S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Theoretical Computer Science, 174:97--121, 1997. 50  Home/Search   Document Details and Download   Summary   Related Articles   Check

....review Ehrenfeucht games and an extended version of Ehrenfeucht games that was introduced by Ajtai and Fagin [AF90] for proving that a problem is not expressible in MonNP. In Sections 3.4 and 3. 5 we discuss two methods that were introduced by Fagin, Stockmeyer and Vardi [FSV93] and Arora and Fagin [AF94], respectively, to simplify the task of proving that Duplicator has a winning strategy in an Ehrenfeucht game. 12 Methods for Proving Inexpressibility in MonNP 3.1 Reductions in the Context of MonNP In general, one of the simplest methods to get inexpressibility results is by reductions. If a ....

....strategy on given finite structures A and A 0 remains, in general, very difficult (see for example [dR87] AF90] One approach to overcome this difficulty is to find general sufficient conditions that assure a winning strategy for Duplicator. Two such conditions were proposed in [FSV93] and [AF94], respectively, and will be discussed in sections 3.4 and 3.5. Another approach will be introduced in Chapter 4 as the central methodical contribution of this thesis. 34 Methods for Proving Inexpressibility in MonNP 3.4 The Approach of Fagin, Stockmeyer and Vardi To show that Duplicator has a ....

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S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Unpublished manuscript, 1994.

.... [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 ....

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Unpublished manuscript, 1994.

....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 ....

Sanjeev Arora and Ronald Fagin. On winning strategies in Ehrenfeucht--Fraiss'e games. Theoretical Computer Science, 174(1--2):97--121, 1997.

....strategy for the duplicator is often very difficult. To simplify such proofs the following approaches have been taken. There have been developed several conditions that assure that the duplicator has a winning strategy on two given structures: the Hanf condition [FSV95] the Arora Fagin condition [AF97] and the condition of Schwentick [Sch96] for a survey see [Fag97] All of these conditions exploit the fact that FO logic can only express combinations of local properties, i.e. properties of regions of bounded size. On the other hand there have been attempts to make the game easier to play for ....

.... One indication in this direction is the fact that many of the inexpressibility proofs that are given in the literature can be proved by using these games (e.g. FSV95, Sch95] One particular example is the proof of Ajtai and Fagin [AF90] which has already been simplified by Arora and Fagin in [AF97]. In both of these proofs the notion of the (r; d) colour of a vertex or an edge is used. As the graphs that were used in those proofs do not contain small cycles all r spheres are rooted trees. The definition assures that if two vertices have the same (r; d) colour their r neighbourhoods have ....

Sanjeev Arora and Ronald Fagin. On winning strategies in Ehrenfeucht--Fraiss'e games. Theoretical Computer Science, 174(1-- 2):97--121, 1997.

....whole graph, cannot help in testing whether a clique has even size or not. We note that Theorem 4. 1 could also be used in several other proofs, like [5] and [11] But it seems to be, in general, incompatible with the technique of Hanf [12] which is described in [11] and the related technique of [2]. We hope that our technique will be useful to obtain other inexpressibility results. Maybe with its help or with the help of some other techniques developed recently (cf. 11, 2] it will be possible to attack the case of existential second order formulas that allow quantification over binary ....

....to be, in general, incompatible with the technique of Hanf [12] which is described in [11] and the related technique of [2] We hope that our technique will be useful to obtain other inexpressibility results. Maybe with its help or with the help of some other techniques developed recently (cf. [11, 2]) it will be possible to attack the case of existential second order formulas that allow quantification over binary relations. Acknowledgements I am very grateful to Clemens Lautemann for uncountably many helpful discussions, valuable suggestions and improvements, and not least because of ....

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Unpublished manuscript, 1994.

.... game on A iff he has a winning strategy in the Ajtai Fagin (l; r) Mon Sigma 1 1 game on A iff A can be characterized by a (l; r) Mon Sigma 1 1 formula [Fag75, AF90] Corresponding statements hold for the other games. For more information on Ehrenfeucht games and their variations see, e.g. [EF95, AF97]) Padding If w is a binary string and m jwj we write w Phi m for the string w10 m Gammajwj Gamma1 , that is the string of length m that has prefix w and a suffix of the form 10 . If G is a graph with universe [n] and m n, we write G Phi m for the graph on [m] that coincides with G on ....

Sanjeev Arora and Ronald Fagin. On winning strategies in Ehrenfeucht--Fraiss'e games. Theoretical Computer Science, 174(1--2):97--121, 1997.

....for the duplicator is often very difficult. To simplify such proofs the following approaches have been taken. ffl There have been developed several conditions that assure that the duplicator has a winning strategy on two given structures: the Hanf condition [FSV95] the Arora Fagin condition [AF97] and the condition of Schwentick [Sch96] for a survey see [Fag97] In many cases, these conditions can be verified easily. All of these conditions exploit the fact that, in a certain sense, first order logic can only express combinations of local properties, i.e. properties of regions of ....

.... One indication in this direction is the fact that many of the inexpressibility proofs that are given in the literature can be proved by using these games (e.g. FSV95, Sch95] One particular example is the proof of Ajtai and Fagin [AF90] which has already been simplified by Arora and Fagin in [AF97]. In both of these proofs the notion of the (r; d) colour of a vertex or an edge is used. As the graphs that were used in those proofs do not contain small cycles all r spheres are rooted trees. The definition assures that if two vertices have the same (r; d) colour their r neighbourhoods have ....

Sanjeev Arora and Ronald Fagin. On winning strategies in Ehrenfeucht--Fraiss'e games. Theoretical Computer Science, 174(1--2):97--121, 1997.

.... Stockmeyer and Vardi [FSV95] and Schwentick [Sch95, Sch96] showed that connectivity is not in monadic NP, even in the presence of various built in relations; Ajtai and Fagin [AF90] showed that directed (s; t) connectivity is not in monadic NP, and their proof was simplified by Arora and Fagin [AF97] Cosmadakis [Cos93] showed that a number of properties, including non 3colorability, are not in monadic NP; Courcelle wrote a sequence of papers, including [Cou94] where he considered expressibility of monadic NP and of a variant where there is a different representation of graphs; Otto [Ott95] ....

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Theoretical Computer Science, 174:97--121, 1997. 50

.... Stockmeyer and Vardi [FSV95] and Schwentick [Sch95, Sch96] showed that connectivity is not in monadic NP, even in the presence of various built in relations; Ajtai and Fagin [AF90] showed that directed (s; t) connectivity is not in monadic NP, and their proof was simplified by Arora and Fagin [AF97] Cosmadakis [Cos93] showed that a number of properties, including non 3colorability, are not in monadic NP; Courcelle wrote a sequence of papers, including [Cou94] where he considered expressibility of monadic NP and of a variant where there is a different representation of graphs; Otto [Ott95] ....

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Theor. Comp. Sci., 174:97--121, 1997.

....the spoiler knows what G 0 and G 1 are before he colors G 0 . This would be disastrous for a duplicator whose coloring strategy is to color G 1 by simply duplicating the coloring for G 0 : if the spoiler knew which edge e were deleted from G 0 to form G 1 = G Gamma e, 5 Arora and the author [AF94] show how to simplify Ajtai and the author s proof of this result. Both papers (Ajtai and Fagin [AF90] and Arora and Fagin [AF94] use exactly the same graphs G0 ; G1 . this might dramatically influence his coloring of G 0 (for example, the spoiler might color the endpoints of e with special ....

....is to color G 1 by simply duplicating the coloring for G 0 : if the spoiler knew which edge e were deleted from G 0 to form G 1 = G Gamma e, 5 Arora and the author [AF94] show how to simplify Ajtai and the author s proof of this result. Both papers (Ajtai and Fagin [AF90] and Arora and Fagin [AF94]) use exactly the same graphs G0 ; G1 . this might dramatically influence his coloring of G 0 (for example, the spoiler might color the endpoints of e with special colors) In the Ajtai Fagin monadic NP game, the spoiler must commit himself to a coloring of G 0 before he knows which edge e is ....

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Research Report RJ 9833, IBM, June 1994. To appear, Theoretical Computer Science.

....In Section 3, we discuss various sufficient conditions for the duplicator to have a winning strategy in a first order Ehrenfeucht Fraiss e game (played over two structures) These include Hanf s condition, as given by Fagin, Stockmeyer and Vardi [FSV95] Section 3. 1) Arora and Fagin s condition [AF94] (Section 3.2) and Schwentick s condition [Sch96b] Section 3.3) These three conditions are compared in Section 3.4. Roughly speaking, we could say that Hanf s condition requires isomorphic neighborhoods in the two structures; Arora and Fagin s condition requires approximately isomorphic ....

....games In this section, we focus on first order Ehrenfeucht Fraiss e games, and give three sufficient conditions for one player (the duplicator) to win. These conditions are based on techniques of Hanf [Han65] and given a new interpretation by Fagin, Stockmeyer and Vardi [FSV95] Arora and Fagin [AF94], and Schwentick [Sch96b] As we shall discuss, such techniques and conditions are valuable tools for obtaining inexpressibility results. We begin with an informal definition of an r round first order Ehrenfeucht Fraiss e game (where r is a positive integer) which we shall call an r game for ....

[Article contains additional citation context not shown here]

S. Arora and R. Fagin. On winning strategies in Ehrenfeucht-Fraiss'e games. Research Report RJ 9833, IBM, June 1994. To appear, Theoretical Computer Science.

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