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Archived from groups: rec.games.diplomacy (More info?)
Jim Burgess wrote:
>
> It is depressing to JUST see messages from the spammers and about
> the spamming. Anyone have any real comments or questions??
>
> I'm looking for articles for the next issue of Diplomacy World,
> just one more month to write them!! Contact me for editorial
> guidance or details.
I won't have the free time to spit and polish any formal essay for a Dip
publication in the next month, but I have been mulling over one highly
theoretical topic for the past few weeks. Perhaps others have discussed
this in other publications; I don't know. (If someone has seen detailed
discussions like this before, I'd be interested in a link or something.
The closest thing I've found is a brief mention of Godel's Theorem
applied to Diplomacy in a thread back in 2001.)
As you may be aware from my recent posts, I have found far too much
personal amusement in the various rule problems of this otherwise very
high-quality game. The rule paradoxes trouble me especially. One day I
got in an e-mail argument with a friend who believes that there really
are no paradoxes in the rules, merely ambiguities. His arguments seemed
spurious to me, though, and I tried to devise how I might prove to him
logically that there really are paradoxes in the rules.
It occurred to me that when you feed a paradoxical situation to a judge
program, it will loop infinitely, never rendering any resolution of the
given orders; whereas with orders that illustrate ambiguities, the judge
program will eventually produce an outcome according to programmer bias.
And then, my mind took a turn for the worse. I began to think of an
adjudicator as a logical reasoning engine, which I suppose is really
what it is, and played several thought experiments.
First, what would happen if the rule paradoxes were fixed? I suppose
that there would be no set of orders that would cause an adjudicator to
loop infinitely. Secondly, why exactly does an adjudicator loop like
that? Because of unresolved circular dependencies, such as "if A then
B; if B then A; if not A then not B; if not B then not A". And then, a
thought experiment I found truly fascinating: what other kinds of
logical problems could be represented as Diplomacy orders on a
particular map with particular units in particular places, and solved
using a judge program?
To get directly to the point, I wonder whether it is possible to encode
any Turing machine, including state diagram and initial "tape" (memory)
contents, as a single Diplomacy map, position, and set of orders;
resolve the orders; and then interpret the resultant position as the
final "tape" contents of the Turing machine. In other words, is the
Diplomacy order resolution process Turing-complete?
This question is quite relevant to the issue of paradoxes, and even
useful. If it were proven that the Diplomacy rules are Turing-complete,
then it must necessarily be true that the rules contain paradoxes, since
Turing machines can loop infinitely. And more importantly, fixing the
Diplomacy rules would render them non-Turing-complete! (Of course, even
if the rules are shown not to be Turing-complete, there still can be
paradoxes.)
I would like to present this problem as a challenge for the Computer
Science variety of Diplomacy players out there. To satisfy the
challenge, some kind of proof by contradiction could be offered that
shows the Diplomacy rules are not Turing-complete, or some algorithm
could be offered for translating the encoding of a Turing machine into a
complete Diplomacy order resolution problem, and for translating the
result back into the final tape contents.
In coming up with an encoding, it should be allowable to have a map of
arbitrary but finite size and structure (not even necessarily a planar
graph, but yet conforming to the rules of land/water adjacencies), and
an arbitrary but finite number of Powers and units (up to the number of
supply centers, of course), with each unit having an arbitrary but legal
location and an arbitrary but valid order.
Also, since all the paradoxes I've ever seen involve the convoy order in
conflict with either the beleaguered garrison rule or, more recently,
the rule that dislodged units do not affect provinces their attackers
come from, it seems natural that these situations would be important for
representing most kinds of iterative/recursive logical operations.
Perhaps it would be helpful to correlate Diplomacy orders resolution to
a known Turing-complete inference engine like Prolog. If any set of
Prolog "rules" can be expressed as a set of Diplomacy orders with map
and units, then Diplomacy is Turing complete.
Or perhaps a highly theoretical language like Unlambda
(http://www.eleves.ens.fr:8080/home/madore/programs/unlambda/) would
actually lend itself to this problem. Unlambda is Turing complete with
only 2 primitive functions and 1 operator, though it includes more
primitives for the pretense of convenience.
It may also be that Diplomacy orders resolution could be more easily
proven equivalent to logic circuits. One can build a Turing complete
system entirely out of 2-input NAND gates and a clock; if some corollary
to the NAND operation can be found within the Diplomacy rules, problem
solved.
Anybody else ever given this question some thought?
--
Will Berry
Co-founder, Second Brain website hosting
http://www.secondbrainhosting.com/
Jim Burgess wrote:
>
> It is depressing to JUST see messages from the spammers and about
> the spamming. Anyone have any real comments or questions??
>
> I'm looking for articles for the next issue of Diplomacy World,
> just one more month to write them!! Contact me for editorial
> guidance or details.
I won't have the free time to spit and polish any formal essay for a Dip
publication in the next month, but I have been mulling over one highly
theoretical topic for the past few weeks. Perhaps others have discussed
this in other publications; I don't know. (If someone has seen detailed
discussions like this before, I'd be interested in a link or something.
The closest thing I've found is a brief mention of Godel's Theorem
applied to Diplomacy in a thread back in 2001.)
As you may be aware from my recent posts, I have found far too much
personal amusement in the various rule problems of this otherwise very
high-quality game. The rule paradoxes trouble me especially. One day I
got in an e-mail argument with a friend who believes that there really
are no paradoxes in the rules, merely ambiguities. His arguments seemed
spurious to me, though, and I tried to devise how I might prove to him
logically that there really are paradoxes in the rules.
It occurred to me that when you feed a paradoxical situation to a judge
program, it will loop infinitely, never rendering any resolution of the
given orders; whereas with orders that illustrate ambiguities, the judge
program will eventually produce an outcome according to programmer bias.
And then, my mind took a turn for the worse. I began to think of an
adjudicator as a logical reasoning engine, which I suppose is really
what it is, and played several thought experiments.
First, what would happen if the rule paradoxes were fixed? I suppose
that there would be no set of orders that would cause an adjudicator to
loop infinitely. Secondly, why exactly does an adjudicator loop like
that? Because of unresolved circular dependencies, such as "if A then
B; if B then A; if not A then not B; if not B then not A". And then, a
thought experiment I found truly fascinating: what other kinds of
logical problems could be represented as Diplomacy orders on a
particular map with particular units in particular places, and solved
using a judge program?
To get directly to the point, I wonder whether it is possible to encode
any Turing machine, including state diagram and initial "tape" (memory)
contents, as a single Diplomacy map, position, and set of orders;
resolve the orders; and then interpret the resultant position as the
final "tape" contents of the Turing machine. In other words, is the
Diplomacy order resolution process Turing-complete?
This question is quite relevant to the issue of paradoxes, and even
useful. If it were proven that the Diplomacy rules are Turing-complete,
then it must necessarily be true that the rules contain paradoxes, since
Turing machines can loop infinitely. And more importantly, fixing the
Diplomacy rules would render them non-Turing-complete! (Of course, even
if the rules are shown not to be Turing-complete, there still can be
paradoxes.)
I would like to present this problem as a challenge for the Computer
Science variety of Diplomacy players out there. To satisfy the
challenge, some kind of proof by contradiction could be offered that
shows the Diplomacy rules are not Turing-complete, or some algorithm
could be offered for translating the encoding of a Turing machine into a
complete Diplomacy order resolution problem, and for translating the
result back into the final tape contents.
In coming up with an encoding, it should be allowable to have a map of
arbitrary but finite size and structure (not even necessarily a planar
graph, but yet conforming to the rules of land/water adjacencies), and
an arbitrary but finite number of Powers and units (up to the number of
supply centers, of course), with each unit having an arbitrary but legal
location and an arbitrary but valid order.
Also, since all the paradoxes I've ever seen involve the convoy order in
conflict with either the beleaguered garrison rule or, more recently,
the rule that dislodged units do not affect provinces their attackers
come from, it seems natural that these situations would be important for
representing most kinds of iterative/recursive logical operations.
Perhaps it would be helpful to correlate Diplomacy orders resolution to
a known Turing-complete inference engine like Prolog. If any set of
Prolog "rules" can be expressed as a set of Diplomacy orders with map
and units, then Diplomacy is Turing complete.
Or perhaps a highly theoretical language like Unlambda
(http://www.eleves.ens.fr:8080/home/madore/programs/unlambda/) would
actually lend itself to this problem. Unlambda is Turing complete with
only 2 primitive functions and 1 operator, though it includes more
primitives for the pretense of convenience.
It may also be that Diplomacy orders resolution could be more easily
proven equivalent to logic circuits. One can build a Turing complete
system entirely out of 2-input NAND gates and a clock; if some corollary
to the NAND operation can be found within the Diplomacy rules, problem
solved.
Anybody else ever given this question some thought?
--
Will Berry
Co-founder, Second Brain website hosting
http://www.secondbrainhosting.com/