Problem Theory

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DOI: 10.6084/m9.figshare.4956353


It presents a problem theory, which is an intuitionist set theory where the resolving subject is a computing device.

A problem is a set defined by intension, its set of solutions is the same set but defined by extension, and to resolve a problem is to calculate its set of solutions, though most times it would be enough to calculate one solution, that is, it would be enough to solve the problem.

Problem solving figure

Following Turing (1936), computing is the model for the resolutions. As a resolver implements a resolution, we model a resolver as a Turing machine. Then we define the range of a resolver as the set of problems that the resolver solves. If the range of resolver A is a superset of the range of resolver B, then resolver A is better (at solving) than resolver B. This allows us to construct a series of improving resolvers.


The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here is why, Why are we Turing complete?

Being Turing complete also means that somehow our brain implements the function that a universal Turing machine implements. The point is that evolution achieved Turing completeness, and then the explanation should be evolutionary, but our explanation is mathematical. The trick is to introduce a mathematical theory of problems, under the basic assumption that solving more problems provides more survival opportunities.

So we build a problem theory by fusing set and computing theories. Then we construct a series of resolvers, where each resolver is defined by its computing capacity, that exhibits the following property: all problems solved by a resolver are also solved by the next resolver in the series if certain condition is satisfied. The last of the conditions is to be Turing complete.

This series defines a resolvers hierarchy that could be seen as a framework for the evolution of cognition. Then the answer to our question would be: to solve most problems. By the way, the problem theory defines adaptation, perception, and learning, and it shows that there are just three ways to resolve any problem: routine, trial, and analogy. And, most importantly, this theory demonstrates how problems can be used to found mathematics and computing on biology.


The series of improving resolvers has five elements: mechanism, adapter, perceiver, learner, and subject. The best resolver is a Turing complete subject, which is computationally equivalent to a universal Turing machine.


Link to the page of my problem theory in figshare, and direct link to the pdf file.

A second version is in the arXiv, with its pdf file.

And there is another version in The Internet Archive.

External references used in this page:

Última actualización: 2019-08-10.

© Ramón Casares 2019