On quantified linear implications

  • Pavlos EirinakisEmail author
  • Salvatore Ruggieri
  • K. Subramani
  • Piotr Wojciechowski


A Quantified Linear Implication (QLI) is an inclusion query over two polyhedral sets, with a quantifier string that specifies which variables are existentially quantified and which are universally quantified. Equivalently, it can be viewed as a quantified implication of two systems of linear inequalities. In this paper, we provide a 2-person game semantics for the QLI problem, which allows us to explore the computational complexities of several of its classes. More specifically, we prove that the decision problem for QLIs with an arbitrary number of quantifier alternations is PSPACE-hard. Furthermore, we explore the computational complexities of several classes of 0, 1, and 2-quantifier alternation QLIs. We observed that some classes are decidable in polynomial time, some are NP-complete, some are coNP-hard and some are \(\boldsymbol{\Pi}_{\textbf 2}^{\textbf P}\) -hard. We also establish the hardness of QLIs with 2 or more quantifier alternations with respect to the first quantifier in the quantifier string and the number of quantifier alternations. All the proofs that we provide for polynomially solvable problems are constructive, i.e., polynomial-time decision algorithms are devised that utilize well-known procedures. QLIs can be utilized as powerful modelling tools for real-life applications. Such applications include reactive systems, real-time schedulers, and static program analyzers.


Quantified Linear Implication Linear constraints Inclusion query over polyhedral sets  Computational complexity Polynomial Hierarchy 

Mathematics Subject Classifications (2010)

68Q17 03D15 90C05 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Pavlos Eirinakis
    • 1
    Email author
  • Salvatore Ruggieri
    • 2
  • K. Subramani
    • 1
  • Piotr Wojciechowski
    • 1
  1. 1.LDCSEEWest Virginia UniversityMorgantownUSA
  2. 2.Dipartimento di InformaticaUniversità di PisaPisaItaly

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