Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination.
Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function.

In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration.

We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of $3.24n$ and $2.59n$ for circuits over $U_2$ and $B_2$, respectively (though the lower bound for the basis $B_2$ is given for a quadratic disperser whose explicit construction is not currently known), and faster than $2^n$ #SAT-algorithms for circuits over $U_2$ and $B_2$ of size at most $3.24n$ and $2.99n$, respectively. Here by $B_2$ we mean the set of all bivariate Boolean functions, and by $U_2$ the set of all bivariate Boolean functions except for parity and its complement.

A critical flaw in the proof of Lemma 11 was fixed.
Many minor fixes and improvements in the other parts of the paper.

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function.

In this paper, we refine the approach by Chen and Kabanets and provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main technical ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount.
This case analysis immediately implies an upper bound for #SAT. To obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration.

We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of $3.24n$ and $2.59n$ for circuits over $U_2$ and $B_2$, respectively (though the lower bound for the basis $B_2$ is given for a quadratic disperser whose explicit construction is not currently known), and faster than $2^n$ #SAT-algorithms for circuits over $U_2$ and $B_2$ of size at most $3.24n$ and $2.99n$, respectively. Here by $B_2$ we mean the set of all bivariate Boolean functions, and by $U_2$ the set of all bivariate Boolean functions except for parity and its complement.

In this revision, we fix a number of typos.

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function.

In this paper, we refine the approach by Chen and Kabanets and provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main technical ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount.
This case analysis immediately implies an upper bound for #SAT. To obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration.

We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of $3.24n$ and $2.59n$ for circuits over $U_2$ and $B_2$, respectively (though the lower bound for the basis $B_2$ is given for a quadratic disperser whose explicit construction is not currently known), and faster than $2^n$ #SAT-algorithms for circuits over $U_2$ and $B_2$ of size at most $3.24n$ and $2.99n$, respectively.