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We introduce a generalized notion of inference system to support more flexible interpretations of recursive definitions. Besides axioms and inference rules with the usual meaning, we allow also coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree. Coaxioms allow us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, whose existence is guaranteed by a smooth extension of classical results. This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, but are rather mixed together.

Coaxioms have been recently introduced to enhance the expressive power of inference systems, by supporting interpretations which are neither purely inductive, nor coinductive. This paper proposes a novel approach based on coaxioms to capture divergence in semantic definitions by allowing inductive and coinductive semantic rules to be merged together for defining a unique semantic judgment. In particular, coinduction is used to derive a special result which models divergence. In this way, divergent, terminating, and stuck computations can be properly distinguished even in semantic definitions where this is typically difficult, as in big-step style. We show how the proposed approach can be applied to several languages; in particular, we first illustrate it on the paradigmatic example of the λ-calculus, then show how it can be adopted for defining the big-step semantics of a simple imperative Java-like language. We provide proof techniques to show classical results, including equivalence with small-step semantics, and type soundness for typed versions of both languages.

We introduce a generalized logic programming paradigm where programs, consisting of facts and rules with the usual syntax, can be enriched by co-facts, which syntactically resemble facts but have a special meaning. As in coinductive logic programming, interpretations are subsets of the complete Herbrand basis, including infinite terms. However, the intended meaning (declarative semantics) of a program is a fixed point which is not necessarily the least, nor the greatest one, but is determined by co-facts. In this way, it is possible to express predicates on non well-founded structures, such as infinite lists and graphs, for which the coinductive interpretation would be not precise enough. Moreover, this paradigm nicely subsumes standard (inductive) and coinductive logic programming, since both can be expressed by a particular choice of co-facts, hence inductive and coinductive predicates can coexist in the same program. We illustrate the paradigm by examples, and provide declarative and operational semantics, proving the correctness of the latter. Finally, we describe a prototype meta-interpreter.

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Recursive definitions of predicates are usually interpreted either inductively or coinductively. Recently, a more powerful approach has been proposed, called flexible coinduction, to express a variety of intermediate interpretations, necessary in some cases to get the correct meaning. We provide a detailed formal account of an extension of logic programming supporting flexible coinduction. Syntactically, programs are enriched by coclauses, clauses with a special meaning used to tune the interpretation of predicates. As usual, the declarative semantics can be expressed as a fixed point which, however, is not necessarily the least, nor the greatest one, but is determined by the coclauses. Correspondingly, the operational semantics is a combination of standard SLD resolution and coSLD resolution. We prove that the operational semantics is sound and complete with respect to declarative semantics restricted to finite comodels.

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Inference systems are a widespread framework used to define possibly recursive predicates by means of inference rules. They allow both inductive and coinductive interpretations that are fairly well-studied. In this paper, we consider a middle way interpretation, called regular, which combines advantages of both approaches: it allows non-well-founded reasoning while being finite. We show that the natural proof-theoretic definition of the regular interpretation, based on regular trees, coincides with a rational fixed point. Then, we provide an equivalent inductive characterization, which leads to an algorithm which looks for a regular derivation of a judgment. Relying on these results, we define proof techniques for regular reasoning: the regular coinduction principle, to prove completeness, and an inductive technique to prove soundness, based on the inductive characterization of the regular interpretation. Finally, we show the regular approach can be smoothly extended to inference systems with corules, a recently introduced, generalised framework, which allows one to refine the coinductive interpretation, proving that also this flexible regular interpretation admits an equivalent inductive characterisation.