Coalgebras are categorical presentations of state-based systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final coalgebra as an object in it. This phenomenon is what Baez and Dolan have called the microcosm principle, a prototypical example of which is “a monoid in a monoidal category.” In this paper we obtain a formalization of the microcosm principle in which such a nested model is expressed categorically as a suitable lax natural transformation. An application of this account is a general compositionality result which supports modular verification of complex systems.

Abstract. We present four coinductive operational semantics for the While language accounting for both terminating and non-terminating program runs: big-step and small-step relational semantics and big-step and small-step functional semantics. The semantics employ traces (possibly infinite sequences of states) to record the states that program runs go through. The relational semantics relate statement-state pairs to traces, whereas the functional semantics return traces for statement-state pairs. All four semantics are equivalent. We formalize the semantics and their equivalence proofs in the constructive setting of Coq. 1

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 (∼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. Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2010.272 1

Abstract. In search for a foundational framework for reasoning about observable behavior of programs that may not terminate, we have previously devised a trace-based big-step semantics for While. In this semantics, both traces and evaluation (relating initial states of program runs to traces they produce) are defined coinductively. On terminating runs, it agrees with the standard inductive state-based semantics. Here we present a Hoare logic counterpart of our coinductive trace-based semantics and prove it sound and complete. Our logic subsumes both the partial correctness Hoare logic and the total correctness Hoare logic: they are embeddable. Since we work with a constructive underlying logic, the range of expressible program properties has a rich structure; in particular, we can distinguish between termination and nondivergence, e.g., unbounded total search fails to be terminating but is nonetheless nondivergent. Our metatheory is entirely constructive as well, and we have formalized it in Coq. 1

Abstract. A theory of traces of computations has emerged within the field of coalgebra, via finality in Kleisli categories. In concurrency theory, traces are traditionally obtained from executions, by projecting away states. These traces and executions are sequences and will be called “thin”. The coalgebraic approach gives rise to both “thin ” and “fat” traces/executions, where in the “fat ” case the structure of computations is preserved. This distinction between thin and fat will be introduced first. It is needed for a theory of schedulers in a coalgebraic setting, of which we only present the very basic definitions and results. 1

We develop a representation theory in which a point of a fractal specified by metric means (by a variant of an iterated function system, IFS) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; an IFS-like metric specification of a fractal is an algebra for the same functor. Relating the two there canonically arises a representation map, much like in America and Rutten’s use of metric enrichment in denotational semantics. A distinctive feature of our framework is that the canonical representation map is bijective. In the technical development, gluing of shapes in a fractal specification is a major challenge. On the metric side we introduce the notion of injective IFS to be used in place of conventional IFSs. On the symbolic side we employ Leinster’s presheaf framework that uniformly addresses necessary identification of streams—such as.0111... =.1000... in the binary expansion of real numbers. Our leading example is the unit interval I = [0, 1].

This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.

Abstract. This paper contributes to the theory of coalgebraic, statebased modelling of components via two additions: a feedback operator in the form of a monoidal trace, and a three-dimensional string calculus for representing and manipulating composite component diagrams. The feedback operator on components is shown to satisfy the trace axioms by Joyal, Street and Verity. As a corollary, we appeal to the microcosm principle and derive a canonical traced monoidal structure on the category of resumptions. This generalises an observation by Abramsky, Haghverdi and Scott. 1

Abstract. This paper extends bialgebraic semantics [1, 2] to take into account notions of behaviour that lead to process equivalences coarser than bisimulation. For that purpose, the requirement of finality for the characterisation of behaviours is relaxed to quasi-finality, which informally consists in relegating to the underlying category the conditions that finality must satisfy in the main category of coalgebras. This setting is then applied to the failure semantics of CSP. 1

In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effect-free coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributing program syntax over effects, again inducing a coalgebraic operational semantics, but this time in the Kleisli category for the monad derived from the algebraic effects. The final coalgebra in this Kleisli category then serves as the denotational model. For it to exist, we ensure that the the Kleisli category is enriched over CPOs by considering the monad of possibly infinite terms, extended with a bottom element. Unlike the effectless setting, not all operational specifications give rise to adequate and compositional semantics. We give a proof of adequacy and compositionality provided the specifications can be described by evaluation-in-context. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.