So you think you can Play this Game

1. So you think you can Play this Game Symposium Driven by Search Univ. Maastricht May 24 2011 Peter van Emde Boas & Lena Kurzen ILLC-FNWI-Univ. of Amsterdam Bronstee.com Software & Services

2. Original Research Topic TACTICS • Game Playability: the relation between complexity aspects of games and human capabilities of actually playing the game. • Interdisciplinary between Formal and Social/Economical science • CREED eventually didn’t participate

3. Complexity • In the context of games various complexity notions are relevant • Size Game Tree • Size Game Graph (State-Space Complexity) • Computational Complexity of Solving Game (End-game Analysis, Winner Determination, Computation Strategy) • Time/Space Complexity Measures

4. Common Belief about Games • (End)-game Analysis of Reasonable Games can be performed in PSPACE • For many Games this problem is in fact PSPACE-hard • Snag: this problem sometimes is even harder (EXPTIME)

5. Why is (End)-game in PSPACE? The Standard Dynamic Programming Algorithm for Backward Induction uses the entire Configuration Graph as a Data Structure: Exponential Space ! Instead we can Use Recursion over Sequences of Moves: The analysis of the Recursive algorithm exposes the implicit assumptions on Reasonable Games which make this approach valid.

6. Playability Implicit Reasonability Assumptions for Game Analysis: We deal with Perfect Information Games represented by a Tree or AcyclicGraph of Configurations Deciding questions like: is p a position ?, is p final ? is p starting position ?, who has to move in p ? is p  q a legal move ? and the generation of successors/predecessors of p are all (computationally) very easy problems ..... Hence we can traverse the tree (graph) in time proportional to its size..... Extra assumption: polynomial bound on length gameplay

7. End-game Analysis in PSPACE? The Recursive method combines recursion (over move sequence) with iteration (over locally legal moves). Space Consumption = O( | Stackframe | . Recursion Depth ) | Stackframe | = O( | Move sequence | + | Configuration| ) Recursion Depth = | Move sequence | = O( Duration Game ) which explains why the game duration should be Polynomial.....

8. Amsterdam Contributions to TACTICS Logic for Cooperation, actions, preferences Complexity Dynamic Epistemic Logic Complexity of playing Eleusis Lena Kurzen 20101216

9. ELEUSIS • Game which models Inductive Inference (Scientific Discovery) • Invented by Robert Abbot in 1956 • Popularized (amongst others) by Martin Gardner (SCIAM 1977) • By its very nature it violates some of the basic reasonability conditions • Deciding whether some move is legal can be hard

10. ELEUSIS • Played with standard decks of cards • first player : God • remaining players : Humans • Scoring based on getting rid of your cards • punishment for wrong moves == drawing extra cards

11. God’s Role in Eleusis • God starts game by inventing a Rule (which he keeps secret) • God checks whether moves played by humans obey the rule; violators must draw extra cards • If some human has declared himself a Prophet, God checks whether the predictions made by this prophet are correct; a false prophet is punished severely

12. Human’s role in Eleusis • If a human can play some of his cards in accordance with the rule he may play one; God checks whether the card indeed is legal; if not the card is placed below the sequence and the human player must draw another card • If a human believes all his cards would violate the rule he can claim so; God checks whether the claim is correct; if not God plays a legal card and the human player is punished

13. Prophets • A human player who believes to have discovered the rule may claim to be a prophet (only once and only if no other prophet is active) • The prophet replaces God as a judge for the moves of other humans, as long as God agrees; however if the prophet gives a false verdict he will be overturned by God and punished severely

14. Further rules • There are special rules enforcing the termination of the game . • The precise rules determining the scoring are irrelevant for the purpose of this talk

15. Example play Human A plays 3 Diamonds; the card is accepted

16. Example play Human B plays 10 clubs; the card is accepted

17. Example play Human C plays 7 Clubs; the card is rejected

18. Example play God moves the 7 Clubs card below the sequence

19. Example play Human A plays Jack of Diamonds; the card is accepted

20. Example play Human B claims that none of his cards can be played in accordance with the rule

21. Example play God finds the King of hearts amongst the cards of B, and plays it; player B is punished

22. Example play Human C plays 3 Clubs; the card is accepted

23. Example play Human A plays 2 Hearts; the card is rejected

24. Example play God places 2 Hearts below the sequence

25. Example play Human B plays 9 Spades; the card is accepted

26. Example play God’s Rule: every card must be followed by one of higher value, but after a King any card can be played.

27. Constraints on Rules • The rule must depend only on the sequence of accepted cards on the table • Excludes rules like: • male humans play black, females play red • if it rains outside play black • only accept cards played by worthy people • Prefix Closure: the initial segments of a legal sequence are legal • Otherwise some legal configuration can become unreachable by a correct game play • Other optional constraints for preventing degenerate plays: E.G., each legal sequence must have a legal extension

28. Formalization If C denotes the set of (52) traditional playing cards a rule can be formalized by a function R : C* X C  {true,false} subject to the condition : If R( <c1,c2,...,ck> , ck+1 ) = true then R( <c1,c2,...,ck-1> , ck ) = true Other than that a rule can be arbitrary

29. Complexity Issues • Since rules can be arbitrary every configuration can be legal (except when at some position some card is both accepted and rejected) • Therefore a crucial parameter: • The class R of rules from which God must select his rule

30. Three decision problems • Q1) Given class R and some configuration of cards C , is there a rule R in R consistent with C ? • Q2) Given Rule R in R and some configuration of cards C , is C consistent with R ? • Q3) Given Rule R in R and some configuration of cards C and some card x, is playing x a legal move ?

31. Interesting Rule classes • k-Bounded context rule: Legality of some card depends on the last k previous cards only (k may be 0 ) • Examples • red, black, red, black, .... • any Ace must be followed by three red cards.... • all figure cards must be black

32. Interesting Rule classes • Periodic rule mod t : there are in fact t rules and the legality of a card in position i depends only on the cards located in positions j ≡ i mod t • Examples • red, black, red, black, .... • on even positions only play figure cards.... • value of card in position i must be within distance 3 of the card in position i - 4

33. A positive result • If R is the class of all k-bounded context rules problem Q1 is solvable in polynomial time • k may be part of the input • idea : the answer is “yes” unless some card is both accepted and rejected when played in identical context’s • remains valid if generalized to periodic k-bounded context rules • Note that the length of a shortest description of some k-bounded context rule may be exponential in k ; so an idea like trying out all possible rules will not work here.

34. Negative results • Since rules can be arbitrary and since cards can be used to encode standard information in a concise way, nobody can prevent you from encoding hard combinatorial problems in rules. • {red, black} encode bits • suits encode symbols from 4-letter alphabet (genetic code) • values encode decimal numbers, leaving the figure cards for coding separators etc...

35. Prefix closure ? • Using these coding tricks the prefix of some code may fail to be a legal code itself. • If the rule requires an encoding of some solvable instance of a combinatorial problem, some prefix may fail to encode an instance at all • Solution: use a signaling card: only when 7 clubs is played check whether the preceding string of cards encodes a solvable instance ....

36. Example of a rule for which Q3 is NP-hard • Encoding Partition problem: use consecutive blocks of cards with value in {A, 2, ..., 10} to code decimal digits and numbers, using figures as separators. • Rule: all cards are OK, but if a red figure card is played the sequence of integers encoded in the prefix must give a solvable instance of the Partition problem.

37. Example of a rule for which Q3 is Undecidable • Encoding PCP problem: use consecutive monochromatic blocks of cards with value in {A, 2, ..., 10} to code string pairs. • Rule: all cards are OK, but if a red figure card is played following a sequence of string pairs then the series of string pairs encoded in the prefix must yield a solvable instance of Post Correspondence Problem.

38. Conclusion • Eleusis is unreasonably flexible – it trivially encodes problems at arbitrary levels of computational complexity • Evident connections to learning theory (which classes of rules can be discovered in the limit, or by approximation (PAC learning)) • Well suited to experiments with real human players: what type of rules are invented by actual humans playing God ?