Dependent Inductive and Coinductive Types are Fibrational Dialgebras

In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories \`a la Martin-L\"of can be interpreted using fibrations. Modern theorem provers, however, are based on more sophisticated type systems that allow the definition of powerful inductive dependent types (known as inductive families) and, somewhat limited, coinductive dependent types. I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors. This description is also a proposal of how coinductive types should be treated in type theories, as they appear here simply as dual of inductive types. Finally, I show how dependent data types correspond to algebras and coalgebras, and give the correspondence to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282

Coinduction in Flow: The Later Modality in Fibrations

This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search. Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results

Composition and Recursion for Causal Structures

Causality appears in various contexts as a property where present behaviour can only depend on past events, but not on future events. In this paper, we compare three different notions of causality that capture the idea of causality in the form of restrictions on morphisms between coinductively defined structures, such as final coalgebras and chains, in fairly general categories. We then focus on one presentation and show that it gives rise to a traced symmetric monoidal category of causal morphisms. This shows that causal morphisms are closed under sequential and parallel composition and, crucially, under recursion

Monoidal Company for Accessible Functors

Distributive laws between functors are a fundamental tool in the theory of coalgebras. In the context of coinduction in complete lattices, they correspond to the so-called compatible functions, which enable enhancements of the coinductive proof technique. Amongst these, the greatest compatible function, called the companion, has recently been shown to satisfy many good properties. Categorically, the companion of a functor corresponds to the final object in a category of distributive laws. We show that every accessible functor on a locally presentable category has a companion. Central to this and other constructions in the paper is the presentation of distributive laws as coalgebras for a certain functor. This functor itself has again, what we call, a second-order companion. We show how this companion interacts with the various monoidal structures on functor categories. In particular, both the first- and second-order companion give rise to monads. We use these results to obtain an abstract GSOS-like extension result for specifications involving the second-order companion

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

In sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen's classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.Comment: Paper presented at the 36th International Conference on Logic Programming (ICLP 2019), University Of Calabria, Rende (CS), Italy, September 2020, 16 page

Session Coalgebras: A Coalgebraic View on Session Types and Communication Protocols

Compositional methods are central to the development and verification of software systems. They allow to break down large systems into smaller components, while enabling reasoning about the behaviour of the composed system. For concurrent and communicating systems, compositional techniques based on behavioural type systems have received much attention. By abstracting communication protocols as types, these type systems can statically check that programs interact with channels according to a certain protocol, whether the intended messages are exchanged in a certain order. In this paper, we put on our coalgebraic spectacles to investigate session types, a widely studied class of behavioural type systems. We provide a syntax-free description of session-based concurrency as states of coalgebras. As a result, we rediscover type equivalence, duality, and subtyping relations in terms of canonical coinductive presentations. In turn, this coinductive presentation makes it possible to elegantly derive a decidable type system with subtyping for $\pi$-calculus processes, in which the states of a coalgebra will serve as channel protocols. Going full circle, we exhibit a coalgebra structure on an existing session type system, and show that the relations and type system resulting from our coalgebraic perspective agree with the existing ones.Comment: 36 pages, submitte

Session Coalgebras: A Coalgebraic View on Regular and Context-Free Session Types

Compositional methods are central to the verification of software systems. For concurrent and communicating systems, compositional techniques based on behavioural type systems have received much attention. By abstracting communication protocols as types, these type systems can statically check that channels in a program interact following a certain protocol—whether messages are exchanged in the intended order. In this article, we put on our coalgebraic spectacles to investigate session types, a widely studied class of behavioural type systems. We provide a syntax-free description of session-based concurrency as states of coalgebras. As a result, we rediscover type equivalence, duality, and subtyping relations in terms of canonical coinductive presentations. In turn, this coinductive presentation enables us to derive a decidable type system with subtyping for the π-calculus, in which the states of a coalgebra will serve as channel protocols. Going full circle, we exhibit a coalgebra structure on an existing session type system, and show that the relations and type system resulting from our coalgebraic perspective coincide with existing ones. We further apply to session coalgebras the coalgebraic approach to regular languages via the so-called rational fixed point, inspired by the trinity of automata, regular languages, and regular expressions with session coalgebras, rational fixed point, and session types, respectively. We establish a suitable restriction on session coalgebras that determines a similar trinity, and reveals the mismatch between usual session types and our syntax-free coalgebraic approach. Furthermore, we extend our coalgebraic approach to account for context-free session types, by equipping session coalgebras with a stack