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Bi-intuitionistic logic is the extension of intuitionistic logic with aconnective dual to implication. Bi-intuitionistic logic was introduced byRauszer as a Hilbert calculus with algebraic and Kripke semantics. But hersubsequent ``cut-free'' sequent calculus for BiInt has recently been shown byUustalu to fail cut-elimination. We present a new cut-free sequent calculus forBiInt, and prove it sound and complete with respect to its Kripke semantics.Ensuring completeness is complicated by the interaction between implication andits dual, similarly to future and past modalities in tense logic. Our calculushandles this interaction using extended sequents which pass information frompremises to conclusions using variables instantiated at the leaves of failedderivation trees. Our simple termination argument allows our calculus to beused for automated deduction, although this is not its main purpose.

A Cut-free Sequent Calculus for Bi-Intuitionistic Logic: Extended Version