Essential Structure of Proofs as a Measure of Complexity

The essential structure of proofs is proposed as the basis for a measure of complexity of formulas in FOL. The motivating idea was the recognition that distinct theorems can have the same derivation modulo some non essential details. Hence the difficulty in proving them is identical and so their complexity should be the same. We propose a notion of complexity of formulas capturing this property. With this purpose, we introduce the notions of schema calculus, schema derivation and description complexity of a schema formula. Based on these concepts we prove general robustness results that relate the complexity of introducing a logical constructor with the complexity of the component schema formulas as well as the complexity of a schema formula across different schema calculi.