Fixpoint Theory -- Upside Down

TL;DR

This paper develops dual proof principles for fixpoints in lattice theory, enabling new methods to establish bounds in probabilistic and metric systems, with applications to algorithms for stochastic games.

Contribution

It introduces dual fixpoint proof rules for showing elements are above the greatest or below the least fixpoint, extending classical coinduction and induction principles.

Findings

Applicable to termination probabilities and behavioral distances

Enables new algorithms for simple stochastic games

Broad applicability to probabilistic automata and metric systems

Abstract

Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $\mathbb{M}^Y$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.