Call-by-Value and Call-by-Name Dual Calculi with Inductive and Coinductive Types

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by translating it into the second-order symmetric lambda calculus. This paper then introduces a call-by-value system and a call-by-name system of the dual calculus with inductive and coinductive types, and shows the duality of call-by-value and call-by-name, their Church-Rosser properties, and their strong normalization. Their strong normalization is proved by translating them into the non-deterministic dual calculus with inductive and coinductive types.Comment: The conference version of this paper has appeared in RTA 200

Decidability for Entailments of Symbolic Heaps with Arrays

This paper presents two decidability results on the validity checking problem for entailments of symbolic heaps in separation logic with Presburger arithmetic and arrays. The first result is for a system with arrays and existential quantifiers. The correctness of the decision procedure is proved under the condition that sizes of arrays in the succedent are not existentially quantified. This condition is different from that proposed by Brotherston et al. in 2017 and one of them does not imply the other. The main idea is a novel translation from an entailment of symbolic heaps into a formula in Presburger arithmetic. The second result is the decidability for a system with both arrays and lists. The key idea is to extend the unroll collapse technique proposed by Berdine et al. in 2005 to arrays and arithmetic as well as double-linked lists.Comment: A submission for the postproceedings of the Continuity, Computability, Constructivity 201

Failure of Cut-Elimination in the Cyclic Proof System of Bunched Logic with Inductive Propositions

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

Cut-Elimination for Cyclic Proof Systems with Inductively Defined Propositions (Theory and Applications of Proof and Computation)

Cyclic proof systems are extensions of the sequent-calculus style proof systems for logics with inductively defined predicates. In cyclic proof systems, inductive reasoning is realized as cyclic structures in proof trees. It has been already known that the cut-elimination property does not hold for the cyclic proof systems of some logics such as the first-order predicate logic and the separation logic. In this paper, we consider the cyclic proof systems with inductively defined propositions (that is, nullary predicates), and prove that the cut-elimination holds for the propositional logic, and it does not hold for the bunched logic