Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

Multiphase ranking functions (M (Forumala Presented). RFs) are used to prove termination of loops in which the computation progresses through a number of phases. They consist of linear functions (Forumala Presented). where (Forumala Presented). decreases during the ith phase. This work provides new insights regarding M (Forumala Presented). RFs for loops described by a conjunction of linear constraints ((Forumala Presented). loops). In particular, we consider the existence problem (does a given (Forumala Presented). loop admit a M (Forumala Presented). RF). The decidability and complexity of the problem, in the case that d is restricted by an input parameter, have been settled in recent work, while in this paper we make progress regarding the existence problem without a given depth bound. Our new approach, while falling short of a decision procedure for the general case, reveals some important insights into the structure of these functions. Interestingly, it relates the problem of seeking M (Forumala Presented). RFs to that of seeking recurrent sets (used to prove nontermination). It also helps in identifying classes of loops for which M (Forumala Presented). RFs are sufficient, and thus have linear runtime bounds. For the depth-bounded existence problem, we obtain a new polynomial-time procedure that can provide witnesses for negative answers as well. To obtain this procedure we introduce a new representation for (Forumala Presented). loops, the difference polyhedron replacing the customary transition polyhedron. We find that this representation yields new insights on M (Forumala Presented). RFs and (Forumala Presented). loops in general, and some results on termination and nontermination of bounded (Forumala Presented). loops become straightforward.