Scrabble is PSPACE-Completevmitsou/scrabble.pdf · Scrabble is PSPACE-Complete Michael Lampis, Valia Mitsou and Karolyna Soltys KTH, GC CUNY, MPI Scrabble is PSPACE-Complete – p.

Scrabble is PSPACE-CompleteMichael 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 Components-Rules

2 players

Board

Bag

Tiles

Rack

Turn


Our Model

Deterministic, full-information version

No special squares on the board

All tiles worth the same amount of points


Results

Deterministic version is PSPACE-Completehardness due to placementharness due to formation

constant-size rackconstant-size wordconstant-size alphabet

One player version is NP-Complete


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 from3-CNF-SAT and

PSPACE-Completeness of 2-player version of Scrabblefrom 3-CNF-QBS.


NP-Completeness due to Placement
















PSPACE-Completeness due to Formation

∃x1∀x2∃x3 . . . φ



φ =(x1 ∨ ¬x2 ∨ ¬x3) ∧(¬x1 ∨ x2 ∨ x4) ∧(¬x2 ∨ x3 ∨ ¬x4)


Dictionary

Words Definition

S(TF )k−1

2 S, F (TF )k−1

2 S, S(FT )k−1

2 F,The literal played has value True.

F (TF )k−3

2 STFTF, F (TF )k−3

2 SFTFT

S(FT )k−1

2 S, T (FT )k−1

2 S, S(TF )k−1

2 T,

The literal played has value False.T (FT )k−3

2 STFTF, T (FT )k−3

2 SFTFT

#AT, #AF First player’s turn to assign truth value.

#BS Second player’s turn to assign truth value.

$$, **, #A, #B, #c, for c ≤ 2kWall 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.



x1 = T, x2 = F



#AT, #AF :Player 1assignstruth value

#BS :Player 2assignstruth value



0**2F01 1 unsatisfied literal

0**1T200 unsatisfied literals0$$1F20

0$$1T20




Bonus

The pass move does not affect the reduction

Zig-zag within the variables to obtain a reduction forconstant size words and rack.



Conclusions

ResultsDeterministic version of Scrabble isPSPACE-CompleteHardness comes from two different aspects of thegame1-player version is NP-Complete

Future WorkFind the minimum alphabet size for which theproblem becomes hardProve that Scrabble is PSPACE-Complete for theEnglish dictionaryStudy the randomized version of the game



Scrabble is PSPACE-Completevmitsou/scrabble.pdf · Scrabble is PSPACE-Complete Michael Lampis, Valia Mitsou and Karolyna Soltys KTH, GC CUNY, MPI Scrabble is PSPACE-Complete – p.

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