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Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

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Static Analysis (SAS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11822))

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Abstract

Multiphase ranking functions (M\(\varPhi \)RFs) are used to prove termination of loops in which the computation progresses through a number of phases. They consist of linear functions \(\langle f_1,\ldots ,f_d \rangle \) where \(f_i\) decreases during the ith phase. This work provides new insights regarding M\(\varPhi \)RFs for loops described by a conjunction of linear constraints (\( SLC \) loops). In particular, we consider the existence problem (does a given \( SLC \) loop admit a M\(\varPhi \)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\(\varPhi \)RFs to that of seeking recurrent sets (used to prove nontermination). It also helps in identifying classes of loops for which M\(\varPhi \)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 \( SLC \) loops, the difference polyhedron replacing the customary transition polyhedron. We find that this representation yields new insights on M\(\varPhi \)RFs and \( SLC \) loops in general, and some results on termination and nontermination of bounded \( SLC \) loops become straightforward.

This work was funded partially by the Spanish MICINN/FEDER, UE project RTI2018-094403-B-C31, the MINECO project TIN2015-69175-C4-2-R, the CM project S2018/TCS-4314 and by the pre-doctoral UCM grant CT27/16-CT28/16.

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Authors and Affiliations

  1. School of Computer Science, The Tel-Aviv Academic College, Tel Aviv, Israel

    Amir M. Ben-Amram

  2. DSIC, Complutense University of Madrid (UCM), Madrid, Spain

    Jesús J. Doménech & Samir Genaim

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  1. Amir M. Ben-Amram
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  2. Jesús J. Doménech
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  3. Samir Genaim
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Correspondence to Samir Genaim .

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  1. University of Colorado, Boulder, CO, USA

    Bor-Yuh Evan Chang

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Ben-Amram, A.M., Doménech, J.J., Genaim, S. (2019). Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets. In: Chang, BY. (eds) Static Analysis. SAS 2019. Lecture Notes in Computer Science(), vol 11822. Springer, Cham. https://doi.org/10.1007/978-3-030-32304-2_22

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  • DOI: https://doi.org/10.1007/978-3-030-32304-2_22

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