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On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

Asier Estevan, Juan-José Minãna, Oscar Valero

TL;DR

The paper addresses weakening Kleene's fixed-point theorem to apply on general partially ordered sets without order-completeness by introducing orbitally $\preceq$-continuous mappings and exploring quasi-metric induced orders. It establishes equivalences that characterize when $Fix(f)\neq\emptyset$ under weaker hypotheses and extends these results to chain-complete and quasi-metric spaces, unifying monotone and non-monotone cases. The authors then apply this framework to asymptotic complexity analysis via Schellekens' complexity space $(\mathcal{C},d_{\mathcal{C}})$, recasting recurrence equations as fixed points of $\Phi_T$ and deriving simultaneous upper and lower bounds for running times, recovering and extending existing methods (e.g., Divide and Conquer, Probabilistic Divide and Conquer) with fewer assumptions. The work provides a cohesive bridge between Denotational Semantics and algorithmic complexity, offering a general fixed-point approach that subsumes prior techniques while enabling broader applicability and sharper asymptotic conclusions. Future work aims to connect orbital continuity with a corresponding topology, potentially yielding a new topological perspective on these fixed-point notions.

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.

On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

TL;DR

The paper addresses weakening Kleene's fixed-point theorem to apply on general partially ordered sets without order-completeness by introducing orbitally -continuous mappings and exploring quasi-metric induced orders. It establishes equivalences that characterize when under weaker hypotheses and extends these results to chain-complete and quasi-metric spaces, unifying monotone and non-monotone cases. The authors then apply this framework to asymptotic complexity analysis via Schellekens' complexity space , recasting recurrence equations as fixed points of and deriving simultaneous upper and lower bounds for running times, recovering and extending existing methods (e.g., Divide and Conquer, Probabilistic Divide and Conquer) with fewer assumptions. The work provides a cohesive bridge between Denotational Semantics and algorithmic complexity, offering a general fixed-point approach that subsumes prior techniques while enabling broader applicability and sharper asymptotic conclusions. Future work aims to connect orbital continuity with a corresponding topology, potentially yielding a new topological perspective on these fixed-point notions.

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.
Paper Structure (4 sections, 26 theorems, 24 equations)

This paper contains 4 sections, 26 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The fixed point theorems
  3. The application
  4. Future Work

Key Result

Theorem 2.1

Let $(X,\preceq)$ be a chain complete partially ordered set and let $f:X\rightarrow X$ be a $\preceq$-continuous mapping. Assume that there exist $x_{0}\in X$ such that $x_{0}\preceq f(x_{0})$. Then, there exist a fixed point $x^{\star}$ which is supremum of the sequence $(f^n(x_0))_{n\in\mathbb{N}^

Theorems & Definitions (47)

  • Theorem 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • proof
  • Example 2.9
  • ...and 37 more