Abstract
The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.
Highlights
Fixed point theory in partially ordered sets plays a central role in the research activity in Mathematics and Computer Science ([8, 10, 12, 19, 25])
The aforesaid result allows to state the so-called Scott’s induction principle which models the meaning of recursive specifications in programming languages as the fixed point of non-recursive monotone self-mappings defined in partially ordered sets, in such a way that the aforesaid fixed point is the supremum of the sequence of successive iterations of the non-recursive mapping acting on a distinguished element of the model
In Scott’s approach, the non-recursive mapping models the evolution of the program execution and the partial order encodes some computational information notion so that each iteration of the mapping matches up with an element of the mathematical model which is greater than those that are associated to the preceding steps of the computational process
Summary
Fixed point theory in partially ordered sets plays a central role in the research activity in Mathematics and Computer Science ([8, 10, 12, 19, 25]). In the original version of the celebrated Kleene fixed point theorem, and in the aforesaid references, the assumed conditions have a global character, i.e., each element of the partially ordered set (the mathematical model) must satisfy them. In the aforementioned real applications, coming, for example, from Denotational Semantics or Logic Programming, to check the aforesaid conditions for all elements of the partially ordered set is unnecessary. In this paper we provide an affirmative answer to the question posed We characterize those properties that a self-mapping must satisfy in order to ensure that its set of fixed points is non-empty when a general partially ordered set is under consideration and no notion of order-completeness is assumed. The new fixed point method preserves the original Scott’s ideas providing a common framework for Denotational Semantics and Asymptotic Complexity of algorithms
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