From: rld@math.ohio-state.edu (Randall Dougherty)
Subject: Re: planar graphs and first-order definability
Date: 7 Sep 1999 20:06:43 GMT
Newsgroups: sci.math
Keywords: Ehrenfeucht-Fraisse games
In article ,
Allan Adler wrote:
>That sounds pretty neat. Where can one find an exposition of
>these Ehrenfeucht-Fraisse games?
One place is chapter 26 of Monk's "Mathematical Logic." (If anyone
knows of an exposition for a wider audience with less notation and
more examples, please let me know.) I think the first place where
it was written about in explicit game form was a paper of Ehrenfeucht:
Fundamenta Mathematicae 49 (1961), 129-141.
Randall Dougherty rld@math.ohio-state.edu
Department of Mathematics, Ohio State University, Columbus, OH 43210 USA
"I have yet to see any problem, however complicated, that when looked at in the
right way didn't become still more complicated." Poul Anderson, "Call Me Joe"
==============================================================================
[From MathSciNet --djr]
[Note: they're actually Ehrenfeucht-Fraïssé games --djr]
82a:03033 03C80 (03C95 68F05 90D45)
Weese, Martin
Generalized Ehrenfeucht games.
Fund. Math. 109 (1980), no. 2, 103--112. [To journal home page]
[...]
In 1961, A. Ehrenfeucht formulated a game-theoretic criterion for
elementary equivalence of two structures. This criterion has since
been generalized to many nonelementary logics, including those with
the quantifiers mentioned above as well as many others.
[...]
Reviewed by John Cowles
_________________________________________________________________
23 #A3666 02.50
Ehrenfeucht, A.
An application of games to the completeness problem for formalized
theories.
Fund. Math. 49 1960/1961 129--141. [To journal home page]
Model-theoretic results are given for the lower predicate calculus
with identity, the set-theoretic $\varepsilon$-relation and any set
$A$ of additional predicates. (A preliminary account was given in
Bull. Acad. Polon. Sci. Cl. III 5 (1957), 35--37, IV [MR 19, 4].)
Given such a formal system, all models are considered in which the
variables range over individuals and finite sets, each predicate of
$A$ is a relation among individuals, and a specified singulary
predicate of $A$ identifies the individuals. Models $M$ and $N$ are
called "indiscernible" if, for any closed formula, satisfaction in $M$
is equivalent to satisfaction in $N$. It is shown that a sufficient,
but not necessary, condition that $M$ and $N$ be indiscernible is
that, for each positive integer $n$, the second player have a winning
strategy in the game $H\sb n(M,N)$ defined thus. $H\sb n(M,N)$ is an
$n$-move game. In each move, the first player chooses one of the two
models and points out a finite sequence of individuals in it; then the
second player pairs each individual of the sequence with an individual
in the other model. Let $(a\sb 1,b\sb 1),\cdots,(a\sb p,b\sb p)$,
where the $a$'s are in $M$ and $b$'s are in $N$, be the sequence of
pairs resulting from a play of the game. The condition for the second
player to win is that, for each predicate $\alpha$ in $A$---say with
$v$ variables and interpretations $\alpha\sb M$ and $\alpha\sb N$ in
the respective models---the following is true: for all $v$-tuples
$(i\sb 1,\cdots,i\sb v)$ of integers from 1 to $p,\alpha\sb M(a\sb
{i\sb 1},\cdots,a\sb {i\sb v})\equiv\alpha\sb N(b\sb {i\sb
1},\cdots,b\sb {i\sb v})$.
A second set of results is obtained regarding satisfaction of a
particular subset of closed formulas (roughly speaking, those that
express statements about individuals). Models $M$ and $N$ are called
"elementarily indiscernible" if, for any formula of this particular
subset, satisfaction in $M$ is equivalent to satisfaction in $N$. A
sufficient condition that $M$ and $N$ be elementarily indiscernible is
that, for each positive integer $n$, the second player have a winning
strategy in the game $G\sb n(M,N)$, which differs from $H\sb n(M,N)$
in that in each move the first player points out a single individual.
This condition is not in general necessary, but it is necessary
whenever $A$ is finite.
The game-theoretic results are applied to three particular formalized
theories of ordinal numbers. For each theory, infinitely many infinite
classes of indiscernible models are found. Moreover, the same
techniques lead to necessary and sufficient conditions that an ordinal
number be definable in the respective theories.
Reviewed by G. F. Rose
_________________________________________________________________
19,4d 02.0X
Ehrenfeucht, A.
Application of games to some problems of mathematical logic. (English.
Russian summary)
Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 35--37, IV,.
This is a preliminary report and the proofs of the theorems contained
in it will be published in Fund. Math. It contains a necessary and
sufficient condition for elementary indiscernibility of models, a
sufficient condition for the indiscernibility of models by means of
finite sets and three theorems, the second of which is the following:
If player II has a winning method in $H\sb n(M',M")$ for every $n$,
then $M'$ and $M"$ are indiscernible by finite sets. Here $H\sb
n(M',M")$ is a game in which the two players I and II make $n$ moves,
and $M'$ and $M"$ are two models for an elementary theory $T$ which
has no terms and a finite number of predicates only. The set of these
predicates is denoted by $P$. In every move the player I points out an
arbitrary finite number $m$ of elements either from $\vert M'\vert $
or from $\vert M"\vert $, where $\vert M\vert $ denotes the set of
individuals of $M$. Player II has to match every individual pointed
out by I with an individual in the other model. Player II wins if for
any predicate $\alpha\in P$ and any finite sequence $k\sb
1,\cdots,k\sb {a(\alpha)}$, where $1\leq k\sb i\leq n$,
$\text{stsf}\sb {M'}\alpha(a'{}\sb {k\sb {{1}}},\cdots,a\sb
{ka(\alpha)})$ if, and only if, $\text{stsf}\sb {M"}\alpha(a\sb {k\sb
{{\bf 1}}}{}",\cdots,a\sb {ka(\alpha)}{}")$. All notations concerning
theories and models are those used by Ehrenfeucht and Mostowski [Fund.
Math. 43 (1956), 50--68; MR 18, 863].
Reviewed by B. Germansky
© Copyright American Mathematical Society 2000