On the complexity of algebraic numbers, and the bit-complexity of straight-line programs¹

Article type: Research Article

Authors: Allender, Eric^{a; *} | Balaji, Nikhil^b | Datta, Samir^c | Pratap, Rameshwar^d

Affiliations: [a] Department of Computer Science, Rutgers University, NJ, USA | [b] Indian Institute of Technology, Delhi, India | [c] Chennai Mathematical Institute & UMI-ReLaX, India | [d] Indian Institute of Technology, Hyderabad, Telangana, India

Correspondence: [*] Corresponding author. allender@cs.rutgers.edu.

Note: [1] Preliminary versions of this material appeared in 2014 and 2012.

Abstract: We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC0 circuits for division, matrix powering, and related problems, where the improvement is in terms of “majority depth” (initially studied by Maciel and Thérien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.

Keywords: Threshold circuits, counting hierarchy, algebraic numbers

DOI: 10.3233/COM-220407

Journal: Computability, vol. 12, no. 2, pp. 145-173, 2023