Friends with benefits: implementing corecursion in foundational proof assistants

We introduce AmiCo, a tool that extends a proof assistant, Isabelle/HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant’s inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user’s specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference

Friends with benefits: implementing corecursion in foundational proof assistants

We introduce AmiCo, a tool that extends a proof assistant, Isabelle/HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant’s inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user’s specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference

The Recursion Scheme from the Cofree Recursive Comonad

AbstractWe instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale

Flexible Coinduction

openRecursive definitions of predicates by means of inference rules are ubiquitous in computer science. They are usually interpreted inductively or coinductively, however there are situations where none of these two options provides the expected meaning. In the thesis we propose a flexible form of coinductive interpretation, based on the notion of corules, able to deal with such situations. In the first part, we define such flexible coinductive interpretation as a fixed point of the standard inference operator lying between the least and the greatest one, and we provide several equivalent proof-theoretic semantics, combining well-founded and non-well-founded derivations. This flexible interpretation nicely subsumes standard inductive and coinductive ones and is naturally associated with a proof principle, which smoothly extends the usual coinduction principle. In the second part, we focus on the problem of modelling infinite behaviour by a big-step operational semantics, which is a paradigmatic example where neither induction nor coinduction provide the desired interpretation. In order to be independent from specific examples, we provide a general, but simple, definition of what a big-step semantics is. Then, we extend it to include also observations, describing the interaction with the environment, thus providing a richer description of the behaviour of programs. In both settings, we show how corules can be successfully adopted to model infinite behaviour, by providing a construction extending a big-step semantics, which as usual only describes finite computations, to a richer one including infinite computations as well. Finally, relying on these constructions, we provide a proof technique to show soundness of a predicate with respect to a big-step semantics. In the third part, we ez face eez the problem of providing an algorithmic support to corules. To this end, we consider the restriction of the flexible coinductive interpretation to regular derivations, analysing again both proof-theoretic and fixed point semantics and developing proof techniques. Furthermore, we show that this flexible regular interpretation can be equivalently characterised inductively by a cycle detection mechanism, thus obtaining a sound and complete (abstract) (semi-)algorithm to check whether a judgement is derivable. Finally, we apply such results to extend logic programming by coclauses, the analogous of corules, defining declarative and operational semantics and proving ez that eez the latter is sound and complete with respect to the regular declarative model, thus obtaining a concrete support to flexible coinduction.openXXXIII CICLO - INFORMATICA E INGEGNERIA DEI SISTEMI/ COMPUTER SCIENCE AND SYSTEMS ENGINEERING - Informatica/computer scienceDagnino, Francesc

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

A coalgebraic perspective on logical interpretations

In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics.Fundação para a Ciência e a Tecnologia (FCT