73 research outputs found
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]
Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs
Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP
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
Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic
branching program (ABP) where each variable occurs in at most one layer. We
give the first polynomial time whitebox identity test for a polynomial computed
by a sum of constantly many ROABPs. We also give a corresponding blackbox
algorithm with quasi-polynomial time complexity . In both the
cases, our time complexity is double exponential in the number of ROABPs.
ROABPs are a generalization of set-multilinear depth- circuits. The prior
results for the sum of constantly many set-multilinear depth- circuits were
only slightly better than brute-force, i.e. exponential-time.
Our techniques are a new interplay of three concepts for ROABP: low
evaluation dimension, basis isolating weight assignment and low-support rank
concentration. We relate basis isolation to rank concentration and extend it to
a sum of two ROABPs using evaluation dimension (or partial derivatives).Comment: 22 pages, Computational Complexity Conference, 201
Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time
Rational Identity Testing (RIT) is the decision problem of determining
whether or not a noncommutative rational formula computes zero in the free skew
field. It admits a deterministic polynomial-time white-box algorithm [Garg,
Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018);
Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen
and Makam (2017)] in the black-box setting, via singularity testing of linear
matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in
the white-box setting follows from the result of Derksen and Makam (2017).
Designing an efficient deterministic black-box algorithm for RIT and
understanding the parallel complexity of RIT are major open problems in this
area. Despite being open since the work of Garg, Gurvits, Oliveira, and
Wigderson (2016), these questions have seen limited progress. In fact, the only
known result in this direction is the construction of a quasipolynomial-size
hitting set for rational formulas of only inversion height two [Arvind,
Chatterjee, Mukhopadhyay (2022)].
In this paper, we significantly improve the black-box complexity of this
problem and obtain the first quasipolynomial-size hitting set for all rational
formulas of polynomial size. Our construction also yields the first
deterministic quasi-NC upper bound for RIT in the white-box setting.Comment: A white-box quasi-NC RIT algorithm has been adde
Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas
In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015
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