Fixpoint Theory – Upside Down

Paolo Baldan,Richard Eggert,Barbara König,Tommaso Padoan

doi:10.1007/978-3-030-71995-1_4

Abstract

Knaster-Tarski’s theorem, characterising the greatest fix- point 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, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

Highlights

A monotone function f : L → L over a complete lattice (L, ), by Knaster-Tarski’s theorem [28], admits a least fixpoint μf and greatest fixpoint νf which are characterised as the least pre-fixpoint and the greatest post-fixpoint, respectively
This immediately gives well-known proof principles for showing that a lattice element l ∈ L is below νf or above μf l f (l)
Simple stochastic games are an important type of games that subsume parity games and the computation of behavioural distances for probabilistic automata

Summary

Introduction

Fixpoints are ubiquitous in computer science as they allow to provide a meaning to inductive and coinductive definitions (see, e.g., [26,23]). The aim of this paper is to present an alternative proof rule for this purpose for functions over lattices of the form L = MY where Y is a finite set and M is an MV-chain, i.e., a totally ordered complete lattice endowed with suitable operations of sum and complement This allows us to capture several examples, ranging from ordinary relations, for dealing with bisimilarity, behavioural metrics, termination probabilities and simple stochastic games. Our idea is to compute the set of states that still has some “wiggle room”, i.e., those states which could reduce their termination probability by δ if all their successors did the same This definition has a coinductive flavour and it can be computed as a greatest fixpoint on the finite powerset 2S of states, instead of on the infinite lattice S[0,1]. Proofs and further material can be found in the full version of the paper [5]

Lattices and MV-Algebras

Non-expansive Functions and Their Approximations

Proof Rules

Termination Probability

Behavioural Metrics for Probabilistic Automata

Bisimilarity

Simple Stochastic Games

Conclusion