The Complexity of Theorem Proving in Circumscription and Minimal Entailment

Abstract

Abstract. 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 [28] and CIRC introduced by Bonatti and Olivetti [8], and the tableaux calculus NTAB suggested by Niemelä [26]. 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 <p CIRC <p MLK of proof systems for circumscription of strictly increasing strength with respect to lengths of proofs.

Tableau vs. Sequent Calculi for Minimal Entailment

“…In particular, Niemelä (1996) introduced a tableau system NTAB for minimal entailment for clausal formulas, and Bonatti and Olivetti (2002) defined an analytic sequent calculus CIRC for circumscription. Building on initial results from (Bonatti and Olivetti 2002) we prove in (Beyersdorff and Chew 2014) that NTAB ≤ p CIRC ≤ p MLK is a chain of proof systems of strictly increasing strength, i.e., in addition to the p-simulations we obtain separations between the proof systems.…”

“…We conclude by mentioning that there are further proof systems for minimal entailment and circumscription, which have been recently analysed from a proofcomplexity perspective (Beyersdorff and Chew 2014). In particular, Niemelä (1996) introduced a tableau system NTAB for minimal entailment for clausal formulas, and Bonatti and Olivetti (2002) defined an analytic sequent calculus CIRC for circumscription.…”

“…We also add more information to the nodes (this can all be done in polynomial time). We start with a fact about LK (for a proof see (Beyersdorff and Chew 2014)).…”

Proof Complexity of Modal Resolution

“…In general, it appears that proof complexity of non-classical logics is at a quite early stage (cf. [23] for a survey), and a number of the existing proof-size lower bounds for some logics, such as for default logic [24,34], autoepistemic logic [9], and circumscription [13], are somewhat ad hoc without using general techniques. It would be interesting to explore existing propositional proof complexity techniques [44] more widely in the context of non-classical logics.…”