Circumscription is one of the main formalisms for non-monotonic reasoning. It uses reasoning with minimal models, the key idea being that minimal models have as few exceptions as possible. In this contribution we provide the first comprehensive proof-complexity analysis of different proof systems for propositional circumscription. In particular, we investigate two sequent-style calculi: MLK defined by Olivetti (J. Autom. Reasoning, 1992) and CIRC introduced by Bonatti and Olivetti (ACM ToCL, 2002), and the tableaux calculus NTAB suggested by Niemelä (TABLEAUX, 1996).

In our analysis we obtain exponential lower bounds for the proof size in NTAB and CIRC and show a polynomial simulation of CIRC by MLK. This yields a chain NTAB < CIRC < MLK of proof systems for circumscription of strictly increasing strength with respect to lengths of proofs.