43 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
Query rewriting over shallow ontologies
We investigate the size of rewritings of conjunctive queries over OWL2QL ontologies of depth 1 and 2 by means of a new hypergraph formalism for computing Boolean functions. Both positive and negative results are obtained. All conjunctive queries over ontologies of depth 1 have polynomial-size nonrecursive datalog rewritings; tree-shaped queries have polynomial-size positive existential rewritings; however, for some queries and ontologies of depth 1, positive existential rewritings can only be of superpolynomial size. Both positive existential and nonrecursive datalog rewritings of conjunctive queries and ontologies of depth 2 suffer an exponential blowup in the worst case, while first-order rewritings can grow superpolynomially unless NP is included in� P/poly
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
Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates
We study the following computational problem: for which values of k, the majority of n bits MAJ_n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ_k o MAJ_k. We observe that the minimum value of k for which there exists a MAJ_k o MAJ_k circuit that has high correlation with the majority of n bits is equal to Theta(sqrt(n)). We then show that for a randomized MAJ_k o MAJ_k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n^(2/3+o(1)). We show a worst case lower bound: if a MAJ_k o MAJ_k circuit computes the majority of n bits correctly on all inputs, then k <= n^(13/19+o(1)). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^(2/3)) can compute MAJ_n correctly on all inputs
Patience of Matrix Games
For matrix games we study how small nonzero probability must be used in
optimal strategies. We show that for nxn win-lose-draw games (i.e. (-1,0,1)
matrix games) nonzero probabilities smaller than n^{-O(n)} are never needed. We
also construct an explicit nxn win-lose game such that the unique optimal
strategy uses a nonzero probability as small as n^{-Omega(n)}. This is done by
constructing an explicit (-1,1) nonsingular nxn matrix, for which the inverse
has only nonnegative entries and where some of the entries are of value
n^{Omega(n)}