24 research outputs found
Deterministic Black-Box Identity Testing -Ordered Algebraic Branching Programs
In this paper we study algebraic branching programs (ABPs) with restrictions
on the order and the number of reads of variables in the program. Given a
permutation of variables, for a -ordered ABP (-OABP), for
any directed path from source to sink, a variable can appear at most once
on , and the order in which variables appear on must respect . An
ABP is said to be of read , if any variable appears at most times in
. Our main result pertains to the identity testing problem. Over any field
and in the black-box model, i.e. given only query access to the polynomial,
we have the following result: read -OABP computable polynomials can be
tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}].
Our next set of results investigates the computational limitations of OABPs.
It is shown that any OABP computing the determinant or permanent requires size
and read . We give a multilinear polynomial
in variables over some specifically selected field , such that
any OABP computing must read some variable at least times. We show
that the elementary symmetric polynomial of degree in variables can be
computed by a size read OABP, but not by a read OABP, for
any . Finally, we give an example of a polynomial and two
variables orders , such that can be computed by a read-once
-OABP, but where any -OABP computing must read some variable at
least $2^n
Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees
We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions:
- An explicit hitting set of quasipolynomial size for UPT circuits,
- An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes),
- An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant.
The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits.
The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016]
Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time
Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of
noncommutative formulas with inverse gates. They introduced the Rational
Identity Testing (RIT) problem which is to decide whether a noncommutative
rational formula computes zero in the free skew field. In the white-box
setting, deterministic polynomial-time algorithms are known for this problem
following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and
Ivanyos, Qiao, and Subrahmanyam (2018).
A central open problem in this area is to design efficient deterministic
black-box identity testing algorithm for rational formulas. In this paper, we
solve this problem for the first nested inverse case. More precisely, we obtain
a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative
rational formulas of inversion height two via a hitting set construction.
Several new technical ideas are involved in the hitting set construction,
including key concepts from matrix coefficient realization theory
(Vol\v{c}i\v{c}, 2018) and properties of cyclic division algebra (Lam, 2001).
En route to the proof, an important step is to embed the hitting set of Forbes
and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra
of small index