T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....MSO; however, none of these properties is expressible in ESO(9 ) even in presence of a successor [13] Therefore, ESO(9 ) and MSO have different expressive power over ordered graphs. Further relevant discussions of ESO and MSO fragments over graphs and general structures can be found in [9, 47, 48, 43, 10]. To the best of our best knowledge, there has been no previous characterization of the regular languages by nonmonadic fragments of ESO. However, many papers cover either extensions or restrictions of MSO or REG. Lynch [34] for example, has studied the logic over strings obtained from ....

T. Schwentick. Graph Connectivity and Monadic NP. In Proc. IEEE Symposium on Foundations of Computer Science (FOCS '94), pages 614--622, 1994.

....sufficient condition for the duplicator to have a winning strategy. In Section 8, we give new inexpressibility results in the presence of certain built in relations. We summarize in Section 9. Other related work. A related development (independent of this paper) is a recent result by Schwentick [Sch94]. He gives another sufficient condition for the duplicator to have a winning strategy, and uses it to show that connectivity is not in monadic NP in the presence of an even larger class of built in relations than Fagin, Stockmeyer, and Vardi had shown. Most importantly, he resolved an open problem ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994. 22

....cannot be defined by FO(C) in the presence of auxiliary relations of moderate degree. 2 In [11] auxiliary relations of moderate degree were shown to be of no help for expressing connectivity of graphs in monadic Sigma 1 1 . This was extended to degrees n o(1) 22] and to a linear order [21]. So one may wonder if a similar program can be carried out for FO(C) There is a significant difference between Facts 1, 2 and 3, and the desired separation for the ordered case: in those Facts, we only deal with auxiliary relations of small degrees these are either constant, or very small ....

T. Schwentick. Graph connectivity and monadic NP. FOCS'94, pages 614--622.

....Corollary 3.6 Transitive closure and deterministic transitive closure are not definable in FO COUNT in the presence of relations of moderate degree. 2 However, the order relation adds all degrees from 0 to the cardinality of the input. Thus, we need a breakthrough like Schwentick s theorem [28] to generalize Corollary 3.4 to the ordered case. Proof of Theorem 3.1 Before giving the proof of Theorem 3.1, we sketch a direct proof that Hanf s locality implies the graph BDP. The proof below completely avoids Lemma 3.11, which is the main technical tool for proving Theorem 3.1, and the ....

T. Schwentick. Graph connectivity and monadic NP. In Proceedings of 35th Symposium on Foundations of Computer Science, pages 614--622, 1994.

....where second order quantifiers may range also over relations of arity higher than 1. As partial results on Fagin s problem we mention his result [Fag75] see also [FSV95] that connectivity of graphs is a monadic property which is not Sigma 1 (i.e. not definable by a Sigma 1 formula) In [Sch94], Schwentick extended this result to graphs with built in order. Within the range of monadic Sigma 1 formulas, Otto [Ott95] showed that the length of the (single) block of leading existential set quantifiers induces a strict hierarchy of properties. From automata theory (cf. Buchi [Buc60] ....

Th. Schwentick. Graph connectivity and monadic NP. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, pages 614--622, 1994.

....none of these properties is expressible in ESO(9 89 ) even in presence of a successor [12] Therefore, ESO(9 89 ) and MSO have different expressive power over ordered graphs. Further relevant work on discussing ESO and MSO fragments over graphs and general structures can be found in [8, 46, 47, 42, 9]. To our best knowledge, there has been no previous characterization of the regular languages by nonmonadic fragments of ESO. However, many papers cover either extensions or restrictions of MSO or REG. Lynch [33] for example, has studied the logic over strings obtained from existential MSO by ....

T. Schwentick. Graph Connectivity and Monadic NP. In Proceedings IEEE FOCS '94, pages 614--622, 1994.

....in the presence of auxiliary relations of moderate degree. 2 In [12] auxiliary relations of moderate degree were shown to be of no help for expressing connectivity of graphs in monadic Sigma 1 1 . Starting from their result, Schwentick extended it to degrees n o(1) 28] and to a linear order [27]. So one may wonder if a similar program can be carried out for FO(C) There is a significant difference between Facts 1, 2 and 3, and the desired separation for the ordered case: in those Facts, we only deal with auxiliary relations of small degrees these are either constant, or very small ....

T. Schwentick. Graph connectivity and monadic NP. In Proceedings of 35th Symposium on Foundations of Computer Science, pages 614--622, 1994.

....also over relations of higher arity than 1. For more background we refer the reader to [4] As partial results on Fagin s problem we mention his result [6] see also [8] that connectivity of graphs is a monadic property which is not Sigma 1 (i.e. not definable by a Sigma 1 formula) In [17], Schwentick extended this result to graphs with built in order. Recent work of Ajtai, Fagin, and Stockmeyer ( 1] announces that monadic logic allows to define complete problems for each level of the polynomial hierarchy. Within the range of monadic Sigma 1 formulas, Otto [15] showed that the ....

T. Schwentick. Graph connectivity and monadic NP. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, pages 614--622, 1994.

....even in the presence of arbitrary built in relations. In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary built in relations of degree n o(1) The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. 1 Introduction Since the result of Fagin [Fag74] that the complexity class NP coincides with the class of all sets of finite structures that can be ....

....of several kinds. It was shown that Graph Connectivity can neither be expressed in MonNP ( Fag75] nor in the presence of a built in successor relation ( dR87] nor in the presence of arbitrary built in relations of degree (log n) o(1) FSV93] nor in the presence of a built in linear order ([Sch94]) Fagin, Stockmeyer and Vardi [FSV93] conjectured that Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built in relations. 1 In fact, Ajtai s result is far more general and not restricted to the monadic case. In this paper we show that Graph Connectivity ....

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T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994. This article was processed using the L a T E X macro package with LLNCS style

....in a definition and to an anonymous referee for many corrections. Last but not least I want to thank Rudiger Schilp for creating the authentic incarnations of Spoiler and Duplicator. This Technical Report is a revised version of the author s Doktorarbeit. Parts of it have been published before in [Sch94] and [Sch95] A preliminary version of it appeared as Technical Report 2 94. 6 Introduction 2 Definitions and Notations Fnan Fnan fflffl 3 Methods for Proving Inexpressibility in MonNP Fnan Fnan fflffl 4 The Extension Theorem # Fnan Fnan ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

....in the proof of our main result) In the case of Monadic NP there are a lot of nonexpressibility results for more natural graph problems. Fagin [Fag75] showed that connectivity of undirected graphs a Monadic coNP property is not in Monadic NP. This result was extended by a variety of papers [dR87, FSV93, Sch94, Sch95, Nur95] to cases where some kinds of built in relations or generalized quantifiers are allowed. Ajtai and Fagin [AF90] proved that directed reachability, in contrast to undirected reachability, is not in Monadic NP. By means of reductions some of these results can be transferred to other problems ....

.... [Cos93] The proofs of the results about connectivity and directed reachability make use of Ehrenfeucht games (see Section 2 below) In many of them sufficient conditions for the existence of a winning strategy for Duplicator, one of the two players in an Ehrenfeucht game, play an important role ([FSV93, AF94, Sch94, Sch95]) One such sufficient condition, the Weak Extension Theorem from [Sch95] will be used in the proof of the main result of this paper. As there has been some success in proving inexpressibility results for Monadic NP, it seems reasonable to turn to the next stage, Binary NP. But for full Binary NP ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

.... of connected graphs (which is in monadic coNP) is not in monadic NP thus showing that monadic NP is not closed under complements [Fag75] Since then more and more powerful methods for using Ehrenfeucht games have been developed and have led to various extensions and strengthenings of this result [AF90, dR87, FSV93, Sch94, Sch95]. However, this success has been confined to the monadic fragment of existential second order logic, and new techniques seem necessary to analyse the expressive power of higher fragments such as binary NP (where second order quantifiers range over binary relations) On the other hand, we know ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

.... in mon Pi 1 1 ) cannot be expressed in mon Sigma 1 1 thus showing that mon Sigma 1 1 is not closed under complements [Fag75] Since then more and more powerful methods for using Ehrenfeucht games have been developed and have led to various extensions and strengthenings of this result [AF90, dR87, FSV93, Sch94, Sch95]. However, this success has been confined to the monadic fragment of Sigma 1 1 , and new techniques seem necessary to analyze the expressive power of higher fragments such as bin Sigma 1 1 (where second order quantifiers range over binary relations) On the other hand, we know that the ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

....built in relations. The following table summarizes some of these results. Most of these results can be essentially proved by adapting Fagin s idea of one cycle vs. successor relations de Rougemont [dR87] relations of degree (log n) o(1) Fagin, Stockmeyer, Vardi [FSV95] linear order Schwentick [Sch94a] relations of degree n o(1) Schwentick [Sch95] trees Kreidler, Seese [KS97] n n o(1) edges Kreidler, Seese [KS97] planar graphs Kreidler Seese [KS98] K l free Kreidler, Seese [KS98] Table 1: A list of built in relations that do not enable MESO logic to express graph connectivity. two ....

....of many useful tools for dealing with Ehrenfeucht games. Besides the already mentioned Ajtai Fagin game, there have been invented several ways to simplify the proof of the existence of a winning strategy for the duplicator in the first order Ehrenfeucht game. We refer the interested reader to [FSV95, AF97, Sch94a, SB98] and, for a survey to [Fag97] A new development in the area of monadic ESO was initiated by Ajtai et al. AFS97] They consider various closures of monadic ESO, e.g. formulas that allow first order quantification in front of existential monadic second order quantifiers and they prove very nice ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

.... It remains to show that the duplicator has a winning strategy in the r round game on (A; 1 ; a) and (A; 2 ; b) The winning strategy of the duplicator will be obtained by transferring the winning strategy on (s; P ) and (s; P 0 ) making use of the gap preserving technique that was invented in [11]. For every fi; fl, with 0 fi; fl 2 r and fi fl 2 r , we define a function f fi;fl from A to f0; n Gamma 1g by f fi;fl (x) i if x is in T i (fi;fl) otherwise. We are going to show that the duplicator can play in such a way that for every i the following conditions ....

T. Schwentick. Graph connectivity and monadic NP. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, pages 614--622, 1994.

....by existential monadic second order logic is not closed under complementation. In fact, the set of connected graphs, although expressible by means of a universal monadic second order sentence, is not contained in monNP [6] a result which has since been refined and extended in a number of ways [4, 1, 7, 11, 12]. On the other hand, the expressive power of sentences with one binary existential second order quantifier is not well understood. We believe that studying semantically restricted versions of this latter logic, will help us understand the limitations, and the expressive power of binary existential ....

T. Schwentick. Graph connectivity and monadic NP. In Proc. 35st IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

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T. Schwentick. Graph connectivity and monadic NP. In Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 614--622, 1994.

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