By V Sankrithi Krishnan

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**Example text**

If D ∪ N is unsatisﬁable, then there exists a clause C ∈ N such that D ∪ {C} is unsatisﬁable. Deﬁnition 39. For a set of variables V and an assignment σ, let Vσ := {{q} | q ∈ V, σ(q) = 1} . Lemma 40. Let D be a set of Horn clauses of size n not containing a negative clause. Let V be a set of variables and p be a variable. Then there exists a monotone circuit C in variables V of size O(n2 ) such that for any assignment σ to V , C outputs 1 if and only if D, Vσ , {¬p} is unsatisﬁable. The proof the two previous lemmas follows from the analysis of the standard satisﬁability algorithm for Horn formulae.

Ak ) → ψ are K-tautologies. Proof. The characteristic set Cπ has size ≤ 3n as Cπ contains n clauses and each clause contains at most three literals. Proof Complexity of Non-classical Logics 35 Let V = { A1 , . . , Ak }. Let C be the monotone circuit from Lemma 40 of size O(n2 ) which outputs 1 if and only if Cπ , Vσ , {¬ ψ} is unsatisﬁable. We note that by the previous Lemma, C will always output 1 on assignments σ which are consistent with ϕ, but we also have to consider other assignments.

Problem I. So far, research on proof complexity of non-classical logics has concentrated on Frege type systems or their equivalent sequent style formulations. Quite in contrast, many results in classical proof complexity concern systems which are motivated by algebra, geometry, or combinatorics. Can we construct algebraic or geometric proof systems for non-classical logics? Problem II. One important tool in the analysis of classically strong systems such as Frege systems is their correspondence to weak arithmetic theories, known as bounded arithmetic (cf.