The Chomsky hierarchy plays a prominent role in the foundations of the theoretical computer science relating classes of formal languages of primary importance. In this paper we use recent developments on coalgebraic and monad-based semantics to obtain a generic notion of a T-automaton, where T is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, valence automata, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for T-automata, and we show by numerous examples that it correctly instantiates for the known classes of machines/languages captured by the Chomsky hierarchy. Moreover, our approach provides new generic techniques for proving expressivity bounds of various machine-based models.
We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or ϵ-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.
We present CoCaml, a functional programming language extending OCaml, which allows us to define functions on coinductive datatypes parameterized by an equation solver. We provide numerous examples that attest to the usefulness of the new programming constructs, including operations on infinite lists, infinitary lambda-terms and p-adic numbers.
Induction is a well-established proof principle that is taught in most undergraduate programs in mathematics and computer science. In computer science, it is used primarily to reason about inductively-defined datatypes such as finite lists, finite trees, and the natural numbers. Coinduction is the dual principle that can be used to reason about coinductive datatypes such as infinite streams or trees, but it is not as widespread or as well understood. In this paper, we illustrate through several examples the use of coinduction in informal mathematical arguments. Our aim is to promote the principle as a useful tool for the working mathematician and to bring it to a level of familiarity on par with induction. We show that coinduction is not only about bisimilarity and equality of behaviors, but also applicable to a variety of functions and relations defined on coinductive datatypes.
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).
We give a new presentation of Brzozowski's algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on the duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations.
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.
Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n, 2n, 3n, ... Paasche's theorem is a generalization of Moessner's; by varying the parameters of the procedure, one can obtain the sequence of factorials 1!, 2!, 3!, ... or the sequence of superfactorials 1!!, 2!!, 3!!, ... Long's theorem generalizes Moessner's in another direction, providing a procedure to generate the sequence a1n-1, (a+d)2n-1, (a+2d)3n-1, ... Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. In this note we give a short and revealing algebraic proof of a general theorem that contains Moessner's, Paasche's, and Long's as special cases. We also prove a generalization that gives new Moessner-type theorems.
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.
Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours.
In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterize weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterize weighted language equivalence.
Relying on the second characterization, we show three different procedures for computing weighted language equivalence. The first one consists in a generalization of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages.
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.
Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization in the system. First, we show that for coalgebras on categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. For instance, when considering epi-mono factorizations in the category of sets and functions, this procedure corresponds to the usual (coalgebraic) minimization algorithm and two states are behaviourally equivalent iff they are mapped to the same state in the minimized coalgebra. Second, motivated by several examples, we consider coalgebras on categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. Finally, we show that the result of the procedure also induces a pseudo-metric on the states, in such a way that the distance between each pair of states is minimized.
Recent approaches to component-based software engineering employ coordinating connectors to compose components into software systems. Reo is a model of component coordination, wherein complex connectors are constructed by composing various types of primitive channels. Reo automata are a simple and intuitive formal model of context-dependent connectors, which provided a compositional semantics for Reo. In this paper, we study Reo automata from a coalgebraic perspective. This enables us to apply the recently developed coalgebraic theory of generalized regular expressions in order to derive a specification language, tailor-made for Reo automata, and sound and complete axiomatizations with respect to three distinct notions of equivalence: (coalgebraic) bisimilarity, the bisimulation notion studied in the original papers on Reo automata and trace equivalence. The obtained language is simple, but nonetheless expressive enough to specify all possible finite Reo automata. Moreover, it comes equipped with a Kleene-like theorem: we provide algorithms to translate expressions to Reo automata and, conversely, to compute an expression equivalent to a state in a Reo automaton.
In this paper we present a compositional semantics for the channel-based coordination language Reo which enables the analysis of quality of service (QoS) properties of service compositions. For this purpose, we annotate Reo channels with stochastic delay rates and explicitly model data-arrival rates at the boundary of a connector, to capture its interaction with the services that comprise its environment. We propose Stochastic Reo automata as an extension of Reo automata, in order to compositionally derive a QoS-aware semantics for Reo. We further present a translation of Stochastic Reo automata to Continuous-Time Markov Chains (CTMCs). This translation enables us to use third- party CTMC verification tools to do an end-to-end performance analysis of service compositions.
Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor $F$ determines both the type of systems ($F$-coalgebras) and a notion of behavioral equivalence ($\sim_F$) amongst them. Many types of transition systems and their equivalences can be captured by a functor $F$. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.
In this paper, we present a systematic way of deriving (1) languages of (generalized) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
Recent approaches to component-based software engineering employ coordinat- ing connectors to compose components into software systems. For maximum flexibility and reuse, such connectors can themselves be composed, resulting in an expressive calculus of connectors whose semantics encompasses complex combinations of synchronisation, mutual exclusion, non-deterministic choice and state-dependent behaviour. A more expressive notion of connector includes also context-dependent behaviour, namely, whenever the choices a connector can take change non-monotonically as the context, given by the pending activity on its ports, changes. Context dependency can express notions of priority and inhibition. Capturing context-dependent behaviour in formal models is non-trivial, as it is unclear how to propagate context information through composition. In this paper we present an intuitive automata-based formal model of context- dependent connectors, and argue that it is superior to previous attempts at such a model for the coordination language Reo.
We study the set TA of infinite binary trees with nodes labelled in a semiring A from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus where definitions are presented as behavioural differential equations. We present a general format for these equations that guarantees the existence and uniqueness of solutions. Although technically not very difficult, the resulting framework has surprisingly nice applications, which is illustrated by various concrete examples.
For polynomial set functors G, we introduce a language of expressions for describing elements of final G-coalgebra. We show that every state of a finite G-coalgebra corresponds to an expression in the language, in the sense that they both have the same semantics. Conversely, we give a compositional synthesis algorithm which transforms every expression into a finite G-coalgebra. The language of expressions is equipped with an equational system that is sound, complete and expressive with respect to G-bisimulation.
We present a novel coalgebraic logic for deterministic Mealy machines that is sound, complete and expressive w.r.t. bisimulation. Every finite Mealy machine corresponds to a finite formula in the language. For the converse, we give a compositional synthesis algorithm which transforms every formula into a finite Mealy machine whose behaviour is exactly the set of causal functions satisfying the formula.
We study the set TA of infinite binary trees with nodes labelled in a semiring A from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus where definitions are presented as behavioural differential equations. We present a general format for these equations that guarantees the existence and uniqueness of solutions. Although technically not very difficult, the resulting framework has surprisingly nice applications, which is illustrated by various concrete examples.
