01836nas a2200157   4500008004100000245003800041210003500079300001400114490000700128520138300135100002201518700002401540700001701564700002501581856007201606       2014                        eng d00aOn quantified linear implications0 aquantified linear implications  a301–3250 v713 aA 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  ΠP2Π2P -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.1 aEirinakis, Pavlos1 aRuggieri, Salvatore1 aSubramani, K1 aWojciechowski, Piotr  uhttps://kdd.isti.cnr.it/publications/quantified-linear-implications