Proof complexity of non-classical logics

Proof complexity is an interdisciplinary area of research utilising techniques from logic, complexity, and combinatorics towards the main aim of understanding the complexity of theorem proving procedures. Traditionally, propositional proofs have been the main object of investigation in proof complexity. Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number of impressive results have been obtained. In these notes we give an introduction to this recent field of proof complexity of non-classical logics. We cover results from proof complexity of modal, intuitionistic, and non-monotonic logics. Some of the results are surveyed, but in addition we provide full details of a recent exponential lower bound for modal logics due to Hrubeš [60] and explain the complexity of several sequent calculi for default logic [16,13]. To make the text self-contained, we also include necessary background information on classical proof systems and non-classical logics.