In the last years a mathematical theory has been developed in order to obtain
a foundation for computing science. In this setting an important progress is
the establishment of mathematical models which are analyze according to their
computational complexity and measure the "distance" between programs and between algorithms.
In 1995, M. Schellekens started the development of a mathematical model to analyze
the algorithmic complexity based on the construction of a quasi-metric defined
on the space of the complexity, providing an adequate computational interpretation
of the fact that a program or an algorithm is more efficient than another in all of its inputs.
This information can be extracted by the asymmetric nature of the model. However,
this framework can not be applied to the analysis of those algorithms whose
complexity depends on two parameters. Motivated by this interesting fact, in this thesis
we will introduce a new complexity quasi-metric space which provides a useful model to
analyze such algorithms. On the other hand, the complexity quasi-metric space does not give
a computational interpretation of the fact that a program or an algorithm is only asymptotically
more efficient than another. The fuzzy (quasi-)metric spaces provide a parameter “t” such that
a suitable use of this ingredient may give rise to extra information on the involved
computational process; thus we will introduce the concept of complexity fuzzy quasi-metric space,
which provides a successful model to interpret the asymptotic efficiency of the complexity
functions.
In this context we will extend the main fixed-point theorems in fuzzy metric spaces,
using an appropriate notion of completeness, and get new ones. Some of these theorems will be
also established in the general framework of the intuitionistic fuzzy quasi-metric spaces,
with less restrictive conditions of contraction.
From the obtained results we deduce a general method to identify the (unique) solution
of recurrence equations associated to certain algorithms, as well as the analysis of
asymptotic efficiency of such algorithms.