Computability Theory
Object 'Computability Theory' belongs to the 'Branches Of Math' theme.Connection to Real Analysis:
One of the biggest, least expected connections between these fields comes from the DPRM-Theorem, also known as The Solution to Hilbert's 10th Problem. This theorem says that the recursively enumerable sets are precisely the diophantine sets. This can be used to prove some otherwise intractable statements in real analysis which don't appear to be related to computability theory at all.
In turn, real analysis contributes back to computability theory by providing concepts like "computable real number", "computable real function", and so on, which is collectively the subject of a subfield sometimes called "computable real analysis".
Connection to Abstract Algebra:In turn, real analysis contributes back to computability theory by providing concepts like "computable real number", "computable real function", and so on, which is collectively the subject of a subfield sometimes called "computable real analysis".
The most famous connection between computability theory and abstract algebra is the celebrated Word Problem: in a group generated by certain generators which are constrained by certain relations, determine when two words are the same. Computability theory is used to show that no general algorithm exists to perform this task.
Connection to Set Theory:Although a basic introduction to computability theory might begin with computable *functions*, emphasis soon shifts toward computable *sets*. Sets of natural numbers provide a rich playground for playing with Turing jumps and such things.
Connection to Number Theory:Computability theory can be used to greatly simplify certain results in number theory. For example, the number-theoretic problem, "Find a sequence of naturals which produces a transcendental continued fraction", is typically done using some rather tricky acrobatics involving Diophantine approximation. This can be simplified tremendously by exploiting a function like the Busy Beaver function from computability theory.
In turn, a great deal of number theory is used in the proof of the DPRM-Theorem, one of the most important results of computability theory. The proof involves a staggering amount of modular arithmetic manipulation.
Connection to Graph Theory:In turn, a great deal of number theory is used in the proof of the DPRM-Theorem, one of the most important results of computability theory. The proof involves a staggering amount of modular arithmetic manipulation.
Many of the prototypical NP-hard problems studied in complexity theory involve graphs: the Traveling Salesman Problem, for example.
Connection to Model Theory:Model theorists are sometimes interested in the computational complexity of the set of statements needed to axiomatize a given theory.
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