Let us fix a prime
We show that for every integer
- NSF-PAR ID:
- 10305476
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 7
- Issue:
- 4
- ISSN:
- 2522-0160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ , by showing that$\mathcal { C}$ admits non-trivial satisfiability and/or$\mathcal { C}$ # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ f (x 1,…,x n ) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ f , and for all “typical” , faster$\mathcal { C}$ # SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ , which may be$f \circ \mathcal { C}$ stronger than itself. In particular:$\mathcal { C}$ # SAT algorithms forn k -size -circuits running in 2$\mathcal { C}$ n /n k time (for allk ) implyN E X P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ # SAT algorithms for -size$2^{n^{{\varepsilon }}}$ -circuits running in$\mathcal { C}$ time (for some$2^{n-n^{{\varepsilon }}}$ ε > 0) implyQ u a s i -N P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ Applying
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