Abstract Filinski and Griffin have independently succeeded in extending the formulae-as-types notion to deal with continuations. Whereas Griffin adopted control operators primitively, Filinski adopted the duality of functions to construct a symmetric lambda-calculus in which continuations are first-class objects. In this paper, we construct a typed lambda-calculus with first-class configurations consisting of expressions and continuations of the same types. Our calculus corresponds to a natural deduction based on Rumfitt’s bilateralism. Function types are represented as the implication and but-not connectives in intuitionistic and paraconsistent logics, respectively. Our calculus is not only logically consistent, but also computationally consistent. Our calculus with call-by-value and call-by-name strategies correspond to Wadler’s call-by-value and call-by-name dual calculi, respectively. Furthermore, we propose a notion of relaxed configurations, which loosely take expressions and continuations of different types. We confirm that the relaxedness defines control operators for delimited continuations.

Cyclic proof systems are sequent-calculus style proof systems that allow circular structures representing induction, and they are considered suitable for automated inductive reasoning. However, Kimura et al. have shown that the cyclic proof system for the symbolic heap separation logic does not satisfy the cut-elimination property, one of the most fundamental properties of proof systems. This paper proves that the cyclic proof system for the bunched logic with only nullary inductive predicates does not satisfy the cut-elimination property. It is hard to adapt the existing proof technique chasing contradictory paths in cyclic proofs since the bunched logic contains the structural rules. This paper proposes a new proof technique called proof unrolling. This technique can be adapted to the symbolic heap separation logic, and it shows that the cut-elimination fails even if we restrict the inductive predicates to nullary ones.