Ranking Function Synthesis for Linear Lasso Programs

TL;DR

This paper introduces a comprehensive method for synthesizing termination arguments for linear lasso programs, extending existing techniques with new mathematical tools, diverse ranking functions, and a unified framework, achieving completeness under certain conditions.

Contribution

It extends termination synthesis methods by incorporating Motzkin's Transposition Theorem, multiple ranking function types, and a unified template approach, providing a complete solution for linear lasso programs.

Findings

Uses Motzkin's Theorem to handle strict inequalities.

Introduces multiphase and piecewise ranking functions.

Proves the method's completeness under specified conditions.

Abstract

The scope of this work is the constraint-based synthesis of termination arguments for the restricted class of programs called linear lasso programs. A termination argument consists of a ranking function as well as a set of supporting invariants. We extend existing methods in several ways. First, we use Motzkin's Transposition Theorem instead of Farkas' Lemma. This allows us to consider linear lasso programs that can additionally contain strict inequalities. Existing methods are restricted to non-strict inequalities and equalities. Second, we consider several kinds of ranking functions: affine-linear, piecewise and lexicographic ranking functions. Moreover, we present a novel kind of ranking function called multiphase ranking function which proceeds through a fixed number of phases such that for each phase, there is an affine-linear ranking function. As an abstraction to the synthesis of specific ranking functions, we introduce the notion ranking function template. This enables us to handle all ranking functions in a unified way. Our method relies on non-linear algebraic constraint solving as a subroutine which is known to scale poorly to large problems. As a mitigation we formalize an assessment of the difficulty of our constraints and present an argument why they are of an easier kind than general non-linear constraints. We prove our method to be complete: if there is a termination argument of the form specified by the given ranking function template with a fixed number of affine-linear supporting invariants, then our method will find a termination argument. To our knowledge, the approach we propose is the most powerful technique of synthesis-based discovery of termination arguments for linear lasso programs and encompasses and enhances several methods having been proposed thus far.