Scrabble is PSPACE-Complete Michael Lampis, Valia Mitsou and Karolyna Soltys KTH, GC CUNY, MPI Scrabble is PSPACE-Complete – p. 1/25 A famous game... Word game played on a grid 150 million sets sold in 121 countries and 29 languages Scrabble is PSPACE-Complete – p. 2/25 Scrabble Components-Rules 2 players Board Bag Tiles Rack Turn Scrabble is PSPACE-Complete – p. 3/25 Our Model Deterministic, full-information version No special squares on the board All tiles worth the same amount of points Scrabble is PSPACE-Complete – p. 4/25 Results Deterministic version is PSPACE-Complete hardness due to placement harness due to formation constant-size rack constant-size word constant-size alphabet One player version is NP-Complete Scrabble is PSPACE-Complete – p. 5/25 3-CNF-Quantified Boolean Satisfaction n variables x1,x2,...,xn FO formula: ∃x1∀x2∃x3 ...φ φ is 3-CNF formula with m clauses QUESTION: Is the FO formula satisfiable? 3-CNF-QBF is PSPACE-Complete. We prove: NP-Completeness of 1-player version of Scrabble from 3-CNF-SAT and PSPACE-Completeness of 2-player version of Scrabble from 3-CNF-QBS. Scrabble is PSPACE-Complete – p. 6/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 7/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 8/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 9/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 10/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 11/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 12/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 13/25 NP-Completeness due to Placement Scrabble is PSPACE-Complete – p. 14/25 PSPACE-Completeness due to Formation ∃x1∀x2∃x3 ...φ Scrabble is PSPACE-Complete – p. 15/25 PSPACE-Completeness due to Formation φ = (x1 ∨¬x2 ∨¬x3) ∧ (¬x1 ∨ x2 ∨ x4) ∧ (¬x2 ∨ x3 ∨¬x4) Scrabble is PSPACE-Complete – p. 16/25 Dictionary Words Definition k−1 k−1 k−1 S(T F ) 2 S, F (T F ) 2 S, S(FT ) 2 F, k−3 k−3 The literal played has value True. F (T F ) 2 STFTF, F (T F ) 2 SFTFT k−1 k−1 k−1 S(FT ) 2 S, T (FT ) 2 S, S(T F ) 2 T, k−3 k−3 T (FT ) 2 STFTF, T (FT ) 2 SFTFT The literal played has value False. #AT, #AF First player’s turn to assign truth value. #BS Second player’s turn to assign truth value. $$, **, #A, #B, #c, for c ≤ 2k Wall word. #5Q#9Q#9Q#5, for Q ∈ {$, *} 0**, 1**, 2**, 0$$, 1$$, 2$$ Word formed during satisfaction phase. 0**1T20, 0$$1T20, 0$$1F20 No unsatisfied literals in the clause so far. 1**2T01, 1$$2T01, One unsatisfied literal in the clause so far. 1$$2F01, 0**2F01 2**0T12, 2$$0T12, Two unsatisfied literals in the clause so far. 2$$0F12, 1**0F12 0120, 1201, 2012 Symbols’ 0, 1, 2 order preserving words. Scrabble is PSPACE-Complete – p. 17/25 PSPACE-Completeness due to Formation x1 = T,x2 = F Scrabble is PSPACE-Complete – p. 18/25 PSPACE-Completeness due to Formation #AT, #AF : Player 1 assigns truth value #BS : Player 2 assigns truth value Scrabble is PSPACE-Complete – p. 19/25 PSPACE-Completeness due to Formation 0**2F01 1 unsatisfied literal 0**1T20 0$$1F20 0 unsatisfied literals 0$$1T20 Scrabble is PSPACE-Complete – p. 20/25 PSPACE-Completeness due to Formation Scrabble is PSPACE-Complete – p. 21/25 Bonus The pass move does not affect the reduction Zig-zag within the variables to obtain a reduction for constant size words and rack. Scrabble is PSPACE-Complete – p. 22/25 Scrabble is PSPACE-Complete – p. 23/25 Conclusions Results Deterministic version of Scrabble is PSPACE-Complete Hardness comes from two different aspects of the game 1-player version is NP-Complete Future Work Find the minimum alphabet size for which the problem becomes hard Prove that Scrabble is PSPACE-Complete for the English dictionary Study the randomized version of the game Scrabble is PSPACE-Complete – p. 24/25 Scrabble is PSPACE-Complete – p. 25/25.