Computer-assisted proof
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 implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Attempts have also been made in the area of
Methods
One method for using computers in mathematical proofs is by means of so-called validated numerics or rigorous numerics. This means computing numerically yet with mathematical rigour. One uses set-valued arithmetic and inclusion principle[clarify] in order to ensure that the set-valued output of a numerical program encloses the solution of the original mathematical problem. This is done by controlling, enclosing and propagating round-off and truncation errors using for example interval arithmetic. More precisely, one reduces the computation to a sequence of elementary operations, say . In a computer, the result of each elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing elementary operations between such intervals of representable numbers.[citation needed]
Philosophical objections
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Computer-assisted proofs are the subject of some controversy in the mathematical world, with Thomas Tymoczko first to articulate objections. Those who adhere to Tymoczko's arguments believe that lengthy computer-assisted proofs are not, in some sense, 'real' mathematical proofs because they involve so many logical steps that they are not practically verifiable by human beings, and that mathematicians are effectively being asked to replace logical deduction from assumed axioms with trust in an empirical computational process, which is potentially affected by errors in the computer program, as well as defects in the runtime environment and hardware.[1]
Other mathematicians believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so that its use can then be regarded as a mere "verification". Arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware can be resolved by providing a formal proof of correctness for the computer program (an approach which was successfully applied to the four-color theorem in 2005) as well as replicating the result using different programming languages, different compilers, and different computer hardware.
Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine-readable form, and then use a
Another argument against computer-aided proofs is that they lack
An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into a
The emerging field of experimental mathematics is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration.
Applications
Theorems proved with the help of computer programs
Inclusion in this list does not imply that a formal computer-checked proof exists, but rather, that a computer program has been involved in some way. See the main articles for details.
- Four color theorem, 1976[2]
- Mitchell Feigenbaum's universality conjecture in non-linear dynamics. Proven by O. E. Lanford using rigorous computer arithmetic, 1982
- Connect Four, 1988 – a solved game
- Non-existence of a finite projective plane of order 10, 1989
- Double bubble conjecture, 1995[3]
- Robbins conjecture, 1996
- Kepler conjecture, 1998 – the problem of optimal sphere packing in a box
- Lorenz attractor, 2002 – 14th of Smale's problems proved by Warwick Tucker using interval arithmetic
- 17-point case of the Happy Ending problem, 2006
- KourilSAT-solver.
- NP-hardness of minimum-weight triangulation, 2008
- AhmedSAT-solvers to prove w(2; 3, 17) = 279 and w(2; 3, 18) = 312 in 2010.[8]
- Optimal solutions for Rubik's Cube can be obtained in at most 20 face moves, 2010[12]
- Minimum number of clues for a solvable Sudoku puzzle is 17, 2012
- In 2014 a special case of the SAT-solver. The full conjecture was later solved by Terence Tao without computer assistance.[13]
- terabytes of data in May 2016.[14]
- Applications to the
- Kazhdan's property (T) for the automorphism group of a free group of rank at least five
- Keller's conjecture in dimension 7 the only remaining case in 2020 with a 200 gigabyte proof[19][20]
- The packing chromatic number of the infinite square grid is 15, by Subercaseaux and Heule in 2023[21][22] (See also: Hadwiger–Nelson problem for the chromatic number of the plane)
See also
- Formal verification
- Logic Theorist
- Mathematical proof
- Metamath
- Model checking
- Seventeen or Bust
- Symbolic computation
- Validated numerics
References
- JSTOR 2025976.
- (PDF) from the original on 2011-08-05
- .
- ^ Kouril, Michal (2006). A Backtracking Framework for Beowulf Clusters with an Extension to Multi-Cluster Computation and Sat Benchmark Problem Implementation (Ph.D. thesis). University of Cincinnati.
- ^
Kouril, Michal (2012). "Computing the van der Waerden number W(3,4)=293". Integers. 12: A46. MR 3083419.
- ^ Kouril, Michal (2015). "Leveraging FPGA clusters for SAT computations". Parallel Computing: On the Road to Exascale: 525–532.
- ^
Ahmed, Tanbir (2009). "Some new van der Waerden numbers and some van der Waerden-type numbers". Integers. 9: A6. S2CID 122129059.
- ^ a b
Ahmed, Tanbir (2010). "Two new van der Waerden numbers w(2;3,17) and w(2;3,18)". Integers. 10 (4): 369–377. S2CID 124272560.
- ^
Ahmed, Tanbir (2012). "On computation of exact van der Waerden numbers". Integers. 12 (3): 417–425. S2CID 11811448.
- ^
Ahmed, Tanbir (2013). "Some More Van der Waerden numbers". Journal of Integer Sequences. 16 (4): 13.4.4. MR 3056628.
- ^
Ahmed, Tanbir; Kullmann, Oliver; Snevily, Hunter (2014). "On the van der Waerden numbers w(2;3,t)". MR 3215454.
- ^ "God's Number is 20". cube20.org. July 2010. Retrieved 2023-10-18.
- PMID 26432222.
- PMID 27251254.
- doi:10.1063/1.527418.
- S2CID 28258285.
- arXiv:1711.08076 [cs.LO].
- ^ "Schur Number Five". www.cs.utexas.edu. Retrieved 2021-10-06.
- PMC 7324133.
- ^ Hartnett, Kevin (2020-08-19). "Computer Search Settles 90-Year-Old Math Problem". Quanta Magazine. Retrieved 2021-10-08.
- arXiv:2301.09757 [cs.DM].
- ^ Hartnett, Kevin (2023-04-20). "The Number 15 Describes the Secret Limit of an Infinite Grid". Quanta Magazine. Retrieved 2023-04-20.
Further reading
- Lenat, D.B. (1976). AM: An artificial intelligence approach to discovery in mathematics as heuristic search (PDF) (PhD). AI Lab., Stanford University. STAN-CS-76-570, Heuristic Programming Project Report HPP-76-8.
- Meyer, K.R.; Schmidt, D.S., eds. (2012). Computer aided proofs in analysis. IMA volumes in mathematics and it's applications. Vol. 28. Springer. ISBN 978-1-4613-9092-3.
- Nakao, M.; Plum, M.; Watanabe, Y. (2019). Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations. Springer Series in Computational Mathematics. Springer. ISBN 9789811376696.
External links
- Lanford, Oscar E. (1982). "A computer-assisted proof of the Feigenbaum conjectures" (PDF). Bull. Amer. Math. Soc. 6 (3): 427–434. .
- Furse, Edmund (1990). Why did AM run out of steam? (Technical report). Department of Computer Studies, University of Glamorgan. CS-90-4. Archived from the original on 2012-07-17. Retrieved 2016-09-06.
{{cite tech report}}
: CS1 maint: bot: original URL status unknown (link) - Begley, S. (April 16, 2018). "Number proofs done by computer might err". Pittsburgh Post-Gazette. Archived from the original on 2018-04-16.
- "A Special Issue on Formal Proof". Notices of the American Mathematical Society. December 2008.