Towards an infinitary logic of domains : Abramsky logic for transition systems

We give a new characterization of sober spaces in terms of their completely distributive lattice of saturated sets. This characterization is used to extend Abramsky's results about a domain logic for transition systems. The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFP-domain obtained as a solution of a recursive domain equation. We prove that the Lindenbaum algebra generated by the infinitary logic is a completely distributive lattice dual to the same SFP-domain. As a consequence soundness and completeness of the infinitary logic is obtained for a class of transition systems that is computational interesting

Coalgebraic semantics of heavy-weighted automata

We study heavy-weighted automata, a generalization of weighted automata in which the weights of the transitions can be any formal power series. We define their semantics in three equivalent ways, and give some examples of how they can provide a more compact representation of certain power series than ordinary weighted automata

Coalgebraic semantics of heavy-weighted automata

We study heavy-weighted automata, a generalization of weighted automata in which the weights of the transitions can be any formal power series. We define their semantics in three equivalent ways, and give some examples of how they can provide a more compact representation of certain power series than ordinary weighted automata

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

Context-free languages, coalgebraically

We give a coalgebraic account of context-free languages using the functor ${\cal D}(X) = 2 \times X^A$ for deterministic automata over an alphabet $A$, in three different but equivalent ways: (i) by viewing context-free grammars as ${\cal D}$-coalgebras; (ii) by defining a format for behavioural differential equations (w.r.t. ${\cal D}$) for which the unique solutions are precisely the context-free languages; and (iii) as the ${\cal D}$-coalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra of all languages, thus paving the way for coinductive proofs of context-free language equivalence. Furthermore, the three characterizations are elementary to the extent that they can serve as the basis for the definition of a general coalgebraic notion of context-freeness, which we see as the ultimate long-term goal of the present study

Coalgebraic semantics of heavy-weighted automata

We study heavy-weighted automata, a generalization of weighted automata in which the weights of the transitions can be any formal power series. We define their semantics in three equivalent ways, and give some examples of how they can provide a more compact representation of certain power series than ordinary weighted automata

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