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International Static Analysis Symposium

SAS 2020: Static Analysis pp 89–112Cite as

Termination of Polynomial Loops

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12389))

Abstract

We consider the termination problem for triangular weakly non-linear loops (twn-loops) over some ring \(\mathcal {S}\) like \(\mathbb {Z}\), \(\mathbb {Q}\), or \(\mathbb {R}\). Essentially, the guard of such a loop is an arbitrary Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment where each \(x_i\) is a variable, \(c_i \in \mathcal {S}\), and each \(p_i\) is a (possibly non-linear) polynomial over \(\mathcal {S}\) and the variables \(x_{i+1},\ldots ,x_{d}\).

We present a reduction from the question of termination to the existential fragment of the first-order theory of \(\mathcal {S}\) and \(\mathbb {R}\). For loops over \(\mathbb {R}\), our reduction entails decidability of termination. For loops over \(\mathbb {Z}\) and \(\mathbb {Q}\), it proves semi-decidability of non-termination.

Furthermore, we present a transformation to convert certain non-twn-loops into twn-form. Then the original loop terminates iff the transformed loop terminates over a specific subset of \(\mathbb {R}\), which can also be checked via our reduction. This transformation also allows us to prove tight complexity bounds for the termination problem for two important classes of loops which can always be transformed into twn-loops.

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 389792660 as part of TRR 248, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 235950644 (Project GI 274/6-2), and by the DFG Research Training Group 2236 UnRAVeL.

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Notes

  1. 1.

    We use row- and column-vectors interchangeably to improve readability.

  2. 2.

    Note that negation is syntactic sugar in our setting, as, e.g., \(\lnot (p > 0)\) is equivalent to \(-p \ge 0\). So w.l.o.g. \(\varphi \) is built from atoms, \(\wedge \), and \(\vee \).

  3. 3.

    In this paper “linear” refers to “linear polynomials” and thus includes affine functions.

  4. 4.

    So we have \(\eta (c \cdot p + c' \cdot p') = c \cdot \eta (p) + c' \cdot \eta (p')\), \(\eta (1)=1\), and \(\eta (p \cdot p') = \eta (p)\cdot \eta (p')\) for all \(c,c' \in \mathcal {S}\) and all \(p,p' \in \mathcal {S}[\vec {x}]\).

  5. 5.

    In other words, we have \(\vec {a}' = (\eta (\vec {x})) \, (\vec {a}) \, (\eta ^{-1}(\vec {x}))\), since \((\eta ^{-1}\circ \widetilde{a}\circ \eta )(\vec {x}) = \eta ^{-1}(\eta (\vec {x})[\vec {x}/\vec {a}]) = \eta (\vec {x})[\vec {x}/\vec {a}][\vec {x} / \eta ^{-1}(\vec {x})] = (\eta (\vec {x}))(\vec {a})(\eta ^{-1}(\vec {x}))\).

  6. 6.

    Our definition of poly-exponential expressions slightly generalizes [15, Def. 9] (e.g., we allow polynomials over the variables \(\vec {x}\) instead of just linear combinations).

  7. 7.

    More precisely, the reduction of Lemma 34 and of the following Theorem 36 takes polynomially many steps in the size of the input of the function in (5).

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Acknowledgments

We thank Alberto Fiori for help with the example loop (13) and Arno van den Essen for useful discussions.

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Authors and Affiliations

  1. Max Planck Institute for Informatics and Saarland Informatics Campus, Saarbrücken, Germany

    Florian Frohn

  2. LuFG Informatik 2, RWTH Aachen University, Aachen, Germany

    Marcel Hark & Jürgen Giesl

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    David Pichardie

  2. LSV, ENS Paris-Saclay, Gif-sur-Yvette, France

    Mihaela Sighireanu

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Frohn, F., Hark, M., Giesl, J. (2020). Termination of Polynomial Loops. In: Pichardie, D., Sighireanu, M. (eds) Static Analysis. SAS 2020. Lecture Notes in Computer Science(), vol 12389. Springer, Cham. https://doi.org/10.1007/978-3-030-65474-0_5

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