Ehrenfeucht Games and Equivalence classes
$begingroup$
One of the fundamental results in Ehrenfeucht games is the equivalence
$mathcal{I}sim_{m}^{sigma}mathcal{J}Leftrightarrowmathcal{I}equiv_{m}mathcal{J}$
with the $sigma$-structures $mathcal{I}$ and $mathcal{J}$ participating (1) to the equivalence relation $sim_m^{sigma}$ if and only if Duplicator wins the Ehrenfeucht game of $m$ rounds and (2) to the equivalence relation $equiv_m$ if and only if they satisfy the same formulas with quantifier rank up to $m$. It is proven that the relation $sim_m^{sigma}$ partitions the set of $sigma$-structures to a finite number of equivalence classes $E_1,E_2,dots,E_r$ and we can pick for each one of those classes $E_i$, a formula $psi_i$ such that $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$. My question is this: these equivalence classes are the same (and equal in number) with the ones produced by the other equivalence relation $equiv_n$ (that also partitions the set of $sigma$-strucures in equivalence classes)? In this case, the characteristic formula $psi_i$ also characterizes and the equivalence classes associated with $equiv_m$? I mean, if we write the conditions (a) $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$ and (b) $mathcal{I},mathcal{J}in E_iLeftrightarrow(mathcal{I}modelspsi_iLeftrightarrowmathcal{J}modelspsi_i)$, then (a) and (b) uses the same $psi_i$?
set-theory game-theory
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
$endgroup$
add a comment |
$begingroup$
One of the fundamental results in Ehrenfeucht games is the equivalence
$mathcal{I}sim_{m}^{sigma}mathcal{J}Leftrightarrowmathcal{I}equiv_{m}mathcal{J}$
with the $sigma$-structures $mathcal{I}$ and $mathcal{J}$ participating (1) to the equivalence relation $sim_m^{sigma}$ if and only if Duplicator wins the Ehrenfeucht game of $m$ rounds and (2) to the equivalence relation $equiv_m$ if and only if they satisfy the same formulas with quantifier rank up to $m$. It is proven that the relation $sim_m^{sigma}$ partitions the set of $sigma$-structures to a finite number of equivalence classes $E_1,E_2,dots,E_r$ and we can pick for each one of those classes $E_i$, a formula $psi_i$ such that $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$. My question is this: these equivalence classes are the same (and equal in number) with the ones produced by the other equivalence relation $equiv_n$ (that also partitions the set of $sigma$-strucures in equivalence classes)? In this case, the characteristic formula $psi_i$ also characterizes and the equivalence classes associated with $equiv_m$? I mean, if we write the conditions (a) $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$ and (b) $mathcal{I},mathcal{J}in E_iLeftrightarrow(mathcal{I}modelspsi_iLeftrightarrowmathcal{J}modelspsi_i)$, then (a) and (b) uses the same $psi_i$?
set-theory game-theory
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
$endgroup$
add a comment |
$begingroup$
One of the fundamental results in Ehrenfeucht games is the equivalence
$mathcal{I}sim_{m}^{sigma}mathcal{J}Leftrightarrowmathcal{I}equiv_{m}mathcal{J}$
with the $sigma$-structures $mathcal{I}$ and $mathcal{J}$ participating (1) to the equivalence relation $sim_m^{sigma}$ if and only if Duplicator wins the Ehrenfeucht game of $m$ rounds and (2) to the equivalence relation $equiv_m$ if and only if they satisfy the same formulas with quantifier rank up to $m$. It is proven that the relation $sim_m^{sigma}$ partitions the set of $sigma$-structures to a finite number of equivalence classes $E_1,E_2,dots,E_r$ and we can pick for each one of those classes $E_i$, a formula $psi_i$ such that $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$. My question is this: these equivalence classes are the same (and equal in number) with the ones produced by the other equivalence relation $equiv_n$ (that also partitions the set of $sigma$-strucures in equivalence classes)? In this case, the characteristic formula $psi_i$ also characterizes and the equivalence classes associated with $equiv_m$? I mean, if we write the conditions (a) $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$ and (b) $mathcal{I},mathcal{J}in E_iLeftrightarrow(mathcal{I}modelspsi_iLeftrightarrowmathcal{J}modelspsi_i)$, then (a) and (b) uses the same $psi_i$?
set-theory game-theory
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
$endgroup$
One of the fundamental results in Ehrenfeucht games is the equivalence
$mathcal{I}sim_{m}^{sigma}mathcal{J}Leftrightarrowmathcal{I}equiv_{m}mathcal{J}$
with the $sigma$-structures $mathcal{I}$ and $mathcal{J}$ participating (1) to the equivalence relation $sim_m^{sigma}$ if and only if Duplicator wins the Ehrenfeucht game of $m$ rounds and (2) to the equivalence relation $equiv_m$ if and only if they satisfy the same formulas with quantifier rank up to $m$. It is proven that the relation $sim_m^{sigma}$ partitions the set of $sigma$-structures to a finite number of equivalence classes $E_1,E_2,dots,E_r$ and we can pick for each one of those classes $E_i$, a formula $psi_i$ such that $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$. My question is this: these equivalence classes are the same (and equal in number) with the ones produced by the other equivalence relation $equiv_n$ (that also partitions the set of $sigma$-strucures in equivalence classes)? In this case, the characteristic formula $psi_i$ also characterizes and the equivalence classes associated with $equiv_m$? I mean, if we write the conditions (a) $mathcal{I}in E_iLeftrightarrowmathcal{I}modelspsi_i$ and (b) $mathcal{I},mathcal{J}in E_iLeftrightarrow(mathcal{I}modelspsi_iLeftrightarrowmathcal{J}modelspsi_i)$, then (a) and (b) uses the same $psi_i$?
set-theory game-theory
set-theory game-theory
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
edited Dec 3 '18 at 20:33
Athanasios Margaris
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
asked Dec 2 '18 at 18:49
Athanasios MargarisAthanasios Margaris
1213
1213
add a comment |
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