We introduce principles 2VR and 2VR which imply reflection for, respectively, the CNFs and the narrow CNFs refutable in the depth-1 propositional LK system PK1. We give a polynomial-size refutation of their negations 2VR and 2VR in the system PK1 and show an exponential lower bound on the size of their resolution refutations. We conjecture that they have no small Res(log) refutations; this woul...

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depth d Frege proofs of m lines can be tr...

In a system of sequents for intuitionistic predicate logic a theorem, which corresponds to Prawitz’s Normal Form Theorem for natural deduction, are proved. In sequent derivations a special kind of cuts, maximum cuts, are defined. Maximum cuts from sequent derivations are connected with maximum segments from natural deduction derivations, i.e., sequent derivations without maximum cuts correspond...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than i...

We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We show that the standard encoding of the PHP as a monotone sequent admits quasipolynomial-size proofs in this system. This result is a consequence of deriving the basic properties of certain quasipolynomial-size monotone formulas computing the boolean...

In their seminal paper Artemov and Protopopescu provide Hilbert formal systems, Brower–Heyting–Kolmogorov Kripke semantics for the logics of intuitionistic belief knowledge. Subsequently Krupski has proved that logic knowledge is PSPACE-complete Su Sano have provided calculi enjoying subformula property. This continues investigations around sequent Intuitionistic Epistemic Logics by providing p...

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which...

In this work we explore the connections between (linear) nested sequent calculi and ordinary sequent calculi for normal and non-normal modal logics. By proposing local versions to ordinary sequent rules we obtain linear nested sequent calculi for a number of logics, including to our knowledge the first nested sequent calculi for a large class of simply dependent multimodal logics, and for many ...