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computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing an algorithm that solves it; a computational
Non-constructive algorithm existence proofs
Non-constructive_algorithm_existence_proofs
Method of proof in mathematics
is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind
Constructive_proof
Theorem which asserts the existence of an object
Such a proof is non-constructive, since the whole approach may not lend itself to construction. In terms of algorithms, purely theoretical existence theorems
Existence_theorem
Philosphical view that existence proofs must be constructive
assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Class of algorithms that find approximate solutions to optimization problems
tour. A classic example of approximation algorithm providing an additive guarantee is the constructive proof of Vizing’s theorem. It shows how to color
Approximation_algorithm
Unsolved problem in computer science
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive
P_versus_NP_problem
Argument that leads to a logical absurdity
freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof. This argument form traces back
Reductio_ad_absurdum
About simultaneous modular congruences
constructions given in § Existence (constructive proof) or § Existence (direct proof). The Chinese remainder theorem can be generalized to non-coprime moduli.
Chinese_remainder_theorem
Logical principle
Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof: According to Brouwer, a statement that
Law_of_excluded_middle
Reasoning for mathematical statements
ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without
Mathematical_proof
Form of mathematical proof
induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs. Despite its name, mathematical
Mathematical_induction
Proof by Alan Turing
machine". In his proof that the Entscheidungsproblem can have no solution, Turing proceeded from two proofs that were to lead to his final proof. His first
Turing's_proof
Method of deriving conclusions
the end of proofs to indicate that the original hypothesis has been demonstrated. There are different strategies used to formulate proofs. For example
Rule_of_inference
Field in logic and theoretical computer science
the existence of a propositional proof system that admits polynomial size proofs for all tautologies is equivalent to NP = co-NP. Contemporary proof complexity
Proof_complexity
Process of repeating items in a self-similar way
can be "solved" to obtain a non-recursive definition (e.g., a closed-form expression). Use of recursion in an algorithm has both advantages and disadvantages
Recursion
Mathematical proof at least partially generated by computer
believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so
Computer-assisted_proof
Axiomatic set theories based on the principles of mathematical constructivism
difference is that the constructive proofs are harder to find. In set theory, a restriction to the constructive reading of existence apriori leads to stricter
Constructive_set_theory
Number that is not a ratio of integers
integers and therefore a rational number. Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab is
Irrational_number
Millennium Prize Problem
problem is phrased as follows: Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
On constructing objects that obey a system of constraints with limited dependence
{A}}}(1-x(A)).} The Lovász Local Lemma is non-constructive because it only allows us to conclude the existence of structural properties or complex objects
Algorithmic Lovász local lemma
Algorithmic_Lovász_local_lemma
Mathematical theory of data types
closely resembles Peano's axioms. In type theory, proofs have types whereas in set theory, proofs are part of the underlying first-order logic. Proponents[who
Type_theory
Measure of algorithmic complexity
formal proofs in S by some procedure def nth_proof(n: int) which takes as input n and outputs some proof. This function enumerates all proofs. Some of
Kolmogorov_complexity
systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic algorithms that explicitly construct the desired
Method of conditional probabilities
Method_of_conditional_probabilities
Approach in philosophy of mathematics and logic
rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3,
Intuitionism
Mathematical model of the physical space
nonconstructive proofs just as sound as constructive ones, they are often considered less elegant, intuitive, or practically useful. Euclid's constructive proofs often
Euclidean_geometry
Limitative results in mathematical logic
completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
determining existence. He provided six such NP-complete search problems, or universal problems. Additionally he found for each of these problems an algorithm that
Cook–Levin_theorem
Branch of mathematical logic
sentences (with parameters). Non-ω models are also useful, especially in the proofs of conservation theorems. Constructive reverse mathematics is a program
Reverse_mathematics
Basic framework of mathematics
self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study
Foundations_of_mathematics
Subfield of mathematics
about intuitionistic proofs to be transferred back to classical proofs. Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach
Mathematical_logic
On solvability of Diophantine equations
Gödel in coding proofs by natural numbers in such a way that the property of being the number representing a proof is algorithmically checkable. Π 1 0
Hilbert's_tenth_problem
Fundamental theorem in mathematical logic
thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkin's proof was simplified by Gisbert Hasenjaeger
Gödel's_completeness_theorem
Complexity class used to classify decision problems
the subset. If the sum is zero, that subset is a proof or witness for the answer is "yes". An algorithm that verifies whether a given subset has sum zero
NP_(complexity)
Branch of mathematics that studies sets
such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes
Set_theory
Algorithm in graph theory
The Misra & Gries edge-coloring algorithm is a polynomial-time algorithm in graph theory that finds an edge coloring of any simple graph. The coloring
Misra & Gries edge-coloring algorithm
Misra_&_Gries_edge-coloring_algorithm
Probability theorem on no events occurring
most commonly used in the probabilistic method, in particular to give existence proofs. There are several different versions of the lemma. The simplest and
Lovász_local_lemma
Mathematical set formed from two given sets
{\displaystyle {\mathcal {P}}} represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms
Cartesian_product
Computation model defining an abstract machine
machine proofs of computability of recursive functions, etc. Knuth, Donald E. (1973). The Art of Computer Programming, Vol. 1: Fundamental Algorithms (2nd ed
Turing_machine
Existence of a line through two points
axioms of constructive analysis, and to adapt Kelly's proof of the theorem to be a valid proof under these axioms. Kelly's proof of the existence of an ordinary
Sylvester–Gallai_theorem
Theorem in topology
and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's
Brouwer_fixed-point_theorem
In mathematics, a statement that has been proven
the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of
Theorem
Theorems that help decompose a finite group based on prime factors of its order
important problem in computational group theory. One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H]
Sylow_theorems
Mathematical function that can be computed by a program
all their corresponding proofs, that prove their computability. This can be done by enumerating all the proofs of the proof system and ignoring irrelevant
Computable_function
Mathematical treatise by Euclid
computer-assisted proofs, and the propositions of the Elements (with some updates to their proofs) have withstood computer checking. Some of the foundational proofs of
Euclid's_Elements
Aesthetic value of mathematics
". His rhetorical device inspired the creation of Proofs from THE BOOK, a collection of such proofs, including many suggested by Erdős himself. In Plato's
Mathematical_beauty
Various systems of symbolic logic
this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of
Intuitionistic_logic
adherents of these schools reject non-constructive proofs, such as using proof by contradiction when showing the existence of an object or when trying to
Philosophy_of_mathematics
Mathematical-logic system based on functions
as used in β-reduction Harrop formula – A kind of constructive logical formula such that proofs are lambda terms Interaction nets Kleene–Rosser paradox
Lambda_calculus
Thesis on the nature of computability
"effective computability" as follows: "Clearly the existence of CC and RC [meaning Church's and Rosser's proofs of the statement that there is no effective method
Church–Turing_thesis
Subfield of automated reasoning and mathematical logic
mathematical proof that was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable
Automated_theorem_proving
Quantifier elimination for semi-algebraic sets
formula of the theory. Although the original proof of the theorem was constructive, the resulting algorithm is galactic, that is, it has a computational
Tarski–Seidenberg_theorem
Measurement/calculations: Yes Platform: Windows, Mac OS, TI-92+, works under Wine Proofs: No The Geometric Supposer Geonext was developed by the University of Bayreuth
List of interactive geometry software
List_of_interactive_geometry_software
23 mathematical problems stated in 1900
systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of
Hilbert's_problems
Modular arithmetic concept
Disquisitiones contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. An equivalent characterization
Primitive_root_modulo_n
Impossible task in computing
problem, which asks for an algorithm to decide whether Diophantine equations have a solution. The non-existence of such an algorithm, established by the work
Entscheidungsproblem
Axioms for the natural numbers
1986, sections 2.3 (p. 464) and 4.1 (p. 471). For formal proofs, see e.g. File:Inductive proofs of properties of add, mult from recursive definitions.pdf
Peano_axioms
Argument in combinatorial game theory
strategy for the first player, and because of this it has been called non-constructive. This raises the question of how to actually compute a winning strategy
Strategy-stealing_argument
rejecting non-constructive proofs such as those involving the law of excluded middle in its full generality. constructive proof A proof that demonstrates
Glossary_of_logic
Function used in computer cryptography
converse is not known to be true, i.e. the existence of a proof that P ≠ NP would not directly imply the existence of one-way functions. In applied contexts
One-way_function
Property of artificial neural networks
variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions
Universal approximation theorem
Universal_approximation_theorem
Theorem in order and lattice theory
for proving the existence of equilibrium states in fields like game theory. It essentially proves that when a system follows simple, non-decreasing rules
Knaster–Tarski_theorem
Existence of values making formula true
Press. Daniel Kroening; Ofer Strichman (2008). Decision Procedures: An Algorithmic Point of View. Springer Science & Business Media. ISBN 978-3-540-74104-6
Satisfiability
Foundational controversy in twentieth-century mathematics
Hilbert had to give up was "constructibility." His proofs would not produce "objects" (except for the proofs themselves – i.e., symbol strings), but rather
Brouwer–Hilbert_controversy
Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology Consistency proof Arithmetization of
List of mathematical logic topics
List_of_mathematical_logic_topics
Relation between algebraic varieties and polynomial ideals
are many known proofs of the theorem. Some are non-constructive, such as the first one. Others are constructive, as based on algorithms for expressing
Hilbert's_Nullstellensatz
Yes/no problem in computer science
in terms of the computational resources needed by the most efficient algorithm for a certain problem. On the other hand, the field of recursion theory
Decision_problem
Study of computable functions and Turing degrees
was demonstrated by Kurt Gödel in the proofs of his completeness theorem and incompleteness theorems. Gödel's proofs show that the set of logical consequences
Computability_theory
Result in combinatorics and graph theory
max-flow min-cut theorem (Ford–Fulkerson algorithm) Dilworth's theorem. In particular, there are simple proofs of the implications Dilworth's theorem ⇔
Hall's_marriage_theorem
Binary sequence
Intuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free
Algorithmically random sequence
Algorithmically_random_sequence
Axiomatization of arithmetic
could be validated for some t {\displaystyle t} . Constructively, this is weaker than the existence claim of such a t {\displaystyle t} . A big part of
Heyting_arithmetic
Basic notion of sameness in mathematics
{\displaystyle f(a)=f(b).} Numerical analysis is the study of constructive methods and algorithms to find numerical approximations (as opposed to symbolic
Equality_(mathematics)
Form of logic that allows quantification over predicates
(Effectiveness) There is a proof-checking algorithm that can correctly decide whether a given sequence of symbols is a proof or not. This corollary is
Second-order_logic
Approach to mathematics using computation
proof of the Kepler conjecture. Various proofs of the four colour theorem. Clement Lam's proof of the non-existence of a finite projective plane of order
Experimental_mathematics
Area of mathematical logic
this to satisfiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i.e., its contrapositive), the
Model_theory
Mathematical result on infinite trees
computability theory. This theorem also has important roles in constructive mathematics and proof theory. Let G {\displaystyle G} be a connected, locally finite
Kőnig's_lemma
Ordered listing of items in collection
countable sets for which an enumeration function can be computed with an algorithm. For avoiding to distinguish between finite and countably infinite set
Enumeration
Mathematical models of strategic interactions
numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games
Game_theory
Finiteness of sets of forbidden graph minors
not provide a concrete polynomial-time algorithm for solving it. Such proofs of polynomiality are non-constructive: they prove polynomiality of problems
Robertson–Seymour_theorem
Hungarian and American mathematician and physicist (1903–1957)
are congruent by translation). His next paper dealt with giving a constructive proof without the axiom of choice that 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}}
John_von_Neumann
Formal semantics for non-classical logic systems
its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose
Kripke_semantics
Number representing a continuous quantity
equation, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof. Hermite (1873) proved that
Real_number
Result in modular arithmetic
branch of analytic number theory. The proof of Hensel's lemma is constructive, and leads to an efficient algorithm for Hensel lifting, which is fundamental
Hensel's_lemma
Xue, 2014) Existence of a non-terminating game of beggar-my-neighbour (Brayden Casella, 2024) The angel problem (Various independent proofs, 2006) Carathéodory
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Abstract strategy board game
end of all possible lines of play". All known proofs of this fact are non-constructive, i.e., the proof gives no indication of what the actual winning
Hex_(board_game)
equations. This theorem is useful to prove the existence of algorithms. However, in practice, the algorithms for the systems are designed directly. A field
Linear_equation_over_a_ring
Analytic function in mathematics
Borwein, Peter (2000). "An Efficient Algorithm for the Riemann Zeta Function" (PDF). In Théra, Michel A. (ed.). Constructive, Experimental, and Nonlinear Analysis
Riemann_zeta_function
Graph theory concept
this proof of existence is non-constructive, and does not lead to an explicit description of the set of forbidden minors or of the algorithm based on
Planar_cover
Branch of mathematics
complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. Numerical analysis is the study of algorithms that use numerical approximation
Mathematical_analysis
Mathematical set of all subsets of a set
axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The
Power_set
Type of logical system
derivations in proof theory. They are also often called proofs but are completely formalized unlike natural-language mathematical proofs. A deductive system
First-order_logic
in the 1950s. The main idea behind list decoding is that the decoding algorithm instead of outputting a single possible message outputs a list of possibilities
List_decoding
Form of second-order logic
in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth
Monadic_second-order_logic
Axiom set used in first-order logic
language is either provable or disprovable from the axioms, and we have an algorithm which decides for any given sentence whether it is provable or not. Early
Tarski's_axioms
Concept in algebraic geometry
higher dimensions is notorious for many incorrect published proofs and announcements of proofs that never appeared. For 3-folds the resolution of singularities
Resolution_of_singularities
Problem of fair division
used in the proof are: The K-K-M-S theorem - a generalization of the K-k-m theorem. Hall's marriage theorem. Their solution is constructive in the same
Rental_harmony
Hypothetical group of multiple universes
describable by constructive mathematics—that is, computer programs. Schmidhuber explicitly includes universe representations describable by non-halting programs
Multiverse
Generalizations in graph theory
each transversal requires at least 2r – 3 vertices. Haxell's proof is not constructive. However, Chidambaram Annamalai proved that a perfect matching
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
calculus. Alan Turing introduces the Turing machine model proves the existence of universal Turing machines, and uses these results to settle the Entscheidungsproblem
Timeline of mathematical logic
Timeline_of_mathematical_logic
Gives conditions for the solvability of quadratic equations modulo prime numbers
published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included
Quadratic_reciprocity
Pair of logical equivalences
readabillity and correctness when designing algorithms. De Morgan's laws are also useful in formal proofs. where transforming logical statements into
De_Morgan's_laws
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
Female
Vietnamese
Vietnamese name NGON means "good communication."
Biblical
same as Non
Girl/Female
Hindu, Indian
Existence
Boy/Male
Greek
Son of Apollo.
Girl/Female
Biblical
Posterity, a fish, eternal.
Girl/Female
Indian
Existence
Male
Norwegian
Danish and Norwegian form of Old Norse Hákon, HÅKON means "high son."
Surname or Lastname
English, German, Dutch, French (Noé, Noë), Spanish (Noé), Catalan (Noè)
English, German, Dutch, French (Noé, Noë), Spanish (Noé), Catalan (Noè) : from the Biblical personal name Noach ‘Noah’, which means ‘comfort’ in Hebrew. According to the Book of Genesis, Noah, having been forewarned by God, built an ark into which he took his family and representatives of every species of animal, and so was saved from the flood that God sent to destroy the world because of human wickedness. The personal name was not common among non-Jews in the Middle Ages, but the Biblical story was an extremely popular subject for miracle plays. In many cases, therefore, the surname probably derives from a nickname referring to someone who had played the part of Noah in a miracle play or pageant, rather than from a personal name.
Female
English
(רï‹×Ÿ) Hebrew unisex name RON means "joy, song." Compare with strictly masculine Ron.
Biblical
posterity; a fish; eternal
Female
English
Variant form of Old English Nona, NONI means "ninth."
Boy/Male
Indian
Existence
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Existence
Boy/Male
American, Australian
Little Son
Girl/Female
Tamil
Existence
Boy/Male
Hindu
Existence
Girl/Female
Australian
Existence
Male
French
French form of Greek Noe, NOÉ means "rest."
Girl/Female
Arabic, Muslim
Existence
Boy/Male
Tamil
Astitva | அஸà¯à®¤à®¿à®¤à¯à®µ
Existence
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
Surname or Lastname
English (Devon)
English (Devon) : apparently a habitational name from a lost or unidentified place in or bordering on Devon.
Girl/Female
Hindu
Required victory, Superior
Boy/Male
Tamil
Paramapurusha | பரமபà¯à®°à¯à®·
The supreme Man
Surname or Lastname
English, Dutch, and Jewish (Ashkenazic)
English, Dutch, and Jewish (Ashkenazic) : patronymic from the personal name Abraham.
Boy/Male
English
Darling, dearly loved, from the Old english 'deorling'.
Male
Babylonian
, Athtor of Yahrak.
Surname or Lastname
English
English : topographic name for someone who lived near an ash tree, or a habitational name from a place named with the Old English word æsc (see Ash). The Anglo-Norman French preposition de ‘of’, ‘from’ has become fused to the name.Americanized spelling of German Dasch.Indian : variant of Das.
Biblical
power; greatness
Male
English
Variant spelling of English Laurence, LAWRENCE means "of Laurentum."
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Desired; Love of Life
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
NON CONSTRUCTIVE-ALGORITHM-EXISTENCE-PROOFS
a.
Having ability to construct or form; employed in construction; as, to exhibit constructive power.
n.
Existence at the same time with another; -- contemporary existence.
n.
The act of constructing vaults; a vaulted construction.
n.
The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition.
a.
Derived from, or depending on, construction or interpretation; not directly expressed, but inferred.
n.
Alt. of Algorithm
n.
Continued or repeated manifestation; occurrence, as of events of any kind; as, the existence of a calamity or of a state of war.
n.
Existence.
n.
The act of constructing; construction.
n.
The state of existing or being; actual possession of being; continuance in being; as, the existence of body and of soul in union; the separate existence of the soul; immortal existence.
a.
No; not. See No, a.
adv.
In a constructive manner; by construction or inference.
n.
That which exists; a being; a creature; an entity; as, living existences.
a.
Having being or existence; existing; being; occurring now; taking place.
n.
Inherent existence; existence possessed by virtue of a being's own nature, and independent of any other being or cause; -- an attribute peculiar to God.
a.
Conveying knowledge; serving to instruct or inform; as, experience furnishes very instructive lessons.
a.
Reconstructing; tending to reconstruct; as, a reconstructive policy.
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
n.
An obstructive person or thing.
n.
Want of being or existence.