20,122 research outputs found
Unary Pushdown Automata and Straight-Line Programs
We consider decision problems for deterministic pushdown automata over a
unary alphabet (udpda, for short). Udpda are a simple computation model that
accept exactly the unary regular languages, but can be exponentially more
succinct than finite-state automata. We complete the complexity landscape for
udpda by showing that emptiness (and thus universality) is P-hard, equivalence
and compressed membership problems are P-complete, and inclusion is
coNP-complete. Our upper bounds are based on a translation theorem between
udpda and straight-line programs over the binary alphabet (SLPs). We show that
the characteristic sequence of any udpda can be represented as a pair of
SLPs---one for the prefix, one for the lasso---that have size linear in the
size of the udpda and can be computed in polynomial time. Hence, decision
problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP
can be converted in logarithmic space into a udpda, and this forms the basis
for our lower bound proofs. We show coNP-hardness of the ordered matching
problem for SLPs, from which we derive coNP-hardness for inclusion. In
addition, we complete the complexity landscape for unary nondeterministic
pushdown automata by showing that the universality problem is -hard, using a new class of integer expressions. Our techniques have
applications beyond udpda. We show that our results imply -completeness for a natural fragment of Presburger arithmetic and coNP lower
bounds for compressed matching problems with one-character wildcards
Minimisation of Multiplicity Tree Automata
We consider the problem of minimising the number of states in a multiplicity
tree automaton over the field of rational numbers. We give a minimisation
algorithm that runs in polynomial time assuming unit-cost arithmetic. We also
show that a polynomial bound in the standard Turing model would require a
breakthrough in the complexity of polynomial identity testing by proving that
the latter problem is logspace equivalent to the decision version of
minimisation. The developed techniques also improve the state of the art in
multiplicity word automata: we give an NC algorithm for minimising multiplicity
word automata. Finally, we consider the minimal consistency problem: does there
exist an automaton with states that is consistent with a given finite
sample of weight-labelled words or trees? We show that this decision problem is
complete for the existential theory of the rationals, both for words and for
trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor
editing changes from previous versio
Implications of quantum automata for contextuality
We construct zero-error quantum finite automata (QFAs) for promise problems
which cannot be solved by bounded-error probabilistic finite automata (PFAs).
Here is a summary of our results:
- There is a promise problem solvable by an exact two-way QFA in exponential
expected time, but not by any bounded-error sublogarithmic space probabilistic
Turing machine (PTM).
- There is a promise problem solvable by an exact two-way QFA in quadratic
expected time, but not by any bounded-error -space PTMs in
polynomial expected time. The same problem can be solvable by a one-way Las
Vegas (or exact two-way) QFA with quantum head in linear (expected) time.
- There is a promise problem solvable by a Las Vegas realtime QFA, but not by
any bounded-error realtime PFA. The same problem can be solvable by an exact
two-way QFA in linear expected time but not by any exact two-way PFA.
- There is a family of promise problems such that each promise problem can be
solvable by a two-state exact realtime QFAs, but, there is no such bound on the
number of states of realtime bounded-error PFAs solving the members this
family.
Our results imply that there exist zero-error quantum computational devices
with a \emph{single qubit} of memory that cannot be simulated by any finite
memory classical computational model. This provides a computational perspective
on results regarding ontological theories of quantum mechanics \cite{Hardy04},
\cite{Montina08}. As a consequence we find that classical automata based
simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently
powerful to simulate quantum contextuality. We conclude by highlighting the
interplay between results from automata models and their application to
developing a general framework for quantum contextuality.Comment: 22 page
Limit Synchronization in Markov Decision Processes
Markov decision processes (MDP) are finite-state systems with both strategic
and probabilistic choices. After fixing a strategy, an MDP produces a sequence
of probability distributions over states. The sequence is eventually
synchronizing if the probability mass accumulates in a single state, possibly
in the limit. Precisely, for 0 <= p <= 1 the sequence is p-synchronizing if a
probability distribution in the sequence assigns probability at least p to some
state, and we distinguish three synchronization modes: (i) sure winning if
there exists a strategy that produces a 1-synchronizing sequence; (ii)
almost-sure winning if there exists a strategy that produces a sequence that
is, for all epsilon > 0, a (1-epsilon)-synchronizing sequence; (iii) limit-sure
winning if for all epsilon > 0, there exists a strategy that produces a
(1-epsilon)-synchronizing sequence.
We consider the problem of deciding whether an MDP is sure, almost-sure,
limit-sure winning, and we establish the decidability and optimal complexity
for all modes, as well as the memory requirements for winning strategies. Our
main contributions are as follows: (a) for each winning modes we present
characterizations that give a PSPACE complexity for the decision problems, and
we establish matching PSPACE lower bounds; (b) we show that for sure winning
strategies, exponential memory is sufficient and may be necessary, and that in
general infinite memory is necessary for almost-sure winning, and unbounded
memory is necessary for limit-sure winning; (c) along with our results, we
establish new complexity results for alternating finite automata over a
one-letter alphabet
Computational Processes and Incompleteness
We introduce a formal definition of Wolfram's notion of computational process
based on cellular automata, a physics-like model of computation. There is a
natural classification of these processes into decidable, intermediate and
complete. It is shown that in the context of standard finite injury priority
arguments one cannot establish the existence of an intermediate computational
process
The Complexity of POMDPs with Long-run Average Objectives
We study the problem of approximation of optimal values in
partially-observable Markov decision processes (POMDPs) with long-run average
objectives. POMDPs are a standard model for dynamic systems with probabilistic
and nondeterministic behavior in uncertain environments. In long-run average
objectives rewards are associated with every transition of the POMDP and the
payoff is the long-run average of the rewards along the executions of the
POMDP. We establish strategy complexity and computational complexity results.
Our main result shows that finite-memory strategies suffice for approximation
of optimal values, and the related decision problem is recursively enumerable
complete
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
Preimage problems for deterministic finite automata
Given a subset of states of a deterministic finite automaton and a word
, the preimage is the subset of all states mapped to a state in by the
action of . We study three natural problems concerning words giving certain
preimages. The first problem is whether, for a given subset, there exists a
word \emph{extending} the subset (giving a larger preimage). The second problem
is whether there exists a \emph{totally extending} word (giving the whole set
of states as a preimage)---equivalently, whether there exists an
\emph{avoiding} word for the complementary subset. The third problem is whether
there exists a \emph{resizing} word. We also consider variants where the length
of the word is upper bounded, where the size of the given subset is restricted,
and where the automaton is strongly connected, synchronizing, or binary. We
conclude with a summary of the complexities in all combinations of the cases
- …