We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its constructive content is exposed via soundness relative to an intuitionistic rst-order logic with recursion controlled by the later modality; and soundness of both proof systems is proven relative to a novel coalgebraic description of complete Herbrand models.We can also view Horn clauses coinductively. The greatest complete Herbrand model for a set P of Horn clauses is the largest set of nite and in nite ground atomic formulae coinductively entailed by P. For example, the greatest complete Herbrand model for the above two clauses is the set N ∞ = N ∪ {nat (s (s (· · · )))}, obtained by taking a backward closure of the above two inference rules on the set of all nite and in nite ground atomic formulae. The greatest Herbrand model is the largest set of nite ground atomic formulae coinductively entailed by P. In our example, it would be given by N already. Finally, one can also consider the least complete Hebrand model, which interprets entailment inductively but over potentially in nite terms. In the case of nat, this interpretation does not di er from N . However, nite paths in coinductive structures like transition systems, for example, require such semantics.The need for coinductive semantics of Horn clauses arises in several scenarios: the Horn clause theory may explicitely de ne a coinductive data structure or a coinductive relation. However, it may also happen that a Horn clause theory, which is not explicitly intended as coinductive, nevertheless gives rise to in nite inference by resolution and has an interesting coinductive model. This commonly happens in type inference. We will illustrate all these cases by means of examples.

We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shuffle, (iii) the infiltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that an operator of the classical difference calculus, the Newton transform, generalises from infinite sequences to weighted languages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into their infiltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them.

One of the first publications of Prakash Panangaden is about compositional semantics of digital networks, back in 1984. Digital networks transform streams of input signals to streams of output signals. If the output streams of the components of the network are functions of their input streams, then the behavior of the entire network can be nicely characterized by a recursive stream function. In this paper we consider signal flow graphs, i.e., open synchronous digital networks with feedbacks, obtained by composing amplifiers, mergers, copiers, and delayers. We give two characterizations of the recursive stream functions computed by signal flow graphs: one algebraic in terms of localization of modules of polynomials, and another coalgebraic in terms of Mealy machines. Our main result is that the two characterizations coincide.

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