On Multiphase-Linear Ranking Functions

Abstract

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

Polynomial reachability witnesses via Stellensätze

“…In practice, our experimental results show that our automated approaches can solve instances beyond the reach of previous methods. An interesting future direction is to incorporate methods such as lexicographic ranking functions [4,17,23] into reachability. Another direction is to consider how our approach can be extended to automate the search for proofs in incorrectness logic [69].…”

“…This is also applicable to other approaches that rely on loop unrolling, such as [2]. • Termination analysis is a special kind of reachability which is usually guaranteed by well-foundedness reasoning such as (lexicographic) ranking functions [4,16,17,23,24,35,36,71,72]. It does not consider target states defined through numerical constraints over program variables.…”

Termination Analysis of Programs with Multiphase Control-Flow

“…For the case of SLC loops, the complexity and algorithmic aspects of the bounded version of the MΦRF problem were settled by Ben-Amram and Genaim [8]. The decision problem is PTIME for SLC loops with rational-valued variables, and coNP-complete for SLC loops with integer-valued variables; synthesising MΦRFs, when they exist, can be performed in polynomial and exponential time, respectively.…”

“…In practice, termination analysis tools search for MΦRFs incrementally, starting by depth 1 and increase the depth until they find one, or reach a predefined limit, after which the returned answer is don't know. Finding a theoretical upper-bound on the depth of a MΦRF, given the loop, would also settle this problem, however, as shown by Ben-Amram and Genaim [8] such bound must depend not only on the number of constraints or variables, as for other classes of LLRFs [3,6,10], but also on the coefficients used in the corresponding constraints. Yuan et al [34] proposed an incomplete method to bound the depth of MΦRFs for SLC loops.…”

The Termination and Complexity Competition

“…Moreover, Ctrl and AProVE also analyzed termination of systems that combine ITSs and TRSs. Here, iRankFinder [19] generates lexicographic combinations of ranking functions and ranks transitions incrementally [7]. Ctrl [37] and AProVE prove termination of TRSs extended by built-in integers by suitable adaptions of termination techniques for ordinary TRSs [24,36].…”