T. Schwentick, On winning Ehrenfeucht games and monadic NP, Annals of Pure and Applied Logic 79 (1996) 61-92. Received 30 August 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....instead of actually having to exhibit a strategy, it is sucient to prove that the two structures chosen by Duplicator are, in some sense, locally equivalent . This is often a considerably easier task. In fact, Theorem 4. 4 also holds over signatures which contain constant symbols (as remarked in [11]) To see this, let = hR 1 ; R 2 ; R r ; C 1 ; C 2 ; C c i be a signature containing the c constant symbols C 1 ; C 2 ; C c , and let A be a structure. Consider now the purely relational signature = hR 1 ; R 2 ; R r ; R 1 ; R 2 ; R c i in ....

T. Schwentick, On winning Ehrenfeucht games and monadic NP, Annals of Pure and Applied Logic 79 (1996) 61-92. Received 30 August 2001.

....instead of actually having to exhibit a strategy, it is sucient to prove that the two structures chosen by Duplicator are, in some sense, locally equivalent . This is often a considerably easier task. In fact, Theorem 9 also holds over signatures which contain constant symbols (as remarked in [11]) To see this, let = hR 1 ; R 2 ; R r ; C 1 ; C 2 ; C c i be a signature containing the c constant symbols C 1 ; C 2 ; C c , and let A be a structure. Consider now the purely relational signature 0 = hR 1 ; R 2 ; R r ; R 0 1 ; R 0 2 ; R 0 c i ....

T. Schwentick, On winning Ehrenfeucht games and monadic NP, Annals of Pure and Applied Logic 79 (1996) 61-92. 13

....he showed that connectivity (of undirected graphs) is not in monadic NP, although it is easy to see that it is in monadic co NP. Since then, monadic NP has been studied by a number of authors. For example, de Rougement [dR87] Fagin, 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 ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79(1):61--92, May 1996.

....in the presence of auxiliary relations of moderate degree. In [9] 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) 31] and to a linear order [32]. So one may wonder if a similar program can be carried out for FO(C) The intuition behind the introduction of a linear order is that it allows us to simulate encodings of structures on the tape of a Turing machine (or the order of inputs of a circuit) While for order invariant properties it ....

....: xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) QED The following is the key lemma, which is proved by a technique reminiscent of that in [32], extended to deal with bijective games. Lemma 3 Let g : N R be nondecreasing and not bounded by a constant. For any A, m 0, a; b 2 A m , and n 0, if a A g;2 n b, then there exists a preorder P on A such that P 2 g and (A; P; a) j bij n (A; P; b) Proof: Let r = 2 n and ....

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T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92.

.... FO count this follows immediately from [27] or [13] and for FO c( Omega Gamma this follows from the result of [7] that generic Boolean queries definable in FO c( Omega Gamma are definable in relational calculus with the order relation, since directed connectivity is not definable in the latter [36]. That testing for chain is not definable in first order logic with built in order relation follows from [13] which shows that this problem is first order complete for deterministic logspace. In the case of monadic Sigma 1 1 , the proof for tc is the same as before, since connectivity is not ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic 79 (1996), 61--92.

....(see, e.g. 5, 13, 14, 27] in connection with capturing PTIME over unordered structures. We proved tight bounds on locality rank for a variety of counting logics, described outputs of local queries, and proved an analog of Gaifman s theorem for FO(Qu ) Continuing the line that started in [10, 26, 28], we gave new winning strategies for the duplicator based on the ideas of locality. We now briefly discuss applications and new directions. Gaifman locality, as defined here, and the BDP, were introduced in connection with the study of expressive power of real life database query languages that ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92.

....notions and the BNDP, and show that the main theorem reduces to proving weak semi locality of a logic. In Section 5, we prove weak semi locality of L 1 (C) in the presence of almost everywhere linear orders, combining the bijective games of [16] and a strategy for the duplicator inspired by [33]. Concluding remarks are given in Section 6. All proofs can be found in the full version [25] 2 Notations Finite Structures and Logics All structures are assumed to be finite. A relational signature oe is a set of relation symbols fR 1 , R l g, with associated arities p i 0. For ....

....1 ; xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) 2 The following is the key lemma, which is proved by a technique reminiscent of that in [33], extended to deal with bijective games. Lemma 2 Let g : N R be nondecreasing and not bounded by a constant. For any A, m 0, a; b 2 A m , and n 0, if a A g;2 n b, then there exists a preorder P on A such that P 2 g and (A; P; a) j bij n (A; P; b) Proof sketch. Let r = ....

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T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92.

....get non definability results in the presence of a linear order. A similar phenomenon can also be seen in many other extensions of first order Ehrenfeucht Fraiss e games, such as the infinite pebble game for the infinitary logic L 1 (see e.g. EF95, Chapter 2] Note however, that Schwentick [Sch96] gave an Ehrenfeucht Fraiss e type game theoretical method that guarantees elementary equivalence of two structures for first order logic even with built in linear order. We solve this problem by requiring that in structures A 6 all neighborhoods N(d; a) are instead defined as neighborhoods in ....

....only if A 6 has length divisible by n 1. Hence B 6 is connected if and only if A 6 2 C 6 n 1 . Therefore is a sentence of FO(D n ) which with the linear order 6 defines the class C 6 n 1 . This is a contradiction according to Lemma 4.12. 2 4.14. Remark. From the work of Schwentick [Sch96] we know that connectivity of graphs is not definable in Mon Sigma 1 1 even with built in linear order. 16 5. Complete trees Consider the logic FO(D n ) with built in linear order augmented with the extra predicate y = nx. We show that this logic is not strong enough to express that the ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79(1):61--92, May 1996.

....results from relations of small (constant or moderate) degree to relations of large (comparable with the size of the input) degree. The concept of moderate degree was introduced in [17] to show that connectivity is not definable in monadic Sigma 1 1 in the presence of those relations. Later, [44] extended to linear orders. Thus, one may ask if a similar avenue of attack on the separation problem can be pursued in the case of FO COUNT. A partial result in this direction was proved recently. Let O k stand for the class of relations which are pre orders hA; OEi (i.e. OE is reflexive and ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92.

....column of R. Winning conditions 2 and 3 are necessary to prove that = q is a congruence relation, i.e. Lemma 19 For t bit grids R; R 0 ; P of equal height, the following holds: If R = q R 0 then RP = q R 0 P . Proof Playing the q rounds according to Schwentick s Extension Theorem ([Sch96]) Duplicator directly obtains the following winning strategy on RP and R 0 P : Let R = R 1 R 2 and R 0 = R 0 1 R 0 2 , where R 2 (and R 0 2 , respectively) consists of the last 2 q columns of R (and R 0 , respectively) Winning condition 3 ensures that R 2 and R 0 2 are coloured ....

Th. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

....in the presence of auxiliary relations of moderate degree. In [9] 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) 32] and to a linear order [33]. So one may wonder if a similar program can be carried out for FO(C) The intuition behind the introduction of a linear order is that it allows us to simulate encodings of structures on the tape of a Turing machine (or the order of inputs of a circuit) While for orderinvariant properties it ....

....: xm ) be a L 1 (C) formula in the language of oe, with all free variables of the first sort. Let (A; a) j bij rk( B; b) where a 2 A m ; b 2 B m . Then A j= a) iff B j= b) 2 The following is the key lemma, which is proved by a technique reminiscent of that in [33], extended to deal with bijective games. Lemma 3 Let g : N R be nondecreasing and not bounded by a constant. For any A, m 0, a; b 2 A m , and n 0, if a A g;2 n b, then there exists a preorder P on A such that P 2 g and (A; P; a) j bij n (A; P; b) Proof. Let r = 2 n ....

[Article contains additional citation context not shown here]

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92.

....as follows: In the second order rounds, on the C part of AC , BC, Duplicator chooses the same colouring as Spoiler; on the A part and the B part of AC , BC Duplicator colours according to her winning strategy on A, B. Playing the first order rounds according to T. Schwentick s Extension Theorem ([SchT96]) Duplicator directly obtains the following winning strategy on AC , BC : Let A = A 1 A 2 (and B = B 1 B 2 ) where A 2 (and B 2 , respectively) consists of the last 2 q columns of A (and B, respectively) Note that winning condition 2 ensures that A 2 and B 2 are coloured identically. Thus on ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic 79 (1996), 61-92.

....he showed that connectivity (of undirected graphs) is not in monadic NP, although it is easy to see that it is in monadic co NP. Since then, monadic NP has been studied by a number of authors. For example, de Rougement [dR87] Fagin, 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 ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79(1):61--92, May 1996.

....winning strategy even on AC ; BC . 4. To each t bit picture A of height m there exists a t bit picture B of height m and width 6 e q;r;t k (m) such that A j k;q;r B. The parts 1 and 2 of Proposition 2 are proved by standard methods; part 3 is an application of Schwentick s Extension Theorem ([Schw96]) and part 4 follows directly from part 3. Before giving a proof for Proposition 2 we will first apply it to prove Theorem 2, i.e. to prove that Sigma k definable functions are at most k fold exponential. Proof of Theorem 2 Let k 1. Let f : N Gamma N be a function defined by a Sigma k ....

....n B 2 q . In the second order rounds, on the C part of AC , BC, Duplicator chooses the same colouring as Spoiler; on the A part and the B part of AC, BC Duplicator colours according to her winning strategy on A, B. Playing the first order rounds according to Schwentick s Extension Theorem ([Schw96]) Duplicator directly obtains the following winning strategy on AC , BC: Let A = A 1 A 2 and B = B 1 B 2 , where A 2 (and B 2 , respectively) consists of the last 2 q columns of A (and B, respectively) Note that winning condition 3 ensures that A 2 and B 2 are coloured identically. Thus on A ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic 79 (1996), 61-92.

....the universe, so that the arguments simultaneously prove the results over the class of ordered structures. In this sense, the situation here for SO(9) contrasts sharply with the case of m. Sigma 1 1 , where it is much more difficult to obtain non definability results over ordered structures (see [25]) On the other hand, the proofs do not carry over to classes with a built in successor relation (which is not definable from a linear order in SO(9) Acknowledgements I am grateful to Prof. Janos Makowsky for many valuable discussions on the subject of this paper, and to Prof. Jorg Flum for ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

....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 the duplicator. An important example is ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

.... 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 is obtained by transferring the winning strategy on (s; P ) and (s; P 0 ) making use of the gap preserving technique that was invented in [14]. For every ; with 0 ; 2 r and 2 r , we define a function f ; from A to f0; n 1g [ f g by f ; x) i if x is in T i ( otherwise. We are going to show that the duplicator can play in such a way that for every i the following conditions hold. 9 ....

T. Schwentick. On winning ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996. 17

....has a winning strategy in the r round 10 M. Grohe and T. Schwentick game on (A; 1 ; a) and (A; 2 ; b) The winning strategy of the duplicator is obtained by transferring the winning strategy on (s; P ) and (s; P 0 ) making use of the gap preserving technique that was invented in [14]. For every ; with 0 ; 2 r and 2 r , we de ne a function f ; from A to f0; n 1g [ f g by f ; x) i if x is in T i ( otherwise. We are going to show that the duplicator can play in such a way that for every i the following conditions hold. 1) ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61-92, 1996.

....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 the duplicator. An important example is ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

....very di#cult. 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 the duplicator. An important example is ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

....s p 1 s 2p # # # # # # # # # # # # G 0 r 1 r p r p 1 r 2p s 1 s p s p 1 s 2p # # # # # # # # # # # # Figure 1: The graphs G and G 0 . Colours are indicated by the different shapes of the vertices. It is now straightforward, by using either the Hanf method [FSV95] or the gap method [Sch96], to prove that the duplicator has a winning strategy for the k round first order game on the two coloured structures. ffl The Hanf method can be applied because in both graphs the same p neighbourhoods occur with the same frequency. Here, a p neighbourhood is a subgraph which is induced by all ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

.... cannot be expressed by a monadic ESO formula, even in the presence of several kinds of built in relations [AF90] Cosmadakis showed how nonexpressibility results can be transfered to other problems via suitable reductions [Cos93] For a generalization of these reductions we refer also to [Sch94b]. With the exception of linear orders all the built in relations that are listed in Table 1 are in fact separable. Hence separable built in relations are at the limit of our ability to prove non expressibility results concerning monadic ESO. Let us call built in relations strong, if they enable ....

T. Schwentick. On Winning Ehrenfeucht Games and Monadic NP. PhD thesis, Universit at Mainz, 1994.

....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 the duplicator. An important example is ....

T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

.... quantification of unary functions; ffl there are a lot of inexpressibility results showing that Graph Connectivity is not expreessible by Mon Sigma 1 1 formulae [Fag75] even in the presence of built in successor relations [dR84] a built in linear order, built in relations of degree n o(1) [Sch96b], or built in trees [KS96] that Directed Reachability is not expressible even in the presence of certain built in relations [AF90] among others. The main tools for these proofs are suitable variants of Ehrenfeucht games. Inexpressibility can be concluded, if it can be shown that Duplicator, one ....

....o(1) Proof. Sketch) Let k 0 and B be a family of built in structures of degree n o(1) We sketch in the following, that no Mon Sigma 1 1 formula can weakly express Graph Connectivity with n k padding in the presence of B. The proof is almost identical to the proof of Theorem 14 in [Sch96b] (Theorem 9 in [Sch95] where it was shown that Graph Connectivity is not expressible by a Mon Sigma 1 1 formula even in the presence of built in relations of degree n o(1) That proof essentially relies on the following property of families B of structures of degree n o(1) For every N ....

T. Schwentick. On winning ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79:61--92, 1996.

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T. Schwentick. On winning Ehrenfeucht games and monadic NP. Annals of Pure and Applied Logic, 79 (1996), 61--92. 22

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