Read-once polynomial identity testing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
On the hardness of the noncommutative determinant
Proceedings of the forty-second ACM symposium on Theory of computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Revisiting the equivalence problem for finite multitape automata
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
Non-commutative arithmetic circuits with division
Proceedings of the 5th conference on Innovations in theoretical computer science
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We continue the study of noncommutative polynomial identity testing initiated by Raz and Shpilka and presentefficient algorithms for the following problems in the noncommutative model: Polynomial identity testing: The algorithm gets as an input an arithmetic circuit with the promise that the polynomial it computes has small degree (for instance, a circuit of logarithmic depth or an arithmetic formula) and determines whether or not the output of the circuit is identically zero (as a formal expression). Unlike the algorithm by Raz and Shpilka, our algorithm is black-box (but randomized with one-sided error) and evaluates the circuit over the ring of matrices. In addition, we present query complexity lower bounds for identity testing and explore the possibility of de-randomizing our algorithm. The analysis of our algorithm uses a noncommutative variant of the Schwartz-Zippel test. Minimizing algebraic branching programs: The algorithm gets as an input an algebraic branching program (ABP) and outputs a smallest equivalent ABP. The algorithm is based on Nisanýs characterization of ABP complexity, and uses as a sub-routine an algorithm for computing linear dependencies amongst arithmetic formulas, a problem previously studied by the authors.