1. More Metagaming General Game Playing Lecture 7
Michael Genesereth / David Haley Spring 2007

2. Overview
Metagaming is the process of analyzing and/or modifying a game during the startclock so as to improve performance once the game begins.
Examples of Metagaming Techniques:
Headstart on Game-Graph Search
Game Optimization
Explicit Propositional / Relational Nets
Compilation
Factoring
Dead State Removal
And so forth

3. Factoring

4. Hodgepodge Hodgepodge = Chess + Othello Analysis of joint game:
Branching factor: a Branching factor: b
Analysis of joint game:
Branching factor as given to players: a*b
Fringe of tree at depth n as given: (a*b)n
Fringe of tree at depth n factored: an+bn

5. Double Tic-Tac-Toe X X O O Double Tic Tac Toe = TTT + TTT
Analysis of joint game:
Branching factor: 81, 64, 49, 36, 25, 16, 9, 4, 1
Branching factor: 9, 8, 7, 6, 5, 4, 3, 2, 1 ( 2) X O X O

6. Buttons and Lights Pressing button a toggles p.
Pressing button b interchanges p and q. p q a b
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, comjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

7. Double Buttons and Lights
Pressing button a toggles p.
Pressing button b interchanges p and q.
Pressing button c toggles s.
Pressing button d interchanges s and t. p q a b c d s t
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, comjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

8. Game Factoring and Its Use
Compute factors:
Behavioral factoring
Goal Factoring
Relative Inertia
Play factors
Reassemble solution:
Append plans
Interleave plans
Parallelize plans with simultaneous actions

9. Behavioral Factoring
A set F of propositions and actions in a propositional net P is a behavioral factor of P if and only if there are no connections between the propositions and actions in F and those in P-F.
Factors of propositional nets can be computed in polynomial time (in terms of the size of the nets).
Basic Idea: For each input, compute all affected propositions (any number of steps). Combine sets into common factor if they intersect.

10. Goal Factoring Simple Case - goal is a conjunction
Partition conjuncts over behavioral factors
Create new goals for each factor
Complex Case - goal a disjunction of conjunctions
Split each conjunct as above
Check for lossless joins, i.e. when recombined,
we get the same results
Good: Bad:
p1 & q1 | p1 & q2 p1 & q1 | p2 & q2

11. Relative Inertia
Show that the propositions in a factor do not change unless one of the actions in the factor is executed.
Easily proven from inertial rules:
next(p) :-
true(p) &
-does(robot,a) &
-does(robot,b)
Note that we must be able to prove this for every proposition in the factor. Note that only actions in the factor can be mentioned.

12. Performance ? (time (genplan propcompbuttons)) 407 states
1,118 milliseconds.
605,728 bytes of memory allocated.
(PROG A B A D E D)
? (time (multiplan propcompbuttons))
14 states
53 milliseconds
22,320 bytes of memory allocated.
Partition time: 1 millisecond.

13. Original Version p' :- a & -p p' :- b & q q' :- a & q q' :- b & p
s' :- a & -s s' :- b & -s
t' :- a & t t' :- b & t
p' :- c & -p p' :- d & q
q' :- c & q q' :- d & p
s' :- c & t s' :- d & t
t' :- c & s t' :- d & s

14. Action Grouping Actions grouped according to behavior.
p' :- {a,c} & -p p' :- {b,d} & q
q' :- {a,c} & q q' :- {b,d} & p
s' :- {a,b} & -s s' :- {c,d} & -s
t' :- {a,b} & t t' :- {c,d} & t
Note the partition on propositions.

15. Import and Export Import: Export: e :- a a :- e & g e :- b b :- e & h
f :- c c :- f & g
f :- d d :- f & h
g :- a
g :- c
h :- b
h :- d

16. Factored Version p' :- e & -p q' :- e & q p' :- f & q q' :- f & p
s' :- g & -s
t' :- g & t
s' :- h & t
t' :- h & s

17. Dead State Removal

18. Latches
A latch is a proposition that persists indefinitely, i.e. once it becomes true, it stays true forever.
next(p) :- true(p)

19. Inhibition
A proposition p inhibits a proposition q if and only if p must be false in every plan to achieve or retain q.
next(q) :-
does(robot,a) &
-true(p)
true(q) &
goal :-
true(r)

20. Dead State Removal
If a latch p inhibits the goal in a game, then it is a good idea to avoid states in which p is true.

21. Untwisty Corridor Roles: Robot Actions: a, b, c, d
Bases: p, q1, …, q8, 1, …, 9
Goal: q8
Actions a, b, c set p.
Once true p remains true.
Action d successively sets q1, …, q8 iff p is false.
Plan: d, d, d, d, d, d, d, d
Game tree size: 349,525

22. Performance ? (time (genplan untwistycorridor)) 349,521 states
1,133,293 milliseconds.
597,682,424 bytes of memory allocated.
(PROG D D D D D D D D)
? (time (strplan untwistycorridor))
9 states
183 milliseconds
54,568 bytes of memory allocated.

23. And So Forth

24. Overview
Game Reformulation is transformation of a game into one or more different games that can be played more efficiently yet yield the same result.
Sample Methods:
Sequential Independence - Bottleneck - Maze
Parallel Independence - Hodgepodge
Single player abstraction
End game book to enlarge goal set - Checkers
Symmetry - Tic-Tac-Toe
Hierarchical Abstraction

25. Metagaming Ideal Trade-off - cost of finding structure vs savings
Find cases where cost varies
with size of description (e.g. proposition set)
rather than with size of expanded game graph
Techniques:
Graph Analysis
Automated Reasoning