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 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 white-box deterministic polynomial-time algorithm for polynomial identity testing (PIT) on unambiguous circuits over ℜ and &complexes.