Their proofs demonstrate a connection between the unsolvability of the decision problem for first-order logic and the unsolvability of the halting problem.
12.
In a 1802 letter to his friend Paolo Ruffini, Abbati extended the proof to the unsolvability of equations of degree greater than five.
13.
In 1959, he completed his doctoral work which involved a proof for lack of local unsolvability for constructing the minimal disjunctive normal form.
14.
Intuitively, this difference in unsolvability is because each instance of the " total machine " problem represents infinitely many instances of the Halting problem.
15.
Turing proved this by establishing the unsolvability of the halting problem, a result with far-ranging implications in both recursion theory and computer science.
16.
Later work by Emil Post and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.
17.
Together with Martin Davis he developed the Davis Putnam algorithm for the Boolean satisfiability problem and he helped demonstrate the unsolvability of Hilbert's tenth problem.
18.
Today recursion theory is mostly concerned with the more refined problem of complexity classes when is a problem efficiently solvable ? and the classification of degrees of unsolvability.
19.
Since the worst case setting often leads to negative results such as unsolvability and intractability, settings with weaker assurances such as average, probabilistic and randomized are also studied.
20.
The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures.
