CNF Encodings of Parity

The minimum number of clauses in a CNF representation of the parity function $x_1 \oplus x_2 \oplus \dotsb \oplus x_n$ is $2^{n-1}$. One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or auxiliary variables). In this paper, we prove the following lower bounds, that almost match known upper bounds, on the number $m$ of clauses and the maximum width $k$ of clauses: 1) if there are at most $s$ auxiliary variables, then $m \ge \Omega\left(2^{n/(s+1)}/n\right)$ and $k \ge n/(s+1)$; 2) the minimum number of clauses is at least $3n$. We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlak, and Zane

SAT-Based Circuit Local Improvement

Finding exact circuit size is notoriously hard. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in the blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of this behavior is that the search space is enormous: the number of circuits of size s is s^?(s), the number of Boolean functions on n variables is 2^(2?). In this paper, we explore the following natural heuristic idea for decreasing the size of a given circuit: go through all its subcircuits of moderate size and check whether any of them can be improved by reducing to SAT. This may be viewed as a local search approach: we search for a smaller circuit in a ball around a given circuit. Through this approach, we prove new upper bounds on the circuit size of various symmetric functions. We also demonstrate that some upper bounds that were proved by hand decades ago, can nowadays be found automatically in a few seconds

SAT-based Circuit Local Improvement

Finding exact circuit size is a notorious optimization problem in practice. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of this behavior is that the search space is enormous: the number of circuits of size $s$ is $s^{\Theta(s)}$, the number of Boolean functions on $n$ variables is $2^{2^n}$. In this paper, we explore the following natural heuristic idea for decreasing the size of a given circuit: go through all its subcircuits of moderate size and check whether any of them can be improved by reducing to SAT. This may be viewed as a local search approach: we search for a smaller circuit in a ball around a given circuit. Through this approach, we prove new upper bounds on the circuit size of various symmetric functions. We also demonstrate that some upper bounds that were proved by hand decades ago, nowadays can be found automatically in a few seconds