Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hardness of Mastermind   Mastermind is a popular board game released in 1971, where a codemaker
chooses a secret pattern of colored pegs, and a codebreaker has to guess it in
several trials. After each attempt, the codebreaker gets a response from the
codemaker containing some information on the number of correctly guessed pegs.
The search space is thus reduced at each turn, and the game continues until the
codebreaker is able to find the correct code, or runs out of trials.
  In this paper we study several variations of #MSP, the problem of computing
the size of the search space resulting from a given (possibly fictitious)
sequence of guesses and responses. Our main contribution is a proof of the
#P-completeness of #MSP under parsimonious reductions, which settles an open
problem posed by Stuckman and Zhang in 2005, concerning the complexity of
deciding if the secret code is uniquely determined by the previous guesses and
responses. Similarly, #MSP stays #P-complete under Turing reductions even with
the promise that the search space has at least k elements, for any constant k.
(In a regular game of Mastermind, k=1.)
  All our hardness results hold even in the most restrictive setting, in which
there are only two available peg colors, and also if the codemaker's responses
contain less information, for instance like in the so-called single-count
(black peg) Mastermind variation.

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The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
2
The categories of this paper are: math
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an