R. Fagin. Easier ways to win logical games. In N. Immerman and Ph. G. Kolaitis, editors, Descriptive Complexity and Finite Models, volume 31 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 1--32. American Mathematical Society, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....FO Ehrenfeucht Fra iss6 games, for short: EF games, were invented by Ehrenfeucht and Fra iss6 in [12, 14] These combinatorial games are particularly useful for investigating what can, and what cannot, be expressed in various logics. A well written survey on EF games is, e.g. given by Fagin in [13]. More details can be found in the textbooks [18, 11] In the present section we will concentrate on the classical, first order r round EF game, which is defined as follows. Let v be a signature and let r be a natural number. The r round EF game is played by two players, the spoiler and the ....

....strategy in the r round EF game on .4 and 3. It is straightforward to see that, for every signature v, the relation my, is an equivalence relation on the set of all v structures. The fundamental use of the game comes from the fact that it characterizes first order logic as follows (cf. e.g. [13, 18, 11]) 4.1 Theorem (Ehrenfeucht, Frass6) Let v be a signature. a) Let r IV and let .4 and 3 be r structures. 4 y, 3 if and only if.4 and 3 satisfy the same FO( sentences of quantifier depth at most r. b) Let be a class of structures and let C . The following are equivalent: i) is not ....

R. Fagin. Easier ways to win logical games. In N. Immerman and P. Kolaitis, editors, Descriptive complexity and finite models, volume 31 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 1 32. American Mathematical Society, 1997.

....4 Games Ehrenfeucht games invented in [Ehr61] building on work of Frasse [Fra54] are an important tool for proving inexpressibility results in Mathematical Logic. In fact, in Finite Model Theory, where only finite structures are considered, they are the major tool available (cf. [Fag97]) To show that a given property P of finite structures is not expressible in FO logic it is enough to prove that the duplicator, one of two players, has a winning strategy in the ordinary FO Ehrenfeucht game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht games are available ....

....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 the invention of the ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

....fl clearly distinguishes C 1 n from C 2 n . This finishes the proof for the case of (deterministic) transitive closure. Now let q 2 SG tree ; we show that q is not verifiable over L. When L = FO count , we can apply the proof of claim 3 in theorem 2, since, by the result of [30] see also [15]) for each k it is possible to find a number r such that any two structures that realize the same number of all r neighborhoods cannot be distinguished by a FO count sentence of quantifier rank k. For FO c( Omega Gamma0 first define an order relation OE on U that is isomorphic to (that is, the ....

....Step 3. The duplicator selects a graph G 0 2 Tree Gamma G and colors its nodes with c colors. Step 4. The spoiler and the duplicator play k rounds of the Ehrehfeucht Fraiss e game on colored G and G 0 . The winner is determined as the winner in Step 4. For more details on games, see [15, 16]. To determine a winner in Step 4, we shall use the criterion below, that follows immediately from Theorem 4.3 of [17] We shall use the notation G 1 d;m G 2 if G 1 and G 2 are two colored graphs, and for every isomorphism type of a d neighborhood of a node, either both graphs have the same ....

R. Fagin. Easier ways to win logical games. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, vol. 31, 1997, pages 1--32.

....compactness fails in the case of finite models [12] to prove results about the limits of expressiveness of first order logic, one has to use Ehrenfeucht Fraisse games. Moreover, Ehrenfeucht Fraisse games are often used as the basic step in other, more sophisticated games for different logics, cf. [14]. For example, playing the Ehrenfeucht Fraisse game is one of the steps in the Ajtai Fagin game for monadic Sigma 1 1 [3] Since playing the game often involves an intricate combinatorial argument, it was suggested by Fagin, Stockmeyer and Vardi in [15] to build a library of winning strategies ....

....case as follows. Theorem 2.4 (Fagin Stockmeyer Vardi [15] Let n 0. Then there exists an integer d 0 such that whenever A d B, then A and B agree on all first order sentences of quantifier rank up to n. 2 It follows from the proof in [15] that d can be taken to be 3 n Gamma1 , see also [14]. This leads to the following definition. Definition 2.5 A sentence Psi is Hanf local if there exists a number d such that any two d equivalent structures agree on Psi. The minimum d for which this holds is called the Hanf locality rank of Psi, and is denoted by hlr( Psi) Thus, ....

R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Descriptive Complexity and Finite Models, AMS 1997.

....for the finite case as follows. Theorem 2.4 (Fagin Stockmeyer Vardi [11] Let n 0. Then there exists an integer d 0 such that whenever A d B, then A and B agree on all first order sentences with qr( n. 2 It follows from the proof in [11] that d can be taken to be 3 n Gamma1 , see also [10]. This leads to the following definition. 5 Definition 2.5 A sentence Psi is Hanf local if there exists a number d such that any two d equivalent structures agree on Psi. The minimum d for which this holds is called the Hanf locality rank of Psi, and is denoted by hlr( Psi) Thus, ....

R. Fagin. Easier ways to win logical games. In Descriptive Complexity and Finite Models, N. Immerman and Ph. Kolaitis, editors, AMS, 1997, pages 1--32. 21

....4 Games Ehrenfeucht games invented in [Ehr61] building on work of Frasse [Fra54] are an important tool for proving inexpressibility results in Mathematical Logic. In fact, in Finite Model Theory, where only finite structures are considered, they are the major tool available (cf. [Fag97]) To show that a given property P of finite structures is not expressible in FO logic it is enough to prove that the duplicator, one of two players, has a winning strategy in the ordinary FO Ehrenfeucht game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht games are available ....

....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 the invention of the ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

....4 Games Ehrenfeucht games invented in [Ehr61] building on work of Frasse [Fra54] are an important tool for proving inexpressibility results in Mathematical Logic. In fact, in Finite Model Theory, where only finite structures are considered, they are the major tool available (cf. [Fag97]) To show that a given property P of finite structures is not expressible in FO logic it is enough to prove that the duplicator, one of two players, has a winning strategy in the ordinary FO Ehrenfeucht game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht games are available ....

....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 the invention of the ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

....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 separation results for some of ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

....4 Games Ehrenfeucht games invented in [Ehr61] building on work of Fraiss e [Fra54] are an important tool for proving inexpressibility results in Mathematical Logic. In fact, in Finite Model Theory, where only finite structures are considered, they are the major tool available (cf. [Fag97]) To show that a given property P of finite structures is not expressible in FO logic it is enough to prove that the duplicator, one of two players, has a winning strategy in the ordinary FO Ehrenfeucht game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht games are available ....

....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 the invention of the ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

....black . 4 Games Ehrenfeucht games invented in [Ehr61] building on work of Fraiss e [Fra54] are an important tool for proving inexpressibility results in Mathematical Logic. In fact, in Finite Model Theory, where only finite structures are considered, they are the major tool available (cf. [Fag97]) To show that a given property P of finite structures is not expressible in first order logic it is enough to prove that the duplicator, one of two players, has a winning strategy in the ordinary first order Ehrenfeucht game for P (for a definition see e.g. EF95] Variants of Ehrenfeucht ....

....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 bounded size. This observation has been made precise by Gaifman ....

R. Fagin. Easier ways to win logical games. In Proceedings of the DIMACS Workshop on Finite Models and Descriptive Complexity. American Mathematical Society, 1997.

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R. Fagin. Easier ways to win logical games. In N. Immerman and Ph. G. Kolaitis, editors, Descriptive Complexity and Finite Models, volume 31 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 1--32. American Mathematical Society, 1997.

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R. Fagin. Easier ways to win logical games. In N. Immerman and Ph. Kolaitis, eds. Descriptive Complexity and Finite Models, AMS, 1997, pages 1-32.

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R. Fagin. Easier ways to win logical games. In N. Immerman and Ph. Kolaitis, eds. Descriptive Complexity and Finite Models, AMS, 1997, pages 1--32.

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R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Finite Model Theory and Descriptive Complexity, 1996.

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R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Descriptive Complexity and Finite Models, AMS 1997. 30

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R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Finite Model Theory and Descriptive Complexity, 1996.

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R. Fagin. Easier ways to win logical games. In Proc. DIMACS Workshop on Finite Model Theory and Descriptive Complexity, 1996.

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