Scrabble is PSPACE-Complete

Transcription

1 Scrabble is PSPACE-Complete Michael Lampis, Valia Mitsou and Karolyna Soltys KTH, GC CUNY, MPI Scrabble is PSPACE-Complete p. 1/25

2 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

3 Scrabble Components-Rules 2 players Board Bag Tiles Rack Turn Scrabble is PSPACE-Complete p. 3/25

4 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

5 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

6 3-CNF-Quantified Boolean Satisfaction n variables x 1,x 2,...,x n FO formula: x 1 x 2 x 3...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

7 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 7/25

8 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 8/25

9 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 9/25

10 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 10/25

11 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 11/25

12 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 12/25

13 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 13/25

14 NP-Completeness due to Placement Scrabble is PSPACE-Complete p. 14/25

15 PSPACE-Completeness due to Formation x 1 x 2 x 3...φ Scrabble is PSPACE-Complete p. 15/25

16 PSPACE-Completeness due to Formation φ = (x 1 x 2 x 3 ) ( x 1 x 2 x 4 ) ( x 2 x 3 x 4 ) Scrabble is PSPACE-Complete p. 16/25

17 Words S(T F ) k 1 2 S, F (T F ) k 1 2 S, S(F T ) k 1 2 F, F (T F ) k 3 2 ST F T F, F (T F ) k 3 2 SF T F T S(F T ) k 1 2 S, T (F T ) k 1 2 S, S(T F ) k 1 2 T, T (F T ) k 3 2 ST F T F, T (F T ) k 3 2 SF T F T #AT, #AF #BS $$, **, #A, #B, # c, for c 2k # 5 Q# 9 Q# 9 Q# 5, for Q {$, *} Dictionary Definition The literal played has value True. The literal played has value False. First player s turn to assign truth value. Second player s turn to assign truth value. Wall word. 0**, 1**, 2**, 0$$, 1$$, 2$$ Word formed during satisfaction phase. 0**1T 20, 0$$1T 20, 0$$1F 20 1**2T 01, 1$$2T 01, 1$$2F 01, 0**2F 01 2**0T 12, 2$$0T 12, 2$$0F 12, 1**0F 12 No unsatisfied literals in the clause so far. One unsatisfied literal in the clause so far. Two unsatisfied literals in the clause so far. 0120, 1201, 2012 Symbols 0, 1, 2 order preserving words. Scrabble is PSPACE-Complete p. 17/25

18 PSPACE-Completeness due to Formation x 1 = T,x 2 = F Scrabble is PSPACE-Complete p. 18/25

19 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

20 PSPACE-Completeness due to Formation 0**2F01 0**1T20 0$$1F20 0$$1T20 1 unsatisfied literal 0 unsatisfied literals Scrabble is PSPACE-Complete p. 20/25

21 PSPACE-Completeness due to Formation Scrabble is PSPACE-Complete p. 21/25

22 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

23 Scrabble is PSPACE-Complete p. 23/25

24 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

25 Scrabble is PSPACE-Complete p. 25/25