M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. JCSS 60(3): 660--716 (2000).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....bounds that show the complexity of S len queries, although within PH, may be prohibitively high. Let MSO(SC) be the class of queries over SC expressible in monadic second order logic. This includes queries of high complexity, namely for each level of the polynomial hierarchy, PH, complete queries [3], in particular, NP complete and coNP complete ones (3 colorability and its complement) Such queries cannot be expressed over arbitrary databases in RC(S len ) e.g. not over nice ones) however, they can be expressed under some additional assumptions. We say that the width of the active domain ....

M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. JCSS 60(3): 660--716 (2000).

....However, we can show the following. Proposition 15 For every i 2 N, there are MSO formulas i and i such that the membership problem for i and i is hard for Sigma and Pi , respectively. In contrast, membership for 2A PAs is ptime complete. Proof. Ajtai, Fagin, and Stockmeyer [4] showed that for every level of the polynomial hierarchy (ph) there is an MSO formula over graphs such that model checking is hard for that level. Hence, it suffices to observe that graphs can readily be encoded as strings. We describe a translation from MSO formulas to MSO formulas and ....

M. Ajtai, R. Fagin, and L. J. Stockmeyer. The Closure of Monadic NP. Journal of Computer and System Sciences, 60(3):660--716, 2000.

....bounds that show the complexity of S len queries, although within PH, may be prohibitively high. Let MSO(SC) be the class of queries over SC expressible in monadic secondorder logic. This includes queries of high complexity, namely for each level of the polynomial hierarchy, PH, complete queries [3], in particular, NP complete and coNP complete ones (3 colorability and its complement) Such queries cannot be expressed over arbitrary databases in RC(S len ) e.g. not over nice ones) however, they can be expressed under some additional assumptions. 40 We say that the width of the active ....

M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. JCSS 60(3): 660-716 (2000).

....However, we can show the following. Proposition 14 For every i # N, there are MSO # formulas # i and # i such that the membership problem for # i and # i is hard for # P i and # P i , respectively. In contrast, membership for 2A PAs is ptime complete. Proof. Ajtai, Fagin, and Stockmeyer [3] showed that for every level of the polynomial hierarchy (ph) there is an MSO formula over graphs such that model checking is hard for that level. Hence, it su#ces to observe that graphs can readily be encoded as strings. We describe a translation from MSO formulas # to MSO # formulas # and a ....

M. Ajtai, R. Fagin, and L. J. Stockmeyer. The Closure of Monadic NP. Journal of Computer and System Sciences, 60(3):660--716, 2000.

....bounds that show the complexity of S len queries, although within PH, may be prohibitively high. Let MSO(SC) be the class of queries over SC expressible in monadic second order logic. This includes queries of high complexity, namely for each level of the polynomial hierarchy, PH, complete queries [2], in particular, NP complete and coNP complete ones (3 colorability and its complement) Such queries cannot be expressed over arbitrary databases in RC(S len ) however, they can be expressed under some additional assumptions. We say that the width of the active domain of a SC database D (over ....

M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. In STOC '98, pages 309-318.

....the lower bound, we shall see below that the complexity of RC(S len ) queries can be quite high. Let MSO(SC) be the class of queries over SC expressible in monadic second order logic. This includes queries of high complexity, namely for each level of the polynomial hierarchy, PH, complete queries [2], in particular, NP complete and coNP complete ones (3 colorability and its complement) We say that the width of the active domain of a SC database D (over ) is k if k is the maximal size of a subset of adom(D) whose elements are pairwise comparable by the pre x relation. It should be noted ....

M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. In STOC '98, pages 309-318.

....the lower bound, we shall see below that the complexity of RC(S len ) queries can be quite high. Let MSO(SC) be the class of queries over SC expressible in monadic second order logic. This includes queries of high complexity, namely for each level of the polynomial hierarchy, PH, complete queries [2], in particular, NP complete and coNP complete ones (3 colorability and its complement) We say that the width of the active domain of a SC database D (over Sigma ) is k if k is the maximal size of a subset of adom(D) whose elements are pairwise comparable by the prefix relation. It should be ....

M. Ajtai, R. Fagin and L. Stockmeyer. The closure of monadic NP. In STOC '98, pages 309--318.

....sets of undirected graphs (or of coloured grids) for the levels of the monadic hierarchy can all be defined with very moderate use of set quantifiers. We show that these witness sets belong to the closure of the class Sigma 1 under first order quantification. This slightly extends the result of [AFS98] which puts these witness sets into the class closed monadic NP (where defining formulas have a quantifier prefix consisting of first order quantifiers and existential set quantifiers in any order) We even show that these witness sets are definable in the extension of first order logic by the ....

....So one has to conclude that the two hierarchies show profound differences and that the present results do not provide any progress for the problem whether the polynomial hierarchy is infinite. A more feasible project would be to investigate the hierarchy built upon closed monadic NP of [AFS98]. Here one allows first order quantifica3 tions for free , i.e. formulas with quantifier prefixes in which both first order and monadic second order quantifiers occur but where the classification into levels refers only to the second order quantifiers. It is open whether in this generalized ....

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M. Ajtai, R. Fagin, and L.J. Stockmeyer. The closure of monadic NP. In F. Chung Graham, editor, The Thirtieth Annual ACM Symposium on Theory of Computing, pages 309--318. SIGACT, 1998.

....Historical Remarks In [Fag75] it was shown that the first level Sigma 1 of the monadic hierarchy over graphs is not closed under complement. The question whether the monadic hierarchy is strict was raised in [Fag95] After that question had been answered affirmatively in [MT97] the authors of [AFS97,AFS98] noted the drawback that the levels of the monadic hierarchy are not closed under first order quantifications, whereas the levels of the polynomial hierarchy (or, by [Imm88] equivalently, of the full second order quantifier alternation hierarchy) are. They raised the question whether also the ....

M. Ajtai, R. Fagin, and L.J. Stockmeyer. The closure of monadic NP. In The Thirtieth Annual ACM Symposium on Theory of Computing. SIGACT, 1998.

.... Delta k 2 . To show this, we consider sets of rectangular grids where the width is a function of the height. If such a set is definable in the first order closure of the boolean closure of Sigma k , then the corresponding function is at most (k 1) fold exponential (this generalizes a result of [AFS97] from k = 1 to arbitrary k) Moreover, we prove that this bound is asymptotically optimal (this solves, for the case k = 1, an open question of [AFS97] 1 Introduction The subject of this paper is monadic second order logic over finite graphs. The monadic second order quantifier alternation ....

.... closure of the boolean closure of Sigma k , then the corresponding function is at most (k 1) fold exponential (this generalizes a result of [AFS97] from k = 1 to arbitrary k) Moreover, we prove that this bound is asymptotically optimal (this solves, for the case k = 1, an open question of [AFS97]) 1 Introduction The subject of this paper is monadic second order logic over finite graphs. The monadic second order quantifier alternation hierarchy 2 1 INTRODUCTION over graphs classifies formulas (and hence graph properties) wrt. the alternation depth of set quantifiers. Sigma k ....

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M. Ajtai, R. Fagin, and L.J. Stockmeyer. The closure of monadic NP. Research Report RJ 10092, IBM, Dec 1997.

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M. Ajtai, R. Fagin, and L. J. Stockmeyer. The Closure of Monadic NP. Journal of Computer and System Sciences, 60(3):660-716, 2000.

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