Ranking functions for size-change termination

Abstract

This article explains how to construct a ranking function for any program that is proved terminating by size-change analysis . The “principle of size-change termination” for a first-order functional language with well-ordered data is intuitive: A program terminates on all inputs, if every infinite call sequence (following program control flow) would imply an infinite descent in some data values. Size-change analysis is based on information associated with the subject program's call-sites. This information indicates, for each call-site, strict or weak data decreases observed as a computation traverses the call-site. The set DESC of call-site sequences for which the size-changes imply infinite descent is ω-regular, as is the set FLOW of infinite call-site sequences following the program flowchart. If FLOW ⊆ DESC (a decidable problem), every infinite call sequence would imply infinite descent in a well-ordering—an impossibility—so the program must terminate. This analysis accounts for termination arguments applicable to different call-site sequences, without indicating a ranking function for the program's termination. In this article, it is explained how one can be constructed whenever size-change analysis succeeds. The constructed function has an unexpectedly simple form; it is expressed using only min, max, and lexicographic tuples of parameters and constants. In principle, such functions can be tested to determine whether size-change analysis will be successful. As a corollary, if a program verified as terminating performs only multiply recursive operations, the function that it computes is multiply recursive. The ranking function construction is connected with the determinization of the Büchi automaton for DESC . While the result is not practical, it is of value in addressing the scope of size-change reasoning. This reasoning has been applied broadly, in the analysis of functional and logic programs, as well as term rewrite systems.