Generalization of Turing computability
of computability relative to a type-2 functional, Kleene showed that a set of natural numbers is hyperarithmetical if and only if it is computable relative
Hyperarithmetical_theory
Mathematical set of all subsets of a set
be applied to the example above, in which S = {x, y, z}, to get the isomorphism with the binary representations of numbers from 0 to 2n − 1, with n being
Power_set
Axiom of set theory
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Axiom_of_choice
Subject of study in ergodic theory
a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms. The first anti-classification
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Theorem
Rham cohomology to the singular cohomology given by integration is an isomorphism. The Poincaré lemma implies that the de Rham cohomology is the sheaf
De_Rham_theorem
Group of even permutations of a finite set
symmetry group of chiral icosahedral symmetry. (See for an indirect isomorphism of PSL2(F5) → A5 using a classification of simple groups of order 60
Alternating_group
In logic, a statement which is always true
for formulas with thousands of propositional variables, as contemporary computing hardware cannot execute the algorithm in a feasible time period. The problem
Tautology_(logic)
Symbol representing a mathematical object
unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple
Variable_(mathematics)
Mathematical group of the homotopy classes of loops in a topological space
isomorphism, this choice makes no difference as long as the space X {\displaystyle X} is path-connected: more precisely, one obtains an isomorphism by
Fundamental_group
Mathematical group
In other words, an automorphism of E / F {\displaystyle E/F} is an isomorphism α : E → E {\displaystyle \alpha :E\to E} such that α ( x ) = x {\displaystyle
Galois_group
Algebraic structure used in topology
mathematical object that encode its properties in a way that is often computable. Cohomology is often related to questions of whether some local property
Cohomology
Structure of a formal language
However, it can also be used as the basis for a parser—a function in computing that determines whether a given string belongs to the language or is grammatically
Formal_grammar
Duality for locally compact abelian groups
{\text{End}}({\widehat {G}})^{\text{op}}} . More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories
Pontryagin_duality
Problem of inverting exponentiation in groups
Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. In cryptography
Discrete_logarithm
Mathematical proposition equivalent to the axiom of choice
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Zorn's_lemma
Basic framework of mathematics
mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer
Foundations_of_mathematics
Convex hull of a finite set of points in a Euclidean space
graph isomorphism problem. However, it is also possible to translate these problems in the opposite direction, showing that polytope isomorphism testing
Convex_polytope
Existence and cardinality of models of logical theories
first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality
Löwenheim–Skolem_theorem
Cohomology with real coefficients computed using differential forms
that for a smooth manifold M {\displaystyle M} , this map is in fact an isomorphism. More precisely, consider the map I : H d R p ( M ) → H p ( M ; R )
De_Rham_cohomology
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
3-volume treatise on mathematics, 1910–1913
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Principia_Mathematica
Formalism in computer science
related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of certain classes
Typed_lambda_calculus
Complexity class of problems
are considered good candidates for being NP-intermediate are the graph isomorphism problem, and decision versions of factoring and the discrete logarithm
NP-intermediate
Isomorphism of projective spaces in geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective
Homography
Relates the homology of a fiber bundle with the homologies of its base and fiber
&\longmapsto &s(\alpha )\smallsmile \pi ^{*}(\beta )\end{array}}} is an isomorphism of H ∗ ( B ) {\displaystyle H^{*}(B)} -modules. In other words, if for
Leray–Hirsch_theorem
Algebraic manipulation of "true" and "false"
Parkes, Alan (2002). Introduction to languages, machines and logic: computable languages, abstract machines and formal logic. Springer. p. 276. ISBN 978-1-85233-464-2
Boolean_algebra
Proposition in mathematical logic
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Continuum_hypothesis
Surjective bounded operator on a Hilbert space preserving the inner product
a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces. Definition 1. A unitary operator is a bounded
Unitary_operator
Logic theorem
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Law_of_noncontradiction
Mathematical function such that every output has at least one input
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Surjective_function
Mathematical use of "there exists"
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Existential_quantification
Term in logic and deductive reasoning
this special sense of completeness, in which the class of models (up to isomorphism) is restricted to the intended one. The original completeness proof applies
Soundness
Size of a possibly infinite set
Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem
Cardinal_number
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
Boy/Male
Muslim
Similar. Comparable.
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Boy/Male
Afghan, Arabic, Celebrity, German, Indian, Muslim, Sindhi
Observer; Supervisor; Little; Insignificant; Warner; Similar; Comparable; Another Name for the Quran; One who Preaches
Boy/Male
Muslim
Similar. Comparable.
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Boy/Male
Arabic, Australian, Muslim
Similar; Comparable; One who Warns
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
Girl/Female
Muslim
Rare, Precious
Girl/Female
Greek Latin
Most beautiful. Calista was a Mythological Arcadian who transformed into a she-bear, then into...
Girl/Female
Christian & English(British/American/Australian)
Small Beauty
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Effective; Efficient; Goddess Durga
Boy/Male
Tamil
Brave, One who fights for peace, Strong, Continuous or ongoing
Girl/Female
British, English
Sweet
Girl/Female
German
Maiden; From the High Tower
Male
German
Short form of German names beginning with Mein-, MEINO means "might, strength."
Boy/Male
Hindu
Radiant
Boy/Male
Gujarati, Indian, Kannada
Lord Indra
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
COMPUTABLE ISOMORPHISM
a.
Capable of existing in harmony; congruous; suitable; not repugnant; -- usually followed by with.
a.
Not compliable; not conformable.
n.
Quality of being imputable.
adv.
In a compatible manner.
a.
Compatible; suitable; consistent.
a.
Not confutable.
n.
The quality of being commutable; interchangeableness.
a.
That may be confuted.
n.
The quality of being imputable; imputableness.
a.
Capable of bending or yielding; apt to yield; compliant.
a.
Capable of being commuted or interchanged.
a.
Not commutable; not capable of being exchanged with, or substituted for, another.
a.
Capable of being attributed; ascribable; imputable.
n.
The quality of being commutable.
a.
Capable of being computed, numbered, or reckoned.
a.
Not computable.
a.
Correspondent; conformable; hence, comparable.
a.
Suitable; consistent.
a.
Such as can be, or is liable to be, combated; as, combatable foes, evils, or arguments.
a.
Comparable.