#### Multiphase-Linear Ranking Functions and their Relation to Recurrent Sets

##### Amir M. Ben-Amram, Jesús J. Doménech, Samir Genaim

Multiphase ranking functions (M$\Phi$RFs) are tuples $\langle f_1,\ldots,f_d \rangle$ 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 M$\Phi$RF asks to determine whether a given SLC loop admits a M$\Phi$RF, and the corresponding bounded decision problem restricts the search to M$\Phi$RFs 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 M$\Phi$RFs to that of seeking recurrent sets (used to prove non-termination). It also helps in identifying classes of loops for which M$\Phi$RFs 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 M$\Phi$RFs and SLC loops in general. For example, a result on bounded SLC loops becomes straightforward.

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