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Hardness of Mastermind Giovanni Viglietta Department of Computer Science, University of Pisa, Italy Venice - June 5 th , 2012 Easy to learn. Easy to play. But not so easy to win. Mastermind commercial, 1981 Hardness of Mastermind

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SLIDE 1

Hardness of Mastermind

Giovanni Viglietta

Department of Computer Science, University of Pisa, Italy

Venice - June 5th, 2012

“Easy to learn. Easy to play. But not so easy to win.” Mastermind commercial, 1981 Hardness of Mastermind

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SLIDE 2

Mastermind is played on a board with colored pegs. A codemaker chooses a secret sequence of colors, and a codebreaker has to guess it in several attempts.

Hardness of Mastermind

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SLIDE 3

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Hardness of Mastermind

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SLIDE 4

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 5

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 6

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 7

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 8

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 9

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 10

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 11

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 12

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 13

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 14

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 15

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 16

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 17

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 18

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 19

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 20

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 21

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 22

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 23

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 24

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 25

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 26

After each guess, the codemaker responds with some black and white pegs.

Black pegs represent correct pegs in the codebreaker’s guess that are also well-placed. White pegs represent pegs in the codebreaker’s guess that are correct but misplaced. Black and white pegs do not mark the positions of the correct pegs in the codebreaker’s guess, but only their amount.

Secret code:

Hardness of Mastermind

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SLIDE 27

Mastermind in bank frauds

The relevance of Mastermind in real-life security issues was pointed out in 2010 by Focardi and Luccio. An insider of a bank who gains access to some switch is able to issue several PIN verification API calls, eventually deducing user PINs, digit by digit. This kind of attack is performed exactly as an extended Mastermind game played between the insider and the bank’s computers.

Hardness of Mastermind

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A feasible heuristic

A systematic study of Mastermind was carried out by Chv´ atal, in a 1983 paper dedicated to Erd˝

  • s on

his 70th birthday. Chv´ atal suggested a simple divide-and-conquer strategy for the codebreaker to guess the code in 2n ⌈log c⌉ + 4n + c

n

  • attempts. Each guess can be computed in polynomial time.

This bound was subsequently lowered by a constant factor, with an improvement on the same basic idea.

Hardness of Mastermind

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SLIDE 29

Mastermind: a piece of cake?

Does Chv´ atal’s strategy trivialize the game?

Hardness of Mastermind

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Mastermind: a piece of cake?

Does Chv´ atal’s strategy trivialize the game? Not really, as long as the number of attempts is critical. The classic (4, 6)-Mastermind is solvable within 5 guesses, while Chv´ atal’s algorithm guesses 18 times. Playing perfectly is still hard.

Hardness of Mastermind

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Exhaustive searches

Another thread of heuristics was started in 1976 by Knuth, who devised a worst-case optimal (w.r.t. the number of guesses) greedy strategy to beat (4, 6)-Mastermind. Every step of the strategy is a brute-force search among all possible guesses and all possible responses of the codemaker. The heuristic is based on choosing the guess that will minimize the number of eligible solutions, in the worst case. This is practical and optimal for (4, 6)-Mastermind, but still infeasible and suboptimal in general. Several other approaches were adopted, most notably genetic algorithms, achieving different performance tradeoffs.

Hardness of Mastermind

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Satisfiability

Mastermind Satisfiability Problem (MSP) Input: (n, c, Q), where Q is any sequence of guesses and responses in (n, c)-Mastermind. Output: YES if there exists a code that is compatible with all the queries in Q, NO otherwise.

Hardness of Mastermind

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SLIDE 33

Satisfiability

Mastermind Satisfiability Problem (MSP) Input: (n, c, Q), where Q is any sequence of guesses and responses in (n, c)-Mastermind. Output: YES if there exists a code that is compatible with all the queries in Q, NO otherwise. In 2005 Stuckman and Zhang proved that MSP is NP-complete. In 2009 Goodrich proved the same result for a variant of Mastermind where the codemaker only responds with black pegs (with an application to genetics).

Hardness of Mastermind

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Solution uniqueness and Restricted Mastermind

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution?

Hardness of Mastermind

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Solution uniqueness and Restricted Mastermind

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? Inspired by Goodrich, we define two variants of MSP.

In MSP-BLACK the codemaker responds only with black pegs. In MSP-WHITE the codemaker responds with (b + w) white pegs whenever in MSP he would have responded (b, w). The codebreaker has to guess the code up to reordering of the pegs.

Hardness of Mastermind

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Solution uniqueness and Restricted Mastermind

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? Inspired by Goodrich, we define two variants of MSP.

In MSP-BLACK the codemaker responds only with black pegs. In MSP-WHITE the codemaker responds with (b + w) white pegs whenever in MSP he would have responded (b, w). The codebreaker has to guess the code up to reordering of the pegs.

c-MSP is always played with c colors, while n is still a variable.

Similarly for c-MSP-BLACK and c-MSP-WHITE.

Hardness of Mastermind

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SLIDE 37

Solution uniqueness and Restricted Mastermind

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? Inspired by Goodrich, we define two variants of MSP.

In MSP-BLACK the codemaker responds only with black pegs. In MSP-WHITE the codemaker responds with (b + w) white pegs whenever in MSP he would have responded (b, w). The codebreaker has to guess the code up to reordering of the pegs.

c-MSP is always played with c colors, while n is still a variable.

Similarly for c-MSP-BLACK and c-MSP-WHITE.

We will determine which of these restrictions are #P-complete.

Hardness of Mastermind

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SLIDE 38

Preliminary results

Observation In (n, c)-Mastermind restricted to white pegs, the codebreaker can guess the code after c − 1 attempts. He tries all possible colors... Although this is suboptimal. Observation #c-MSP-WHITE ∈ FP. There are only n+c−1

c−1

  • = Θ(nc−1) possible codes to check.

Observation #(c − 1)-MSP pars #c-MSP. Add the guess ccc · · · with response (0, 0). This holds also for #c-MSP-BLACK and #c-MSP-WHITE.

Hardness of Mastermind

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SLIDE 39

Hardness of Mastermind

Theorem #1-IN-3-SAT pars      #MSP-WHITE #2-MSP-BLACK #2-MSP. Hence all these variations are #P-complete.

Hardness of Mastermind

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SLIDE 40

#1-IN-3-SAT pars #MSP-WHITE

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: x, ¯ x, y, ¯ y, z, ¯ z, w, ¯ w, ⋆. Code length: 4.

Hardness of Mastermind

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SLIDE 41

#1-IN-3-SAT pars #MSP-WHITE

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: x, ¯ x, y, ¯ y, z, ¯ z, w, ¯ w, ⋆. Code length: 4. Queries:

⋆ ⋆ ⋆ ⋆ with score (0)

Hardness of Mastermind

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SLIDE 42

#1-IN-3-SAT pars #MSP-WHITE

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: x, ¯ x, y, ¯ y, z, ¯ z, w, ¯ w, ⋆. Code length: 4. Queries:

⋆ ⋆ ⋆ ⋆ with score (0) x x ¯ x ¯ x with score (1) y y ¯ y ¯ y with score (1) z z ¯ z ¯ z with score (1) ww ¯ w ¯ w with score (1)

Hardness of Mastermind

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SLIDE 43

#1-IN-3-SAT pars #MSP-WHITE

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: x, ¯ x, y, ¯ y, z, ¯ z, w, ¯ w, ⋆. Code length: 4. Queries:

⋆ ⋆ ⋆ ⋆ with score (0) x x ¯ x ¯ x with score (1) y y ¯ y ¯ y with score (1) z z ¯ z ¯ z with score (1) ww ¯ w ¯ w with score (1) x ¯ y z ⋆ with score (1) ¯ x y w ⋆ with score (1) y ¯ z ¯ w ⋆ with score (1)

Hardness of Mastermind

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SLIDE 44

#1-IN-3-SAT pars #2-MSP-BLACK

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: •, ◦. Code length: 8.

Hardness of Mastermind

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SLIDE 45

#1-IN-3-SAT pars #2-MSP-BLACK

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: •, ◦. Code length: 8. Queries: x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4)

Hardness of Mastermind

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SLIDE 46

#1-IN-3-SAT pars #2-MSP-BLACK

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: •, ◦. Code length: 8. Queries: x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4)

x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4)
  • (4)
  • (4)
  • (4)

Hardness of Mastermind

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SLIDE 47

#1-IN-3-SAT pars #2-MSP-BLACK

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: •, ◦. Code length: 8. Queries: x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4)

x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4)
  • (4)
  • (4)
  • (4)

x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (3)
  • (3)
  • (3)

Hardness of Mastermind

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SLIDE 48

#1-IN-3-SAT pars #2-MSP

Let ϕ = (x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w). Colors: •, ◦. Code length: 8. Queries: x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4, 0)

x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (4, 2)
  • (4, 2)
  • (4, 2)
  • (4, 2)

x ¯ x y ¯ y z ¯ z w ¯ w Score

  • (3, 4)
  • (3, 4)
  • (3, 4)

Hardness of Mastermind

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SLIDE 49

Solutions

(x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ w) ∧ (y ∨ ¬z ∨ ¬w) x y z w T T T T T T T F T T F T T T F F T F F T F T T T F T T F F F T F F F F T F F F F x x y y z z w w

  • Hardness of Mastermind
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SLIDE 50

Corollaries

In a real game of Mastermind we would know that our queries are satisfiable. Can we use this information to compute the size of the solution space?

In general, is it easier to compute the number of solutions, knowing that they are at least k?

Hardness of Mastermind

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SLIDE 51

Corollaries

In a real game of Mastermind we would know that our queries are satisfiable. Can we use this information to compute the size of the solution space?

In general, is it easier to compute the number of solutions, knowing that they are at least k?

Let #MATCH be the problem of counting all the matchings (perfect and imperfect) in a given graph. Lemma (Valiant, 1979) #MATCH is #P-complete under Turing reductions.

Hardness of Mastermind

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SLIDE 52

Corollaries

In a real game of Mastermind we would know that our queries are satisfiable. Can we use this information to compute the size of the solution space?

In general, is it easier to compute the number of solutions, knowing that they are at least k?

Let #MATCH be the problem of counting all the matchings (perfect and imperfect) in a given graph. Lemma (Valiant, 1979) #MATCH is #P-complete under Turing reductions. Then #SAT T #MATCH pars #MSP. The graphs with fewer than k edges are solved by hand; the others (which have at least k matchings) are mapped to #MSP. Corollary #MSP restricted to instances with at least k solutions is #P-hard.

Hardness of Mastermind

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SLIDE 53

Corollaries

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution?

Hardness of Mastermind

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SLIDE 54

Corollaries

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? If we are given k 1 solutions, can we tell if there are more?

Hardness of Mastermind

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SLIDE 55

Corollaries

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? If we are given k 1 solutions, can we tell if there are more? Corollary No, it is NP-complete. Not only solving Mastermind puzzles is hard, but designing puzzles around a solution is hard.

Hardness of Mastermind

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SLIDE 56

Corollaries

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? If we are given k 1 solutions, can we tell if there are more? Corollary No, it is NP-complete. Not only solving Mastermind puzzles is hard, but designing puzzles around a solution is hard. What if we know that the solution is unique? Can we find it?

Hardness of Mastermind

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SLIDE 57

Corollaries

Problem (Stuckman–Zhang, 2005) Can we detect MSP instances with a unique solution? If we are given k 1 solutions, can we tell if there are more? Corollary No, it is NP-complete. Not only solving Mastermind puzzles is hard, but designing puzzles around a solution is hard. What if we know that the solution is unique? Can we find it? Corollary No, it is NP-hard under randomized Turing reductions.

Hardness of Mastermind

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SLIDE 58

Open problems

#c-MSP-WHITE ∈ FP when c is a constant. # k √n -MSP-WHITE is #P-complete for every k 1. Problem What is the lowest order of growth of c(n) such that #c(n)-MSP-WHITE is #P-complete?

Hardness of Mastermind

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SLIDE 59

Open problems

Solving MSP is a sub-step of several heuristics.

But is it really necessary?

MASTERMIND Input: (n, c, Q, k). Output: YES if the codebreaker has a strategy to guess the code within k attempts, given the set of queries Q. NO otherwise.

Hardness of Mastermind

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SLIDE 60

Open problems

Solving MSP is a sub-step of several heuristics.

But is it really necessary?

MASTERMIND Input: (n, c, Q, k). Output: YES if the codebreaker has a strategy to guess the code within k attempts, given the set of queries Q. NO otherwise. MASTERMIND ∈ PSPACE, due to Chv´ atal’s strategy. Problem Is MASTERMIND PSPACE-complete?

Hardness of Mastermind