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A Graphical Theorem of the Alternative for UTVPI Constraints

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Theoretical Aspects of Computing - ICTAC 2015 (ICTAC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9399))

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Abstract

There exist quite a few theorems of the alternative for linear systems in the literature, with Farkas’ lemma being the most famous. All these theorems have the following form: We are given two closely related linear systems such that one and exactly one has a solution. Some specialized classes of linear systems can also be represented using graphical structures and the corresponding theorems of the alternative can then be stated in terms of properties of the graphical structure. For instance, it is well-known that a system of difference constraints (DCS) can be represented as a constraint network such that the DCS is feasible if and only if there does not exist a negative cost cycle in the network. In this paper, we provide a new graphical constraint network representation of Unit Two Variable Per Inequality (UTVPI) constraints. This constraint network representation permits us to derive a theorem of the alternative for the feasibility of UTVPI systems. UTVPI constraints find applications in a number of domains, including but not limited to program verification, abstract interpretation, and array bounds checking. Theorems of the Alternative find primary use in the design of certificates in certifying algorithms. It follows that our work is important from this perspective.

This research was supported in part by the National Science Foundation through Award CCF-1305054.

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Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    K. Subramani & Piotr Wojciechowski

Authors
  1. K. Subramani
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  2. Piotr Wojciechowski
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Correspondence to Piotr Wojciechowski .

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Editors and Affiliations

  1. University of LĂĽbeck, LĂĽbeck, Schleswig-Holstein, Germany

    Martin Leucker

  2. Pontificia Universidad Javeriana-Cali, Cali, Colombia

    Camilo Rueda

  3. Pontificia Universidad Javeriana-Cali, CNRS LIX École Polytechnique de Paris, Palaiseau Cedex, France

    Frank D. Valencia

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Subramani, K., Wojciechowski, P. (2015). A Graphical Theorem of the Alternative for UTVPI Constraints. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_20

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  • DOI: https://doi.org/10.1007/978-3-319-25150-9_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25149-3

  • Online ISBN: 978-3-319-25150-9

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