We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λµ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a restricted form of first-class continuations). Keywords: Curry-Howard isomorphism, Subtractive Logic, control operators, coroutines. 1

Abstract. Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri’s general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for counter-model extraction from a failed proof attempt. 1

Abstract. Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be singleconclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Goré et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cut-free labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus. 1