Coalgebraic logic and synthesis of Mealy machines

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

Monoidal computer III: A coalgebraic view of computability and complexity

Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations. In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi

Non-Deterministic Kleene Coalgebras

In this paper, we present a systematic way of deriving (1) languages of (generalised) 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

Angluin learning via logic

In this paper we will provide a fresh take on Dana Angluin's algorithm for learning using ideas from coalgebraic modal logic. Our work opens up possibilities for applications of tools & techniques from modal logic to automata learning and vice versa. As main technical result we obtain a generalisation of Angluin's original algorithm from DFAs to coalgebras for an arbitrary finitary set functor T in the following sense: given a (possibly infinite) pointed T-coalgebra that we assume to be regular (i.e. having an equivalent finite representation) we can learn its finite representation by asking (i) "logical queries" (corresponding to membership queries) and (ii) making conjectures to which the teacher has to reply with a counterexample. This covers (a known variant) of the original L* algorithm and the learning of Mealy/Moore machines. Other examples are bisimulation quotients of (probabilistic) transition systems

Regular expressions for polynomial coalgebras

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

Symbolic Synthesis of Mealy Machines from Arithmetic Bistream Functions

In this paper, we describe a symbolic synthesis method which given an algebraic expression that specifies a bitstream function f, constructs a (minimal) Mealy machine that realises f. The synthesis algorithm can be seen as an analogue of Brzozowski’s construction of a finite deterministic automaton from a regular expression. It is based on a coinductive characterisation of the operators of 2-adic arithmetic in terms of stream differential equations.

Coalgebra for the working software engineer

Often referred to as ‘the mathematics of dynamical, state-based systems’, Coalgebra claims to provide a compositional and uniform framework to spec ify, analyse and reason about state and behaviour in computing. This paper addresses this claim by discussing why Coalgebra matters for the design of models and logics for computational phenomena. To a great extent, in this domain one is interested in properties that are preserved along the system’s evolution, the so-called ‘business rules’ or system’s invariants, as well as in liveness requirements, stating that e.g. some desirable outcome will be eventually produced. Both classes are examples of modal assertions, i.e. properties that are to be interpreted across a transition system capturing the system’s dynamics. The relevance of modal reasoning in computing is witnessed by the fact that most university syllabi in the area include some incursion into modal logic, in particular in its temporal variants. The novelty is that, as it happens with the notions of transition, behaviour, or observational equivalence, modalities in Coalgebra acquire a shape . That is, they become parametric on whatever type of behaviour, and corresponding coinduction scheme, seems appropriate for addressing the problem at hand. In this context, the paper revisits Coalgebra from a computational perspective, focussing on three topics central to software design: how systems are modelled, how models are composed, and finally, how properties of their behaviours can be expressed and verified.Fuzziness, as a way to express imprecision, or uncertainty, in computation is an important feature in a number of current application scenarios: from hybrid systems interfacing with sensor networks with error boundaries, to knowledge bases collecting data from often non-coincident human experts. Their abstraction in e.g. fuzzy transition systems led to a number of mathematical structures to model this sort of systems and reason about them. This paper adds two more elements to this family: two modal logics, framed as institutions, to reason about fuzzy transition systems and the corresponding processes. This paves the way to the development, in the second part of the paper, of an associated theory of structured specification for fuzzy computational systems