The Algebra of Iterative Constructions

2026-05-04

AI Summary

This work addresses the constructive problem of fixed-point iteration for continuous self-maps on complete lattices by introducing a purely algebraic reasoning framework, termed AIC, which eschews traditional index-based analytic methods in favor of equational logic to express fixed-point iterations as algebraic identities. The framework not only provides an algebraic reconstruction of classical results such as the Tarski–Kantorovich principle and its k-induction generalizations but also yields a novel fixed-point theorem based on limsup and liminf constructions. Furthermore, it reveals the incompleteness of finite axiom systems for this setting and demonstrates the necessity of infinitary axioms. All theoretical developments are formalized in Isabelle/HOL and fully verified automatically via Sledgehammer, thereby establishing precise completeness boundaries for the AIC system.

Abstract

Fixed points are a recurring theme in computer science and are often constructed as limits of suitably seeded fixed point iterations. We present the algebra of iterative constructions (AIC) -- a purely algebraic approach to reasoning about fixed point iterations of continuous endomaps on complete lattices. AIC allows derivations of constructive fixed point theorems via equational logic and avoids explicit computations with indices. For example, $$F \,\Diamond\, F^{*} \bot = \Diamond\, F^{*} \bot$$ states in AIC that $\sup_n F^n (\bot)$ -- a construction known from the Kleene fixed point theorem -- is a fixed point of $F$. We demonstrate the applicability of AIC by providing algebraic proofs of several well- and less-well-known fixed point theorems: Among others, we prove the Tarski-Kantorovich principle -- a generalization of the Kleene fixed point theorem -- as well as a fixed point-theoretic generalization of $k$-induction --a technique used in software verification. We moreover present a novel fixed point theorem. Under suitable continuity conditions, it obtains fixed points as lattice-theoretic limit inferiors and limit superiors of iterating an endomap on an arbitrary seed element. We have mechanized our algebra in Isabelle/HOL. Isabelle's sledgehammer tool is able to find proofs of the above fixed point theorems fully automatically. Finally, we investigate the completeness of our axiomatization of AIC. We prove that our finite set of finitary axioms is (a) sound but incomplete for standard models of AIC (sequences of elements from a complete lattice) and that (b) a different finite set of infinitary axioms is complete. We also prove that infinitary axioms are unavoidable: there exists no complete axiomatization of standard models given by finitely many finitary axioms.

Problem

• fixed point
• complete lattice
• algebraic reasoning
• axiomatization
• iteration

Innovation

• Algebra of Iterative Constructions
• fixed point theorems
• complete lattices
• equational logic
• automated theorem proving