Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)

TL;DR

This paper demonstrates that fixed-points in the Intuitionistic Propositional Calculus are definable without fixed-point operators, providing an axiomatization and an algorithm to eliminate fixed-points, simplifying the $alculus.

Contribution

It introduces an axiomatization and an algorithm for fixed-point elimination in the IPC, showing that the $alculus is trivial and fixed-points can be removed.

Findings

Fixed-points in IPC are definable by formulas, making the $alculus trivial.

An algorithm computes fixed-point free equivalents of $alculus$ formulas.

Upper bounds for the number of iterations to reach fixed-points are established, with some being polynomial and optimal.

Abstract

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-calculus based on intuitionistic logic is trivial, every $\mu$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.