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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, So you think you can Play this Game Peter van Emde Boas & Lena Kurzen ILLC-FNWI-Univ. of Amsterdam Bronstee.com Software & Services Symposium Driven by Search Univ. Maastricht May

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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)

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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.

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Playability Implicit Reasonability Assumptions for Game Analysis: We deal with Perfect Information Games represented by a Tree or Acyclic Graph 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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.....

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Amsterdam Contributions to TACTICS Lena Kurzen Logic for Cooperation, actions, preferences Complexity Dynamic Epistemic Logic Complexity of playing Eleusis

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human A plays 3 Diamonds; the card is accepted

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human B plays 10 clubs; the card is accepted

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human C plays 7 Clubs; the card is rejected

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play God moves the 7 Clubs card below the sequence

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human A plays Jack of Diamonds; the card is accepted

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human B claims that none of his cards can be played in accordance with the rule

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play God finds the King of hearts amongst the cards of B, and plays it; player B is punished

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human C plays 3 Clubs; the card is accepted

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human A plays 2 Hearts; the card is rejected

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play God places 2 Hearts below the sequence

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play Human B plays 9 Spades; the card is accepted

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Example play God’s Rule: every card must be followed by one of higher value, but after a King any card can be played.

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Formalization If C denotes the set of (52) traditional playing cards a rule can be formalized by a function R R : C* X C {true,false} subject to the condition : RR If R(, c k+1 ) = true then R(, c k ) = true Other than that a rule can be arbitrary

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, Three decision problems RQ1) Given class R and some configuration of cards C, is there a rule R in R consistent with C ? R RQ2) Given Rule R in R and some configuration of cards C, is C consistent with R ? RQ3) Given Rule R in R and some configuration of cards C and some card x, is playing x a legal move ?

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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.

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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...

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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....

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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.

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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.

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Peter van Emde Boas: So you think you can play this game ? Symposium Driven by Search, Maastricht, 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 ?

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