31 research outputs found
Tropical Effective Primary and Dual Nullstellens\"atze
Tropical algebra is an emerging field with a number of applications in
various areas of mathematics. In many of these applications appeal to tropical
polynomials allows to study properties of mathematical objects such as
algebraic varieties and algebraic curves from the computational point of view.
This makes it important to study both mathematical and computational aspects of
tropical polynomials.
In this paper we prove a tropical Nullstellensatz and moreover we show an
effective formulation of this theorem. Nullstellensatz is a natural step in
building algebraic theory of tropical polynomials and its effective version is
relevant for computational aspects of this field.
On our way we establish a simple formulation of min-plus and tropical linear
dualities. We also observe a close connection between tropical and min-plus
polynomial systems
Tree-like Queries in OWL 2 QL: Succinctness and Complexity Results
This paper investigates the impact of query topology on the difficulty of
answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first
contribution is to clarify the worst-case size of positive existential (PE),
non-recursive Datalog (NDL), and first-order (FO) rewritings for various
classes of tree-like conjunctive queries, ranging from linear queries to
bounded treewidth queries. Perhaps our most surprising result is a
superpolynomial lower bound on the size of PE-rewritings that holds already for
linear queries and ontologies of depth 2. More positively, we show that
polynomial-size NDL-rewritings always exist for tree-shaped queries with a
bounded number of leaves (and arbitrary ontologies), and for bounded treewidth
queries paired with bounded depth ontologies. For FO-rewritings, we equate the
existence of polysize rewritings with well-known problems in Boolean circuit
complexity. As our second contribution, we analyze the computational complexity
of query answering and establish tractability results (either NL- or
LOGCFL-completeness) for a range of query-ontology pairs. Combining our new
results with those from the literature yields a complete picture of the
succinctness and complexity landscapes for the considered classes of queries
and ontologies.Comment: This is an extended version of a paper accepted at LICS'15. It
contains both succinctness and complexity results and adopts FOL notation.
The appendix contains proofs that had to be omitted from the conference
version for lack of space. The previous arxiv version (a long version of our
DL'14 workshop paper) only contained the succinctness results and used
description logic notatio
Multiparty Karchmer - Wigderson Games and Threshold Circuits
We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013)
On the Succinctness of Query Rewriting over OWL 2 QL Ontologies with Shallow Chases
We investigate the size of first-order rewritings of conjunctive queries over
OWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computing
Boolean functions. Both positive and negative results are obtained. Conjunctive
queries over ontologies of depth 1 have polynomial-size nonrecursive datalog
rewritings; tree-shaped queries have polynomial positive existential
rewritings; however, in the worst case, positive existential rewritings can
only be of superpolynomial size. Positive existential and nonrecursive datalog
rewritings of queries over ontologies of depth 2 suffer an exponential blowup
in the worst case, while first-order rewritings are superpolynomial unless
. We also analyse rewritings of
tree-shaped queries over arbitrary ontologies and observe that the query
entailment problem for such queries is fixed-parameter tractable
Exponential Lower Bounds and Separation for Query Rewriting
We establish connections between the size of circuits and formulas computing
monotone Boolean functions and the size of first-order and nonrecursive Datalog
rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower
bounds and separation results from circuit complexity to prove similar results
for the size of rewritings that do not use non-signature constants. For
example, we show that, in the worst case, positive existential and nonrecursive
Datalog rewritings are exponentially longer than the original queries;
nonrecursive Datalog rewritings are in general exponentially more succinct than
positive existential rewritings; while first-order rewritings can be
superpolynomially more succinct than positive existential rewritings
One-Way Communication Complexity of Partial XOR Functions
Boolean function for is an XOR function if
for some function on input bits, where
is a bit-wise XOR. XOR functions are relevant in communication complexity,
partially for allowing Fourier analytic technique. For total XOR functions it
is known that deterministic communication complexity of is closely related
to parity decision tree complexity of . Montanaro and Osbourne (2009)
observed that one-sided communication complexity of
is exactly equal to nonadaptive parity decision tree complexity
of . Hatami et al. (2018) showed that unrestricted
communication complexity of is polynomially related to parity decision tree
complexity of .
We initiate the studies of a similar connection for partial functions. We
show that in case of one-sided communication complexity whether these measures
are equal, depends on the number of undefined inputs of . On the one hand,
if and is undefined on at most
, then .
On the other hand, for a wide range of values of
and (from constant to ) we provide partial functions
for which . In particular, we
provide a function with an exponential gap between the two measures. Our
separation results translate to the case of two-sided communication complexity
as well, in particular showing that the result of Hatami et al. (2018) cannot
be generalized to partial functions.
Previous results for total functions heavily rely on Boolean Fourier analysis
and the technique does not translate to partial functions. For the proofs of
our results we build a linear algebraic framework instead. Separation results
are proved through the reduction to covering codes