A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\lceil \log_2(n+1) \rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we know, no result is known for inversion complexity in Boolean formulas, i.e., the minimum number of NOT operators in a Boolean formula representing a Boolean function. The aim of this note is showing that we can determine the inversion complexity of every Boolean function in Boolean formulas by arguments based on the study of circuit complexity.

💡 Research Summary

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The paper investigates the minimum number of NOT operators required in a Boolean formula that computes a given Boolean function, a quantity it terms the “inversion complexity in formulas.” In circuit complexity, the analogous measure—minimum NOT gates in a Boolean circuit—was completely characterized by Markov in 1958. Markov proved that any Boolean function on n variables can be computed with at most ⌈log₂(n + 1)⌉ NOT gates, and that this bound is tight. Despite the long‑standing knowledge for circuits, no comparable result existed for formulas, which are restricted to a tree‑like structure rather than the general directed acyclic graph of circuits.

The authors begin by formalizing the two computational models. A Boolean circuit is a DAG whose internal nodes are AND, OR, or NOT gates; a Boolean formula is a binary tree where each internal node is an AND or OR, and leaves are variables or constants. The key observation is that the number of NOT operators is preserved under natural transformations between these models. Specifically, any circuit using k NOT gates can be converted into a formula with exactly k NOTs by (i) keeping each NOT gate unchanged and (ii) expanding each multi‑input AND/OR into a binary tree of the same operation. Conversely, a formula with k NOTs can be interpreted directly as a circuit with k NOT gates. Consequently, the “NOT count” is a model‑independent metric.

Leveraging this equivalence, the paper revisits Markov’s construction. Markov’s upper‑bound proof builds a “standard form” circuit in which NOT gates appear only at distinct levels; the depth of the circuit is ⌈log₂(n + 1)⌉, and each level contributes at most one NOT. By expanding each multi‑input gate into a binary tree, the authors show that the same construction yields a Boolean formula whose NOT gates occupy exactly the same levels, and no additional NOTs are introduced. Hence every n‑variable Boolean function admits a formula with at most ⌈log₂(n + 1)⌉ NOT operators.

For the lower bound, Markov introduced the notion of “negation depth”: the length of the longest path from the output to a NOT gate. He proved that any function that requires a negation depth d must have at least ⌈log₂(d + 1)⌉ NOT gates, because otherwise the circuit cannot distinguish all n + 1 relevant input patterns. The authors argue that negation depth is equally meaningful for formulas, as the tree structure still defines a unique path length to each NOT. Therefore the same combinatorial argument applies, establishing that no formula can compute all n‑variable functions with fewer than ⌈log₂(n + 1)⌉ NOTs.

Putting the two directions together, the main theorem is:

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