13,053 research outputs found
Deterministic Automata for Unordered Trees
Automata for unordered unranked trees are relevant for defining schemas and
queries for data trees in Json or Xml format. While the existing notions are
well-investigated concerning expressiveness, they all lack a proper notion of
determinism, which makes it difficult to distinguish subclasses of automata for
which problems such as inclusion, equivalence, and minimization can be solved
efficiently. In this paper, we propose and investigate different notions of
"horizontal determinism", starting from automata for unranked trees in which
the horizontal evaluation is performed by finite state automata. We show that a
restriction to confluent horizontal evaluation leads to polynomial-time
emptiness and universality, but still suffers from coNP-completeness of the
emptiness of binary intersections. Finally, efficient algorithms can be
obtained by imposing an order of horizontal evaluation globally for all
automata in the class. Depending on the choice of the order, we obtain
different classes of automata, each of which has the same expressiveness as
CMso.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Complexity of Problems of Commutative Grammars
We consider commutative regular and context-free grammars, or, in other
words, Parikh images of regular and context-free languages. By using linear
algebra and a branching analog of the classic Euler theorem, we show that,
under an assumption that the terminal alphabet is fixed, the membership problem
for regular grammars (given v in binary and a regular commutative grammar G,
does G generate v?) is P, and that the equivalence problem for context free
grammars (do G_1 and G_2 generate the same language?) is in
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
A complex analogue of Toda's Theorem
Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time
hierarchy, , is contained in the class \mathbf{P}^{#\mathbf{P}},
namely the class of languages that can be decided by a Turing machine in
polynomial time given access to an oracle with the power to compute a function
in the counting complexity class #\mathbf{P}. This result, which illustrates
the power of counting is considered to be a seminal result in computational
complexity theory. An analogous result (with a compactness hypothesis) in the
complexity theory over the reals (in the sense of Blum-Shub-Smale real machines
\cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete
case, which relied on sophisticated combinatorial arguments, the proof in
\cite{BZ09} is topological in nature in which the properties of the topological
join is used in a fundamental way. However, the constructions used in
\cite{BZ09} were semi-algebraic -- they used real inequalities in an essential
way and as such do not extend to the complex case. In this paper, we extend the
techniques developed in \cite{BZ09} to the complex projective case. A key role
is played by the complex join of quasi-projective complex varieties. As a
consequence we obtain a complex analogue of Toda's theorem. The results
contained in this paper, taken together with those contained in \cite{BZ09},
illustrate the central role of the Poincar\'e polynomial in algorithmic
algebraic geometry, as well as, in computational complexity theory over the
complex and real numbers -- namely, the ability to compute it efficiently
enables one to decide in polynomial time all languages in the (compact)
polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational
Mathematic
Algebraic properties of structured context-free languages: old approaches and novel developments
The historical research line on the algebraic properties of structured CF
languages initiated by McNaughton's Parenthesis Languages has recently
attracted much renewed interest with the Balanced Languages, the Visibly
Pushdown Automata languages (VPDA), the Synchronized Languages, and the
Height-deterministic ones. Such families preserve to a varying degree the basic
algebraic properties of Regular languages: boolean closure, closure under
reversal, under concatenation, and Kleene star. We prove that the VPDA family
is strictly contained within the Floyd Grammars (FG) family historically known
as operator precedence. Languages over the same precedence matrix are known to
be closed under boolean operations, and are recognized by a machine whose pop
or push operations on the stack are purely determined by terminal letters. We
characterize VPDA's as the subclass of FG having a peculiarly structured set of
precedence relations, and balanced grammars as a further restricted case. The
non-counting invariance property of FG has a direct implication for VPDA too.Comment: Extended version of paper presented at WORDS2009, Salerno,Italy,
September 200
Automata for Unordered Trees
International audienceWe present a framework for defining automata for unordereddata trees that is parametrized by the way in which multisets of children nodes are described. Presburger tree automata and alternatingPresburger tree automata are particular instances. We establish the usual equivalence in expressiveness of tree automata and MSO for the automata defined inour framework.We then investigate subclasses of automata for unordered treesfor which testing language equivalence is in P-time. For this we start from automata in our framework that describe multisets of childrenby finite automata, and propose two approaches of how todo this deterministically. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers fromcoNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending onthe choice of the order, we obtain different classes of automata, eachof which has the same expressiveness as Counting MSO
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