The Hardness of Finding Linear Ranking Functions for Lasso Programs

TL;DR

Determining the existence of linear ranking functions for lasso programs with preconditions is computationally hard, with proven EXPSPACE-hardness over integers and PSPACE-hardness over rationals, highlighting the complexity of loop termination analysis.

Contribution

This paper establishes new complexity lower bounds for deciding linear ranking functions in lasso programs with preconditions, extending understanding of the problem's computational difficulty.

Findings

EXPSPACE-hardness over integers for linear ranking functions

PSPACE-hardness over rationals for linear ranking functions

Complexity bounds for invariant-supported ranking functions

Abstract

Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linear-constraint loop with a precondition, specifically with partially-specified input, has a linear ranking function is EXPSPACE-hard over the integers, and PSPACE-hard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear ranking-function supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACE-hard for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of natural numbers as invariants.