Abstract
We study two register arithmetic computation and skew arithmetic circuits. Our main results are the following:
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For commutative computations, we present an exponential circuit size lower bound for a model of 2-register straight-line programs (SLPs) which is a universal model of computation (unlike width-2 algebraic branching programs that are not universal [AW11]).
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For noncommutative computations, we show that Coppersmith’s 2-register SLP model [BOC88], which can efficiently simulate arithmetic formulas in the commutative setting, is not universal. However, assuming the underlying noncommutative ring has quaternions, Coppersmith’s 2-register model can simulate noncommutative formulas efficiently.
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We consider skew noncommutative arithmetic circuits and show:
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An exponential separation between noncommutative monotone circuits and noncommutative monotone skew circuits.
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We define k-regular skew circuits and show that (k + 1)-regular skew circuits are strictly powerful than k-regular skew circuits, where \(k\leq \frac{n}{\omega(\log n)}\).
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Arvind, V., Raja, S. (2014). The Complexity of Bounded Register and Skew Arithmetic Computation. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_49
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DOI: https://doi.org/10.1007/978-3-319-08783-2_49
Publisher Name: Springer, Cham
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