The mathematical structure generated on a set of problems by the reductions of a particular type generally forms a preorder, whose equivalence classes may be used to define degrees of unsolvability and complexity classes.
22.
He was in 1983 an invited speaker at the International Congress of Mathematicians in Warsaw and gave a talk " The Degrees of Unsolvability : the Ordering of Functions by Relative Computability ".
23.
This is due to the unsolvability of the word problem for groups, or more precisely, the triviality problem ( given a finite presentation for a group, is it the trivial group ? ).
24.
For first-order syntactical unification, Martelli and Montanari gave an algorithm that reports unsolvability or computes a complete and minimal singleton substitution set containing the so-called "'most general unifier " '.
25.
An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is "'algorithmically unsolvable "'if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem.
26.
Cooper graduated from Jesus College, Oxford in 1966, and in 1970 received his PhD from University of Leicester under the supervision of Reuben Goodstein and C . E . M . Yates, with a thesis entitled " Degrees of Unsolvability ".
27.
Two sets are "'Turing equivalent "'if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent.
28.
However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ( for reasons somewhat analogous to the unsolvability of the halting problem ); various more-concrete ways of defining ordinals that definitely have notations are available.
29.
Strangely ( perhaps World War II intervened ) it took Post some ten years to dissect it in the " Appendix " to his paper " Recursive Unsolvability of a Problem of Thue ", 1947 ( reprinted in " Undecidable ", p . 293 ).
30.
At the suggestion of Alonzo Church, Emil Post in a paper published in 1947, first proved " a certain Problem of Thue " to be unsolvable, what Martin Davis states as " . . . the first unsolvability proof for a problem from classical mathematics-- in this case the word problem for semigroups . " ( Undecidable p . 292)
