140 research outputs found
2048 is (PSPACE) Hard, but Sometimes Easy
We prove that a variant of 2048, a popular online puzzle game, is
PSPACE-Complete. Our hardness result holds for a version of the problem where
the player has oracle access to the computer player's moves. Specifically, we
show that for an game board , computing a sequence of
moves to reach a particular configuration from an initial
configuration is PSPACE-Complete. Our reduction is from
Nondeterministic Constraint Logic (NCL). We also show that determining whether
or not there exists a fixed sequence of moves of length that results in a
winning configuration for an game board is fixed-parameter
tractable (FPT). We describe an algorithm to solve this problem in
time.Comment: 13 pages, 11 figure
The subpower membership problem for semigroups
Fix a finite semigroup and let be tuples in a direct
power . The subpower membership problem (SMP) asks whether can be
generated by . If is a finite group, then there is a
folklore algorithm that decides this problem in time polynomial in . For
semigroups this problem always lies in PSPACE. We show that the SMP for a full
transformation semigroup on 3 letters or more is actually PSPACE-complete,
while on 2 letters it is in P. For commutative semigroups, we provide a
dichotomy result: if a commutative semigroup embeds into a direct product
of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P;
otherwise it is NP-complete
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Trading Determinism for Time in Space Bounded Computations
Savitch showed in that nondeterministic logspace (NL) is contained in
deterministic space but his algorithm requires
quasipolynomial time. The question whether we can have a deterministic
algorithm for every problem in NL that requires polylogarithmic space and
simultaneously runs in polynomial time was left open.
In this paper we give a partial solution to this problem and show that for
every language in NL there exists an unambiguous nondeterministic algorithm
that requires space and simultaneously runs in
polynomial time.Comment: Accepted in MFCS 201
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
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