This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.

Abstract. We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sections. Soundness is guaranteed by a well-foundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables. A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a non-cyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition. 1

We argue that, by supporting a mixture of “compositional” and “structural” styles of proof, sequent-based proof systems provide a useful framework for the formal verification of processes. As a worked example, we present a sequent calculus for establishing that processes from a process algebra satisfy assertions in Hennessy-Milner logic. The main novelty lies in the use of the operational semantics to derive introduction rules, on the left and right of sequents, for the operators of the process calculus. This gives a generic proof system applicable to any process algebra with an operational semantics specified in the GSOS format. Using a general algebraic notion of GSOS model, we prove a completeness theorem for the cut-free fragment of the proof system, thereby establishing the admissibility of the cut rule. Under mild (and necessary) conditions on the process algebra, an ω-completeness result, relative to the “intended” model of closed process terms, follows.

Abstract. We present a framework for inductive definitions in the logic of bunched implications, BI, and formulate two sequent calculus proof systems for inductive reasoning in this framework. The first proof system adopts a traditional approach to inductive proof, extending the usual sequent calculus for predicate BI with explicit induction rules for the inductively defined predicates. The second system allows an alternative mode of reasoning with inductive definitions by cyclic proof. In this system, the induction rules are replaced by simple case-split rules, and the proof structures are cyclic graphs formed by identifying some sequent occurrences in a derivation tree. Because such proof structures are not sound in general, we demand that cyclic proofs must additionally satisfy a global trace condition that ensures soundness. We illustrate our inductive definition framework and proof systems with simple examples which indicate that, in our setting, cyclic proof may enjoy certain advantages over the traditional induction approach. 1