Non-commutative computations: lower bounds and polynomial identity testing

In the setting of non-commutative arithmetic computations, we define a class of circuits that generalize 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 isomorphic). 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 R and C, thus providing the largest class of circuits so far in a noncommutative setting for which we can derandomize PIT. ∗Univ Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089 CNRS, F-75205 Paris, France. Email: guillaume.lagarde@liafa.univ-paris-diderot.fr. †Univ Paris Diderot, Sorbonne Paris Cité, IMJ-PRG, UMR 7586 CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France. Email: malod@math.univ-paris-diderot.fr. ‡Univ Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089 CNRS, F-75205 Paris, France. Email: sylvain.perifel@liafa.univ-paris-diderot.fr.