Program termination analysis in polynomial time

Abstract

A size-change termination algorithm takes as input abstract information about a program in the form of size-change graphs and uses it to determine whether any infinite computation would imply that some data decrease in size infinitely. Since such an infinite descent is presumed impossible, this proves program termination. The property of the graphs that implies program termination is called SCT. There are many examples of practical programs whose termination can be verified by creating size-change graphs and testing them for SCT.The size-change graph abstraction is useful because the graphs often carry sufficient information to deduce termination, and at the same time are simple enough to be analyzed automatically. However, there is a tradeoff between the completeness and efficiency of this analysis, and complete algorithms in the literature can easily be pushed to an exponential combinatorial search by certain patterns in the graph structures.We therefore propose a novel algorithm to detect common forms of parameter-descent behavior efficiently. Specifically, we target lexicographic descent, multiset descent, and min- and max-descent. Our algorithm makes it possible to verify practical instances of SCT while guarding against unwarranted combinatorial search. It has worst-case time complexity cubic in the input size, and its effectiveness is demonstrated empirically using a test suite of over 90 programs.