Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of $3\frac1{86}n-o(n)$. All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of $cn$ beyond a certain constant $c$, where $c$ depends only on the number of substitutions made at a single step of the induction.

Proved a limitation of the gate elimination method (with any subadditive measure) that performs almost the maximal N/2 substitutions. Added a lower bound on decision tree complexity.

Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of $3\frac1{86}n-o(n)$. All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of $cn$ beyond a certain constant $c$, where $c$ depends only on the number of substitutions made at a single step of the induction.