01216nas a2200157 4500008004100000245005100041210004400092300001400136490000800150520073400158100002400892700002200916700001700938700002500955856007800980 2014 eng d00aOn the complexity of quantified linear systems0 acomplexity of quantified linear systems a128–1340 v5183 aIn this paper, we explore the computational complexity of the conjunctive fragment of the first-order theory of linear arithmetic. Quantified propositional formulas of linear inequalities with (k−1) quantifier alternations are log-space complete in ΣkP or ΠkP depending on the initial quantifier. We show that when we restrict ourselves to quantified conjunctions of linear inequalities, i.e., quantified linear systems, the complexity classes collapse to polynomial time. In other words, the presence of universal quantifiers does not alter the complexity of the linear programming problem, which is known to be in P. Our result reinforces the importance of sentence formats from the perspective of computational complexity.1 aRuggieri, Salvatore1 aEirinakis, Pavlos1 aSubramani, K1 aWojciechowski, Piotr uhttps://kdd.isti.cnr.it/publications/complexity-quantified-linear-systems01836nas 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