The size-change principle for program termination

Abstract

The "size-change termination" principle for a first-order functional language with well-founded data is: a program terminates on all inputs if every infinite call sequence (following program control flow) would cause an infinite descent in some data values.Size-change analysis is based only on local approximations to parameter size changes derivable from program syntax. The set of infinite call sequences that follow program flow and can be recognized as causing infinite descent is an ω -regular set, representable by a Büchi automaton. Algorithms for such automata can be used to decide size-change termination. We also give a direct algorithm operating on "size-change graphs" (without the passage to automata).Compared to other results in the literature, termination analysis based on the size-change principle is surprisingly simple and general: lexical orders (also called lexicographic orders), indirect function calls and permuted arguments (descent that is not in-situ ) are all handled automatically and without special treatment , with no need for manually supplied argument orders, or theorem-proving methods not certain to terminate at analysis time.We establish the problem's intrinsic complexity . This turns out to be surprisingly high, complete for PSPACE, in spite of the simplicity of the principle. PSPACE hardness is proved by a reduction from Boolean program termination. An ineresting consequence: the same hardness result applies to many other analyses found in the termination and quasitermination literature.

Quasi-terminating logic programs for ensuring the termination of partial evaluation

“…For this purpose, we adapt the notions of size-change graph and size-change termination [22]. Then, we show that size-change termination does not generally imply the (strong) termination of SLD computations.…”

“…The size-change principle [22] is a recent technique originally aimed at analyzing the termination of functional programs. Intuitively speaking, it consists in tracing size changes of function arguments when going from one function call to another by means of so-called size-change graphs.…”

“…Intuitively speaking, it consists in tracing size changes of function arguments when going from one function call to another by means of so-called size-change graphs. Then, assuming that the measure of size gives rise to a well-founded order, the following principle applies [22]: If every infinite computation would give rise to an infinitely decreasing value sequence (according to the size-change graphs), then no infinite computation is possible. Now, we adapt the definitions of size-change graph and maximal multigraph to logic programs; we mainly follow the definitions in [36] for rewrite systems, which in turn originates from the original definitions in [22] for first-order functional programs.…”

“…Following the spirit of [22], we say that a program is size-change terminating if every maximal multigraph contains at least one strictly decreasing parameter:…”

“…We formalize our quasi-termination analysis by means of size-change graphs [22], which (together with monotonicity constraints [4]) are currently used by an increasing number of termination analyzers for different programming languages. Basically, one should first approximate the transition relation of the program by means of sizechange graphs; then, we compute the closure of this set under an appropriate composition operator; finally, the program terminates if every idempotent graph in this closure includes a strictly decreasing parameter.…”