Fixed-point elimination in the intuitionistic propositional calculus

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, revealing the triviality of the $$-calculus.

Contribution

It introduces an axiomatization and an algorithm for fixed-point elimination in the intuitionistic propositional calculus, showing the triviality of the $$-calculus and providing bounds on fixed-point computations.

Findings

Fixed-points are definable by formulas of IPC.

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

Upper bounds for fixed-point iteration steps are polynomial and optimal for some formulas.

Abstract

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic 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 equivalent 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.