No 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.

In this paper we study the complexity of the Linear Ranking problem: given a loop, described by linear constraints over a finite set of integer variables, is there a linear ranking function for this loop? While existence of such a function implies termination, this problem is not equivalent to termination. When the variables range over the rationals or reals, the Linear Ranking problem is known to be PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity of the Linear Ranking problem has not yet been determined. We show that it is coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear ranking functions, both for the general case and the special PTIME cases.

Multiphase ranking functions (MΦRFs) 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 MΦRF 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 MΦRFs. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions, LLRF , 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 MΦRFs, and thus the questions of complexity of detection and synthesis, and of resulting iteration bounds, are also answered for this class. * amirben@mta.ac.il

Multiphase ranking functions (MΦRFs) are tuples f 1 , . . . , f d of linear functions that are often used to prove termination of loops in which the computation progresses through a number of "phases". Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (Single-Path Constraint loops). The decision problem existence of a MΦRF asks to determine whether a given SLC loop admits a MΦRF; and the corresponding bounded decision problem restricts the search to MΦRFs of depth d, where the parameter d is part of the input. The algorithmic and complexity aspects of the bounded problem have been completely settled in a recent work. In this paper we make progress regarding the existence problem, without a given depth bound. In particular, we present an approach that reveals some important insights into the structure of these functions. Interestingly, it relates the problem of seeking MΦRFs to that of seeking recurrent sets (used to prove non-termination). It also helps in identifying classes of loops for which MΦRFs are sufficient. Our research has led to a new representation for single-path loops, the difference polyhedron replacing the customary transition polyhedron. This representation yields new insights on MΦRFs and SLC loops in general. For example, a result on bounded SLC loops becomes straightforward.