On Multiphase-Linear Ranking Functions

TL;DR

This paper provides a comprehensive analysis of multiphase-linear ranking functions for loop termination proofs, offering polynomial-time solutions for rational and real variables, and establishing complexity bounds for integer variables, along with new iteration bounds.

Contribution

It introduces complete solutions for the existence and synthesis of multiphase-linear ranking functions, including bounds on iterations, and relates lexicographic ranking functions to multiphase functions for loops.

Findings

Polynomial-time solutions for rational and real variables.

coNP-completeness for integer variables.

Linear bounds on loop iterations.

Abstract

Multiphase ranking functions ($\mathit{M{\Phi}RFs}$) were proposed as a means to prove the termination of a loop in which the computation progresses through a number of "phases", and the progress of each phase is described by a different linear ranking function. Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (single-path loops). We provide a complete polynomial-time solution to the problem of existence and of synthesis of $\mathit{M{\Phi}RF}$ of bounded depth (number of phases), when variables range over rational or real numbers; a complete solution for the (harder) case that variables are integer, with a matching lower-bound proof, showing that the problem is coNP-complete; and a new theorem which bounds the number of iterations for loops with $\mathit{M{\Phi}RFs}$. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions, $\mathit{LLRFs}$, more expressive than types of lexicographic functions for which complete solutions have been given so far. We prove that for the above type of loops, lexicographic functions can be reduced to $\mathit{M{\Phi}RFs}$, and thus the questions of complexity of detection and synthesis, and of resulting iteration bounds, are also answered for this class.