A Coalgebraic Perspective on Minimization and Determinization

Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by co-recursion. In this (entirely self-contained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment -- from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the input-output behavior of the sy...

ArXiv, 2018

We consider conditional transition systems, that model software product lines with upgrades, in a coalgebraic setting. By using Birkhoff's duality for distributive lattices, we derive two equivalent Kleisli categories in which these coalgebras live: Kleisli categories based on the reader and on the so-called lattice monad over $\mathsf{Poset}$. We study two different functors describing the branching type of the coalgebra and investigate the resulting behavioural equivalence. Furthermore we show how an existing algorithm for coalgebra minimisation can be instantiated to derive behavioural equivalences in this setting.

Information and Computation, 2012

Weighted automata are a generalization of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence.

. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and co-induction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...

We develop rules for coalgebras in type theory, and give meaning explanations for them. We show that elements of coalgebras are determined by their elimination rules, whereas the introduction rules can be considered as derived. This is in contrast with algebraic data types, for which the opposite is true: elements are determined by their introduction rules, and the elimination rules can be considered as derived. In this sense, the function type from the logical framework is more like a coalgebraic data type, the elements of which are determined by the elimination rule. We illustrate why the simplest form of guarded recursion is nothing but the introduction rule originating from the formulation of coalgebras in category theory. We discuss restrictions needed in order to preserve decidability of equality. Dedicated to Per Martin-Löf on the occasion of his retirement. 1.1 Introduction Most programs in computing are interactive programs. This means that they are not batch programs, which, once started, are guaranteed to terminate after a certain amount of time and deliver their result. They are programs which keep running and interacting with user input, until they are terminated by the user. Such programs correspond to non-well-founded trees: Nodes are labelled by commands and the branching degree of a node labelled by a command is the set of responses to this command. A computation which goes on for ever corresponds to an infinite path in this tree. More details of this can be found in a series of articles by the author and Peter Hancock [22, 23, 24, 25, 26]. Colists are simple trees with branching degrees 0 or 1, and for ease of presentation, we restrict ourselves in this article to colists.

Theoretical Computer Science, 2002

The paper presents a ÿrst reconstruction of Hoare's theory of CSP in terms of partial automata and related coalgebras. We show that the concepts of processes in Hoare (Communicating Sequential Processes, Prentice-Hall, Englewood Cli s, NJ, 1985) are strongly related to the concepts of states for special, namely, ÿnal partial automata. Moreover, we show how the deterministic and nondeterministic operations in can be interpreted in a compatible way by constructions on the semantical level of automata. Based on this, we are able to interpret ÿnite process expressions as representing ÿnite partial automata with designated initial states. In such a way we provide a new method for solving recursive process equations which is based on the concept of ÿnal automata. The coalgebraic reconstruction of CSP allows us to use coinduction as a new proof principle. To make evident the usefulness of this principle we prove some example laws from Hoare (1985).

The Coalgebraic Logic seminar took place between the 6th and 9th of December 2009 at the Leibniz Forschungszentrum Schloss Dagstuhl. The event was very well received, with more than 35 scientists from Europe, the United States, Canada and China attending, and with more than thirty presentations during the three days. Given the unforeseen enthusiasm of the meeting's reception, we thought that it would be a good idea for us to put together a special journal issue dedicated to this topic. We were also encouraged by the positive response we received from Guiseppe Longo, who indicated his interest in a special issue for Mathematical Structures in Computer Science, and here we are.

2010

Electronic Notes in Theoretical Computer Science, 1999

We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for non-deterministic choice. In Set, this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category of the non-empty powerset monad. The corresponding final coalgebra semantics is fully abstract with respect to trace equivalence.

Electronic Notes in Theoretical Computer Science, 2009

A transition system can be presented either as a binary relation or as a coalgebra for the powerset functor, each representation being obtained from the other by transposition. More generally, a coalgebra for a functor F generalises transition systems in the sense that a shape for transitions is determined by F, typically encoding a signature of methods and observers. This paper explores such a duality to frame in purely relational terms coalgebraic refinement, showing that relational (data) refinement of transition relations, in its two variants, downward and upward (functional) simulations, is equivalent to coalgebraic refinement based on backward and forward morphisms, respectively. Going deeper, it is also shown that downward simulation provides a complete relational rule to prove coalgebraic refinement. With such a single rule the paper defines a pre-ordered calculus for refinement of coalgebras, with bisimilarity as the induced equivalence. The calculus is monotonic with respect to the main relational operators and arbitrary relator F, therefore providing a framework for structural reasoning about refinement.