Paulo Jesus

Recursive functions defined on a coalgebraic datatype may not converge if there are cycles in the input, that is, if the input object is not well-founded. Even so, there is often a useful solution; for example, the free variables of an infinitary lambda-term, or the expected running time of a finite-state probabilistic protocol. Theoretical models of recursion schemes have been well studied under the names well-founded coalgebras, recursive coalgebras, corecursive algebras, and Elgot algebras. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g.\ when the domain is well-founded. If the domain is not well-founded, there can be multiple solutions. The standard semantics of recursive programs gives a particular solution, namely the least solution in a flat Scott domain, which may not be the one we want. Unfortunately, current programming languages provide no support for specifying alternatives. In this paper we give numerous examples in which it would be useful to do so and propose programming language constructs that would allow the specification of alternative solutions and methods to compute them. The programmer must essentially provide a way to solve equations in the codomain. We show how to implement these new constructs as an extension of existing languages. We also prove some theoretical results characterizing well-founded coalgebras that slightly extend results of Adamek, Luecke, and Milius (2007).

Replicating data under Eventual Consistency (EC) allows any replica to accept updates without remote synchronisation. This ensures performance and scalability in largescale distributed systems (e.g., clouds). However, published EC approaches are ad-hoc and error-prone. Under a formal Strong Eventual Consistency (SEC) model, we study sufficient conditions for convergence. A data type that satisfies these conditions is called a Conflictfree Replicated Data Type (CRDT). Replicas of any CRDT are guaranteed to converge in a self-stabilising manner, despite any number of failures. This paper formalises two popular approaches (state- and operation-based) and their relevant sufficient conditions. We study a number of useful CRDTs, such as sets with clean semantics, supporting both add and remove operations, and consider in depth the more complex Graph data type. CRDT types can be composed to develop large-scale distributed applications, and have interesting theoretical properties.

Distributed data aggregation is an important task, allowing the decentralized determination of meaningful global properties, that can then be used to direct the execution of other applications. The resulting values result from the distributed computation of functions like COUNT, SUM and AVERAGE. Some application examples can found to determine the network size, total storage capacity, average load, majorities and many others.
In the last decade, many different approaches have been proposed, with different trade-offs in terms of accuracy, reliability, message and time complexity. Due to the considerable amount and variety of aggregation algorithms, it can be difficult and time consuming to determine which techniques will be more appropriate to use in specific settings, justifying the existence of a survey to aid in this task.
This work reviews the state of the art on distributed data aggregation algorithms, providing three main contributions. First, it formally defines the concept of aggregation, characterizing the different types of aggregation functions. Second, it succinctly describes the main aggregation techniques, organizing them in a taxonomy. Finally, it provides some guidelines toward the selection and use of the most relevant techniques, summarizing their principal characteristics.

This report presents a typed, “pointfree” generalization of relational data depen- dency theory expressed not in the standard set-theoretic way, “a` la Codd”, but in the calculus of binary relations which, initiated by De Morgan in the 1860s, is the core of modern algebra of programming.
Contrary to the intuition that a binary relation is just a particular case of an n- ary relation, this report shows the effectiveness of the former in “explaining” and reasoning about the latter. Data dependency theory, which is central to relational database design, is addressed in pointfree calculational style instead of reasoning about (sets of) tuples in conventional “implication-first” logic style.
It turns out that the theory becomes more general, more structured and sim- pler. Elegant expressions replace lengthy formulæ and easy-to-follow calculations replace pointwise proofs with lots of “· · ·” notation, case analysis and natural lan- guage explanations for “obvious” steps. In particular, attributes are generalized to arbitrary (observation) functions and the principle of lossless decomposition is established for arbitrary such functions.
The report concludes by showing how the proposed generalization of data dependency theory paves the way to interesting synergies with other branches of computer science, namely formal modeling and transition systems theory. A number of open topics for research in the field are proposed as future work.

We give a new, significantly shorter proof of the completeness of the left-handed star rule of Kleene algebra. The proof reveals the rich interaction of algebra and coalgebra in the theory.

A notion of generalized regular expressions for a large class of systems modeled as coalgebras, and an analogue of Kleene’s theorem and Kleene algebra, were recently proposed by a subset of the authors of this paper. Examples of the systems covered include infinite streams, deterministic automata, Mealy machines and labelled transition sytems. In this paper, we present a novel algorithm to decide whether two expressions are bisimilar or not. The procedure is implemented in the automatic theorem prover CIRC, by reducing coinduction to an entailment relation between an algebraic specification and an appropriate set of equations. We illustrate the generality of the tool with three examples: infinite streams of real numbers, Mealy machines and labelled transition systems.