ECCC-Report TR16-094https://eccc.weizmann.ac.il/report/2016/094Comments and Revisions published for TR16-094en-usMon, 18 Jul 2016 09:04:22 +0300
Comment 1
| Non-commutative computations: lower bounds and polynomial identity testing |
Guillaume Lagarde,
Guillaume Malod
https://eccc.weizmann.ac.il/report/2016/094#comment1In the setting of non-commutative arithmetic computations, we define a class of circuits that gener-
alize algebraic branching programs (ABP). This model is called unambiguous because it captures the
polynomials in which all monomials are computed in a similar way (that is, all the parse trees are iso-
morphic).
We show that unambiguous circuits of polynomial size can compute polynomials that require ABPs
of exponential size, and that they are incomparable with skew circuits.
Generalizing a result of Nisan [17] on ABPs, we provide an exact characterization of the complexity
of any polynomial in our model, and use it to prove exponential lower bounds for explicit polynomials
such as the determinant.
Finally, we give a deterministic polynomial-time algorithm for polynomial identity testing (PIT) on unambiguous circuits over $\mathbb{R}$ and $\mathbb{C}$, thus providing the largest class of circuits so far in a non-commutative setting for which we can derandomize PIT.Mon, 18 Jul 2016 09:04:22 +0300https://eccc.weizmann.ac.il/report/2016/094#comment1
Paper TR16-094
| Non-commutative computations: lower bounds and polynomial identity testing |
Guillaume Lagarde,
Guillaume Malod
https://eccc.weizmann.ac.il/report/2016/094In the setting of non-commutative arithmetic computations, we define a class of circuits that gener-
alize algebraic branching programs (ABP). This model is called unambiguous because it captures the
polynomials in which all monomials are computed in a similar way (that is, all the parse trees are iso-
morphic).
We show that unambiguous circuits of polynomial size can compute polynomials that require ABPs
of exponential size, and that they are incomparable with skew circuits.
Generalizing a result of Nisan [17] on ABPs, we provide an exact characterization of the complexity
of any polynomial in our model, and use it to prove exponential lower bounds for explicit polynomials
such as the determinant.
Finally, we give a deterministic polynomial-time algorithm for polynomial identity testing (PIT) on unambiguous circuits over $\mathbb{R}$ and $\mathbb{C}$, thus providing the largest class of circuits so far in a non-commutative setting for which we can derandomize PIT.Mon, 06 Jun 2016 18:12:07 +0300https://eccc.weizmann.ac.il/report/2016/094