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Mathematical proof at least partially generated by computer
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been
Computer-assisted_proof
Reasoning for mathematical statements
or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that
Mathematical_proof
Interactive theorem prover software
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal
Proof_assistant
Type of mathematical proof
prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color
Proof_by_exhaustion
Proof that is not easily verified by hand
Tymoczko in 1979 in criticism of Kenneth Appel and Wolfgang Haken's computer-assisted proof of the four color theorem, and has since been applied to other
Non-surveyable_proof
a proof for 22 moves. Finally, in 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge gave the final computer-assisted proof that
Optimal solutions for the Rubik's Cube
Optimal_solutions_for_the_Rubik's_Cube
Sequence of numbers consisting of 1 and -1
This was the best such bound available at the time. Their proof relied on a SAT-solver computer algorithm whose output takes up 13 gigabytes of data, more
Sign_sequence
Planar maps require at most four colors
to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human
Four_color_theorem
Electronic circuit that behaves chaotically
determined by the particular values of the circuit components. A computer-assisted proof of chaotic behavior (more precisely, of positive topological entropy)
Chua's_circuit
Can one split the integers into two sets such that every Pythagorean triple spans both?
Heule, Oliver Kullmann and Victor W. Marek in May 2016 through a computer-assisted proof, which showed that such a coloring is only possible up to the number
Boolean Pythagorean triples problem
Boolean_Pythagorean_triples_problem
American mathematician
This provided the first mathematical proof of the rigidity conjectures of Feigenbaum. The proof was computer assisted. The hyperbolicity of the fixed point
Oscar_Lanford
Math theorem about sphere packing
had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations
Kepler_conjecture
Two-player board game
brute-force analysis was not deemed feasible given the game's complexity and the computer technology available at the time. Connect Four has since been solved with
Connect_Four
Branch of game theory about two-player sequential games with perfect information
both sides leading to a draw; however, this result required a computer-assisted proof. Many real-world games remain too complex for complete analysis
Combinatorial_game_theory
Scientific area at the interface between computer science and mathematics
Automated theorem prover Computer-assisted proof Computational algebraic geometry Computer algebra system Differential analyser Proof checker Model checker
Computer_algebra
Mathematics term
on a computer. Notably, this method has confirmed property (T) for the automorphism group of the free group of rank at least 5. No human proof is known
Kazhdan's_property_(T)
American computer scientist
first player. Using 1500 hours of computer time, Patashnik's proof is an early example of a computer-assisted proof. In 1985, Patashnik created the bibliography-system
Oren_Patashnik
One of several theorems in different areas of mathematics
to Issai Schur. The Wikibook Combinatorics has a page on the topic of: Proof of Schur's theorem In Ramsey theory, Schur's theorem states that for every
Schur's_theorem
American software developer (born 1970)
employed at Google. He holds a PhD in computer science from UC Berkeley. He also made a computer-assisted proof system similar to Metamath: Ghilbert.
Raph_Levien
Method for bounding the errors of numerical computations
the idea in spring 1958, and a year later he published an article about computer interval arithmetic. Its merit was that starting with a simple principle
Interval_arithmetic
Conjecture about prime numbers, proof under review
+ 3). In 2013, Harald Helfgott released a supposed proof of Goldbach's weak conjecture. The proof was accepted for publication in the Annals of Mathematics
Goldbach's_weak_conjecture
Proposition in mathematics that is unproven
proof was not accepted by mathematicians at all because the computer-assisted proof was infeasible for a human to check by hand. However, the proof has
Conjecture
Mathematical constants related to chaotic behavior
thesis). University of Melbourne. Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434
Feigenbaum_constants
Result in dynamical systems
Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20. V. I. Arnold, "Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic
Kolmogorov–Arnold–Moser theorem
Kolmogorov–Arnold–Moser_theorem
Formal language and associated computer program
formal language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. Several databases of proved theorems
Metamath
Logic-based number-placement puzzle
thanks to the efforts of the Hong Kong judge Wayne Gould, who devised a computer program to rapidly produce unique puzzles. Number puzzles appeared in newspapers
Sudoku
Computer program for the Boolean satisfiability problem
better bound. SAT solvers have been used to assist in proving mathematical theorems through computer-assisted proof. In Ramsey theory, several previously unknown
SAT_solver
Geometric concept of a 2D space with "points at infinity" adjoined
This follows since dualizing each statement in the proof "in C" gives a statement of the proof "in C*." In the projective plane C, it can be shown that
Projective_plane
Geometry problem on tiling by hypercubes
of the conjecture in seven dimensions. Finally, a 200-gigabyte computer-assisted proof in 2019 used Keller graphs to establish that the conjecture holds
Keller's_conjecture
Concept in dynamical systems
collaboration with the author." Iii, Oscar E. Lanford (May 1982). "A computer-assisted proof of the Feigenbaum conjectures". Bulletin (New Series) of the American
Feigenbaum_function
23 mathematical problems stated in 1900
and transcendence of certain numbers. 8. Problems of prime numbers. 9. Proof of the most general law of reciprocity in any number field. 10. Determination
Hilbert's_problems
Reasoning Newsletter. 35: 1–3. Kolata, Gina (December 10, 1996). "Computer Math Proof Shows Reasoning Power". The New York Times. Retrieved 10 December
Robbins_algebra
Concept in quantum information theory
mutually unbiased bases, form a set that can be used in a Kochen–Specker proof. However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete
Mutually_unbiased_bases
Mathematical software
Noah (2023). "Computer Assisted Proofs and Automated Methods in Mathematics Education". Electronic Proceedings in Theoretical Computer Science. 375: 2–23
Computer_algebra_system
Fixed number that has received a name
Publishers. ISBN 0-632-00768-0. Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434
Mathematical_constant
Type of measurement in quantum mechanics
{\displaystyle \left\{D_{j,k}|\phi \rangle \right\}_{j,k=1}^{d}} is a SIC-POVM. The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open
SIC-POVM
Approach to mathematics using computation
proof. Use of computer programs to check a large but finite number of cases to complete a computer-assisted proof by exhaustion Thomas Hales's proof of
Experimental_mathematics
replacements for all builtin integer types. Safe numerics on GitHub Computer-assisted proof Interval arithmetic Affine arithmetic INTLAB (Interval Laboratory)
Validated_numerics
2008 mathematics book
duality with Poincaré duality, and views the book's attitude towards computer-assisted proof as "unnecessarily dismissive", he nevertheless concludes that the
Euler's_Gem
French academic and programmer
the TLA+ proof manager which supports the incremental development and checking of hierarchically structured computer-assisted proofs. The proof manager
Damien_Doligez
18 mathematical problems stated in 1998
attractor? Resolved. Result: Yes, solved by Warwick Tucker using a computer-assisted proof combined with normal form techniques. 2002 15th Do the Navier–Stokes
Smale's_problems
to which releasing a product with known issues is acceptable. Computer-assisted proof: Philosophical objections Philosophy of artificial intelligence
Philosophy of computer science
Philosophy_of_computer_science
Mathematical phenomenon
differ; therefore, this is not a rigorous mathematical proof of the phenomenon. A computer assisted proof of noise-induced order for the Matsumoto-Tsuda map
Noise-induced_order
Text-comparison software
Computer-assisted reviewing (CAR) tools are pieces of software based on text-comparison and analysis algorithms. These tools focus on the differences
Computer-assisted_reviewing
Topics referred to by the same term
refer to: China Association for Promoting Democracy CAPD library (Computer Assisted Proofs in Dynamics), a software library Central auditory processing disorder
CAPD
C++ module software library
The CAPD library (Computer Assisted Proofs in Dynamics) is a software library that aims to provide a set of flexible C++ modules designed for rigorous
CAPD_library
Point set triangulation minimizing total length
satisfies exactly one literal in each clause. The proof uses complex gadgets, and involves computer assistance to verify the correct behavior of these
Minimum-weight_triangulation
Abstract strategy game
by Oren Patashnik in 1980. Patashnik used a computer-assisted proof that consumed 1500 hours of computer time. The strategy comprised move choices for
3D_tic-tac-toe
Infinite sum
important in numerical analysis (especially validated numerics and computer-assisted proof). It can be used to prove convergence and to analyze rates of convergence
Series_(mathematics)
Tiling of the plane by pentagons
are considered distinct. In July 2017 Michaël Rao completed a computer-assisted proof showing that there are no other types of convex pentagons that
Pentagonal_tiling
Unsolved problem in graph theory
doi:10.1016/0012-365X(82)90079-6, MR 0676860 Kardoš, F. (2020), "A computer-assisted proof of the Barnette-Goodey Conjecture: not only fullerene graphs are
Barnette's_conjecture
Iterative method in conformal mapping
method Galerkin method Petrov–Galerkin method Validated numerics Computer-assisted proof Integrable algorithm Method of fundamental solutions Related Numerical
Schwarz_alternating_method
Decomposition in multilinear algebra
Then, the following statement was proved to be correct using a computer-assisted proof for all spaces of dimension Π < 15000 {\displaystyle \Pi <15000}
Tensor_rank_decomposition
Field of knowledge
rigorous proofs can require hundreds of pages to express, such as the 255-page Feit–Thompson theorem. The emergence of computer-assisted proofs has allowed
Mathematics
United States Ramon E. Moore prize University of Texas at El Paso Computer-assisted proof United States Richard C. DiPrima Prize Society for Industrial and
List_of_mathematics_awards
Automated reasoning Mathematics Automated theorem prover Computer-assisted proof – Computer algebra General Problem Solver Expert system – Decision support
Outline of artificial intelligence
Outline_of_artificial_intelligence
Mathematical treatise by Euclid
rigor arose, computer-assisted proofs, and the propositions of the Elements (with some updates to their proofs) have withstood computer checking. Some
Euclid's_Elements
Subfield of automated reasoning and mathematical logic
theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science. While
Automated_theorem_proving
6-regular graph with 57 vertices and 171 edges
to (i+1,k) when (k−j)3 = 26i. Coolsaet, K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel Graph." Designs, Codes and Crypt
Perkel_graph
AI program
for mathematical concepts to the search for useful heuristics. Computer-assisted proof Automated theorem proving Symbolic mathematics Experimental mathematics
Automated_Mathematician
Geometric system with a finite number of points
The non-existence of a finite plane of order 10 was proven in a computer-assisted proof that finished in 1989 – see (Lam 1991) for details. The next smallest
Finite_geometry
Mathematician
on computer calculations to verify analytical inequalities that were an essential ingredient in her proof, making the proof a computer-assisted proof. Romik's
Dan_Romik
Mathematical problem solved in 1967
the earliest examples of a computer-assisted proof. It was uncommon in the 1960s for mathematicians to rely on computers for research, but Boyce, then
Common_fixed_point_problem
Proof assistant
mathematical assertions, mechanical checking of proofs of these assertions, assists in finding formal proofs using proof automation routines and extraction of a
Rocq
Resolved problem in plane geometry
are no equichordal curves with excentricities > 0.33. The proof is mildly computer-assisted. In 1992, Schäfke and Volkmer showed that there is at most
Equichordal_point_problem
Korst, J.; van Leeuwen, J.; Spieksma, F. C. R. (2012-02-01). "Computer-assisted proof of performance ratios for the Differencing Method". Discrete Optimization
Balanced_number_partitioning
Dutch computer scientist
the packing chromatic number of the infinite square grid is 15. Computer-assisted proof#Applications Calmthout, Martijn van (June 6, 2016). "Bewijs dat
Marijn_Heule
value problems. Computer arithmetic, 225–286. L.B. Rall: Automatic Differentiation: Techniques and Applications, Lecture Notes in Computer Science 120, Springer
INTLAB
Kind of proof calculus
branch, known as dependent type theory, is used in a number of computer-assisted proof systems. Dependent type theory allows quantifiers to range over
Natural_deduction
Proof assistant program
Jape is the most popular program for "computer-assisted logic teaching" that involves exercises in developing proofs in mathematical logic. Jape was created
Jape_(software)
Protection of computer systems from information disclosure, theft or damage
related attempts. Hardware-based or assisted computer security also offers an alternative to software-only computer security. Using devices and methods
Computer_security
Test to determine whether a user is human
numbers from a distorted image. Because the test is administered by a computer, in contrast to the standard Turing test that is administered by a human
CAPTCHA
Surgical procedure
surgery or robot-assisted surgery is any type of surgical procedure that is performed with the use of robotic systems. Robotically assisted surgery was developed
Robotic_surgery
Assistive devices for people with disabilities
use. Wikimedia Commons has media related to Assistive technology. Accessibility Adaptive fashion Assisted living Augmentative and alternative communication
Assistive_technology
Computer hardware technology that uses quantum mechanics
benchmark-level quantum advantage should not be interpreted as proof that quantum computers are already broadly useful across practical computing workloads
Quantum_computing
Norwegian mathematician
computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning". arXiv:2110.15734v1 [math.OC]. Colbrook, Matthew
Anders_C._Hansen
item Computer-aided design Computer-aided manufacturing Computer-assisted proof Computer-assisted surgery Computer Audition Computer cluster Computer Engineering
Index_of_robotics_articles
1989 nonfiction book by Clifford Stoll
Maze of Computer Espionage is a 1989 book written by Clifford Stoll. It is his first-person account of the hunt for Markus Hess, a computer hacker who
The_Cuckoo's_Egg_(book)
Academic discipline
Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three
Logic_in_computer_science
Overview of and topical guide to computer science
Computability theory – What is calculable with the current models of computers. Proofs developed by Alan Turing and others provide insight into the possibilities
Outline_of_computer_science
American mathematician
problem, making this thesis one of the earliest examples of a computer-assisted proof. His 1981 paper, "Baxter Permutations and Functional Composition
William_Martin_Boyce
and imagining of what computers could do. ~ Items marked with a tilde are circa dates. Biography portal Lists portal Computer Pioneer Award IEEE John
List of pioneers in computer science
List_of_pioneers_in_computer_science
American mathematician
mathematics into computer science and philosophy. Notably, he explored the use of automated theorem proving and computer-assisted proofs in graph theory
Landon_Rabern
Limitative results in mathematical logic
ISBN 978-3-540-28372-0. S2CID 15610367. Paulson, Lawrence (2014). "A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Set of all computer components critical to its security
an exhaustive examination of its code base (by means of manual or computer-assisted software audit or program verification) becomes feasible. The term
Trusted_computing_base
This glossary of computer science is a list of definitions of terms and concepts used in computer science, its sub-disciplines, and related fields, including
Glossary_of_computer_science
Method for numerical differential equations
solution or the model data. Non-linear models for which such convergence proof of the GDM have been carried out comprise: the Stefan problem which is modelling
Gradient discretisation method
Gradient_discretisation_method
Area of mathematics
number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Computational
Computational_mathematics
English computer scientist (1912–1954)
(/ˈtjʊərɪŋ/; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist
Alan_Turing
Connection between brain and computer
external device, most commonly a computer or robotic limb. BCIs are often directed at researching, mapping, assisting, augmenting, or repairing human cognitive
Brain–computer_interface
American mathematician
assistant to Stephen Wolfram, assisting with work on Wolfram's book, A New Kind of Science. Among other things, he developed a proof showing that the Rule 110
Matthew_Cook
Cryptography secured against quantum computers
Post-quantum cryptography (PQC), sometimes referred to as quantum-proof, quantum-safe, or quantum-resistant, is the development of cryptographic algorithms
Post-quantum_cryptography
Adhering absolutely to certain constraints with consistency
modelled as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check some proofs mechanically. Formal rigour is the
Rigour
Overview of and topical guide to algorithms
algorithms includes their design, proof of correctness, efficiency, computational complexity, and implementation in computer programs. Algorithm — finite sequence
Outline_of_algorithms
Method of deriving conclusions
inference are relevant to many areas, such as proofs in mathematics and automated reasoning in computer science. Their conceptual and psychological underpinnings
Rule_of_inference
Use of various control systems for operating equipment
reliability. Agent-assisted automation refers to automation used by call center agents to handle customer inquiries. The key benefit of agent-assisted automation
Automation
1993 video game
enjoy this." In 1996, Computer Gaming World declared Star Wars Chess the 49th-worst computer game ever released stating: "Proof that there really is no
Star_Wars_Chess
Practice and study of secure communication techniques
End-to-End Encryption (E2EE), and Zero Knowledge Proofs (ZKP). Estimates suggest that a quantum computer could reduce the effort required to break today’s
Cryptography
Interactive theorem proving system
"Computed-assisted simulation proofs," Fifth Conference on Computer-Aided Verification (CAV '03), Costas Courcoubetis (editor), Lecture Notes in Computer Science
Larch_Prover
Processing of natural language by a computer
is the processing of natural language information by a computer. NLP is a subfield of computer science and is closely associated with artificial intelligence
Natural_language_processing
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
Boy/Male
Hindu
Computer
Boy/Male
Muslim
Helper. Assister.
Boy/Male
Arabic, Muslim
Compiler of Hadith
Boy/Male
Biblical
He that assists or is assisted.
Boy/Male
Arabic
Assister of the Faith
Boy/Male
Muslim
Compiler of Hadith
Biblical
he that assists or is assisted
Biblical
he that assists or is assisted
Girl/Female
Indian
Assisted, Victorious
Boy/Male
Afghan, Arabic, Australian, Iranian, Muslim, Parsi
Assister; Friend
Boy/Male
Australian, Finnish
Help; Assist
Girl/Female
Muslim
To compete with pride
Boy/Male
Hindu, Indian, Sanskrit
Compiler of the Vedas
Boy/Male
Biblical
He that assists or is assisted.
Boy/Male
Indian, Sanskrit
Unattained; Cannot be Competed with
Girl/Female
Arabic, Muslim
To Compete with Pride
Girl/Female
Muslim
Assisted, Victorious
Boy/Male
Arabic
Assister of the Faith
Boy/Male
Tamil
Computer
Girl/Female
Hindu, Indian, Latin
Young Girls who Assisted at Pagan Religious Ceremonies
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
Girl/Female
Haryanvi, Indian, Kannada, Telugu
Fearless; Surprise; Very Nice
Girl/Female
American, Australian, British, Christian, English, Finnish, Gujarati, Hebrew, Indian, Malayalam, Sindhi, Swedish
Grace; Favoured; God is Gracious; God has Shown Favour; Diminutive of Ann Frequently Used as an Independent Name; Unique
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Tamil, Telugu
Blessings
Girl/Female
Hindu
One who is endowed with immense capabilities, Name of Goddess Saraswati
Girl/Female
French
Grace. Famous bearer: 17th century aristrocat Ninon de Lenclos was famous for her wit and beauty.
Girl/Female
English American
From the linden tree island.
Boy/Male
Norse
Bear.
Boy/Male
Tamil
Gold or Lord Buddha, Early winter
Boy/Male
Tamil
Shaarwin | ஷாரà¯à®µà¯€à®¨Â
Victory
Boy/Male
Tamil
Maharanth | மஹாரஂத
Pollen inside a flower
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
COMPUTER ASSISTED-PROOF
p. pr. & vb. n.
of Compute
n.
One who reckons or computes; also, a book of calculations, tables, etc., to assist in reckoning.
imp. & p. p.
of Commute
n.
One who computes.
n.
One who composes or writes a book; a composer, as distinguished from an editor, translator, or compiler.
n.
A preparation of fruit in sirup in such a manner as to preserve its form, either whole, halved, or quartered; as, a compote of pears.
v. i.
To pay, or arrange to pay, in gross instead of part by part; as, to commute for a year's travel over a route.
n.
A composer or compiler of hymns; one versed in hymnology.
n.
One who commutes; especially, one who commutes in traveling.
imp. & p. p.
of Compute
v. i.
To calculate; to compute.
v. i.
To contend emulously; to seek or strive for the same thing, position, or reward for which another is striving; to contend in rivalry, as for a prize or in business; as, tradesmen compete with one another.
imp. & p. p.
of Compete
imp. & p. p.
of Assist
v. t.
To exchange; to put or substitute something else in place of, as a smaller penalty, obligation, or payment, for a greater, or a single thing for an aggregate; hence, to lessen; to diminish; as, to commute a sentence of death to one of imprisonment for life; to commute tithes; to commute charges for fares.
v. t.
To compute erroneously.
n.
A computer.
n.
A assister.
v. i.
To be present as a spectator; as, to assist at a public meeting.
n.
Compiler.