366 research outputs found
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
On the complexity of solving linear congruences and computing nullspaces modulo a constant
We consider the problems of determining the feasibility of a linear
congruence, producing a solution to a linear congruence, and finding a spanning
set for the nullspace of an integer matrix, where each problem is considered
modulo an arbitrary constant k>1. These problems are known to be complete for
the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case
that k is prime (Buntrock et al, 1992). By considering variants of standard
logspace function classes --- related to #L and functions computable by UL
machines, but which only characterize the number of accepting paths modulo k
--- we show that these problems of linear algebra are also complete for
{coMod_k L} for any constant k>1.
Our results are obtained by defining a class of functions FUL_k which are low
for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the
case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1
reductions (including {Mod_k L} oracle reductions). In addition to the results
above, we briefly consider the relationship of the class FUL_k for arbitrary
moduli k to the class {F.coMod_k L} of functions whose output symbols are
verifiable by {coMod_k L} algorithms; and consider what consequences such a
comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L}
= {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to
presentation, new observations regarding the prospect of oracle closures.
Comments welcom
The descriptive complexity approach to LOGCFL
Building upon the known generalized-quantifier-based first-order
characterization of LOGCFL, we lay the groundwork for a deeper investigation.
Specifically, we examine subclasses of LOGCFL arising from varying the arity
and nesting of groupoidal quantifiers. Our work extends the elaborate theory
relating monoidal quantifiers to NC1 and its subclasses. In the absence of the
BIT predicate, we resolve the main issues: we show in particular that no single
outermost unary groupoidal quantifier with FO can capture all the context-free
languages, and we obtain the surprising result that a variant of Greibach's
``hardest context-free language'' is LOGCFL-complete under quantifier-free
BIT-free projections. We then prove that FO with unary groupoidal quantifiers
is strictly more expressive with the BIT predicate than without. Considering a
particular groupoidal quantifier, we prove that first-order logic with majority
of pairs is strictly more expressive than first-order with majority of
individuals. As a technical tool of independent interest, we define the notion
of an aperiodic nondeterministic finite automaton and prove that FO
translations are precisely the mappings computed by single-valued aperiodic
nondeterministic finite transducers.Comment: 10 pages, 1 figur
Completeness Results for Parameterized Space Classes
The parameterized complexity of a problem is considered "settled" once it has
been shown to lie in FPT or to be complete for a class in the W-hierarchy or a
similar parameterized hierarchy. Several natural parameterized problems have,
however, resisted such a classification. At least in some cases, the reason is
that upper and lower bounds for their parameterized space complexity have
recently been obtained that rule out completeness results for parameterized
time classes. In this paper, we make progress in this direction by proving that
the associative generability problem and the longest common subsequence problem
are complete for parameterized space classes. These classes are defined in
terms of different forms of bounded nondeterminism and in terms of simultaneous
time--space bounds. As a technical tool we introduce a "union operation" that
translates between problems complete for classical complexity classes and for
W-classes.Comment: IPEC 201
Branching-time model checking of one-counter processes
One-counter processes (OCPs) are pushdown processes which operate only on a
unary stack alphabet. We study the computational complexity of model checking
computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from
the modal mu-calculus for this problem. First, we analyze the periodic
behaviour of CTL over OCPs and derive a model checking algorithm whose running
time is exponential only in the number of control locations and a syntactic
notion of the formula that we call leftward until depth. Thus, model checking
fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This
generalizes a result of the first author, Mayr, and To for the expression
complexity of CTL's fragment EF. Second, we prove that already over some fixed
OCP, CTL model checking is PSPACE-hard. Third, we show that there already
exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To
obtain the latter result, we employ two results from complexity theory: (i)
Converting a natural number in Chinese remainder presentation into binary
presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable.
We demonstrate that our approach can be used to obtain further results. We show
that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus
establishing a matching lower bound and answering an open question of the first
author, Mayr, and To. We moreover show that the following problem is hard for
PSPACE: Given a one-counter Markov decision process, a set of target states
with counter value zero each, and an initial state, to decide whether the
probability that the initial state will eventually reach one of the target
states is arbitrarily close to 1. This improves a previously known lower bound
for every level of the Boolean hierarchy by Brazdil et al
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
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