1,601 research outputs found
Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting
In this paper we study lifted inference for the Weighted First-Order Model
Counting problem (WFOMC), which counts the assignments that satisfy a given
sentence in first-order logic (FOL); it has applications in Statistical
Relational Learning (SRL) and Probabilistic Databases (PDB). We present several
results. First, we describe a lifted inference algorithm that generalizes prior
approaches in SRL and PDB. Second, we provide a novel dichotomy result for a
non-trivial fragment of FO CNF sentences, showing that for each sentence the
WFOMC problem is either in PTIME or #P-hard in the size of the input domain; we
prove that, in the first case our algorithm solves the WFOMC problem in PTIME,
and in the second case it fails. Third, we present several properties of the
algorithm. Finally, we discuss limitations of lifted inference for symmetric
probabilistic databases (where the weights of ground literals depend only on
the relation name, and not on the constants of the domain), and prove the
impossibility of a dichotomy result for the complexity of probabilistic
inference for the entire language FOL
Algebraic Independence and Blackbox Identity Testing
Algebraic independence is an advanced notion in commutative algebra that
generalizes independence of linear polynomials to higher degree. Polynomials
{f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent
if there is no non-zero polynomial F such that F(f_1, ..., f_m) = 0. The
transcendence degree, trdeg{f_1, ..., f_m}, is the maximal number r of
algebraically independent polynomials in the set. In this paper we design
blackbox and efficient linear maps \phi that reduce the number of variables
from n to r but maintain trdeg{\phi(f_i)}_i = r, assuming f_i's sparse and
small r. We apply these fundamental maps to solve several cases of blackbox
identity testing:
(1) Given a polynomial-degree circuit C and sparse polynomials f_1, ..., f_m
with trdeg r, we can test blackbox D := C(f_1, ..., f_m) for zeroness in
poly(size(D))^r time.
(2) Define a spsp_\delta(k,s,n) circuit C to be of the form \sum_{i=1}^k
\prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree
at most \delta. For k = 2 we give a poly(sn\delta)^{\delta^2} time blackbox
identity test.
(3) For a general depth-4 circuit we define a notion of rank. Assuming there
is a rank bound R for minimal simple spsp_\delta(k,s,n) identities, we give a
poly(snR\delta)^{Rk\delta^2} time blackbox identity test for spsp_\delta(k,s,n)
circuits. This partially generalizes the state of the art of depth-3 to depth-4
circuits.
The notion of trdeg works best with large or zero characteristic, but we also
give versions of our results for arbitrary fields.Comment: 32 pages, preliminary versio
On Sequences, Rational Functions and Decomposition
Our overall goal is to unify and extend some results in the literature
related to the approximation of generating functions of finite and infinite
sequences over a field by rational functions. In our approach, numerators play
a significant role. We revisit a theorem of Niederreiter on (i) linear
complexities and (ii) ' minimal polynomials' of an infinite sequence,
proved using partial quotients. We prove (i) and its converse from first
principles and generalise (ii) to rational functions where the denominator need
not have minimal degree. We prove (ii) in two parts: firstly for geometric
sequences and then for sequences with a jump in linear complexity. The basic
idea is to decompose the denominator as a sum of polynomial multiples of two
polynomials of minimal degree; there is a similar decomposition for the
numerators. The decomposition is unique when the denominator has degree at most
the length of the sequence. The proof also applies to rational functions
related to finite sequences, generalising a result of Massey. We give a number
of applications to rational functions associated to sequences.Comment: Several more typos corrected. To appear in J. Applied Algebra in
Engineering, Communication and Computing. The final publication version is
available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
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