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Redwood City, California: Benjamin/Cummings Publishing Company, Inc. Appendix C includes impossibility of algorithms deciding if a grammar contains ambiguities, and impossibility of verifying program correctness by an algorithm as example of Halting Problem. Halava, Vesa (1997).
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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Classic examples of wicked problems include economic, environmental, and political issues. A problem whose solution requires a great number of people to change their mindsets and behavior is likely to be a wicked problem. Therefore, many standard examples of wicked problems come from the areas of public planning and policy.
A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics.This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems.
For example, some unicellular organisms have genomes much larger than that of humans. Cole's paradox: Even a tiny fecundity advantage of one additional offspring would favor the evolution of semelparity. Gray's paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.
Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Gödel's Incompleteness Theorem. [5] In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term.
"Undecidable", sometimes also used as a synonym of independent, something that can neither be proved nor disproved within a mathematical theory Undecidable figure , a two-dimensional drawing of something that cannot exist in 3d, such as appeared in some of the works of M. C. Escher