Weak Pullbacks Research Articles

A complete logic for behavioural equivalence in coalgebras of finitary set functors

This paper presents a sound and complete sequent-style deduction system for determining behavioural equivalence in coalgebras of finitary set functors preserving weak pullbacks. We also prove soundness without the weak pullback requirement. Finitary set functors are investigated because they are quotients of polynomial functors: the polynomial functor provides a ready-made signature and the quotient provides necessary additional axioms. We also show that certain operations on functors can be expressed with uniform changes to the presentations of the input functors, making this system compositional for a range of widely-studied classes of functors, including the Kripke polynomial functors. Our system has roots in the FLR0 proof system of Moschovakis et al., particularly as used by Moss, Wennstrom, and Whitney for non-wellfounded sets. Similarities can also be drawn to expression calculi in the style of Bonsangue, Rutten, and Silva.

Positive fragments of coalgebraic logics

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories $\mathsf{Set}$ of sets and $\mathsf{Pos}$ of posets: Every functor $\mathsf{Set} \to \mathsf{Pos}$ has a $\mathsf{Pos}$-enriched left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$. Functors arising in this way are said to have a presentation in discrete arities. In the case that $\mathsf{Set} \to \mathsf{Pos}$ is actually $\mathsf{Set}$-valued, we call the corresponding left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$ its posetification. A $\mathsf{Set}$-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A $\mathsf{Pos}$-functor with a presentation in discrete arities preserves surjections. The inclusion $\mathsf{Set} \to \mathsf{Pos}$ is dense. A functor $\mathsf{Pos} \to \mathsf{Pos}$ has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets'. A functor $\mathsf{Pos} \to \mathsf{Pos}$ is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves $\mathsf{Pos}$-enriched sifted colimits.

Lax extensions of coalgebra functors and their logic

We discuss the use of relation lifting in the theory of set-based coalgebra and coalgebraic logic. On the one hand we prove that the neighborhood functor does not extend to a relation lifting of which the associated notion of bisimilarity coincides with behavioral equivalence. On the other hand we argue that relation liftings may be of use for many other functors that do not preserve weak pullbacks, such as the monotone neighborhood functor. We prove that for any relation lifting L that is a lax extension extending the coalgebra functor T and preserving diagonal relations, L-bisimilarity captures behavioral equivalence. We also show that a finitary T admits such an extension iff it has a separating set of finitary monotone predicate liftings. Finally, we present the coalgebraic logic, based on a cover modality, for an arbitrary lax extension.

Coalgebras for Fuzzy Transition Systems

This paper studies a coalgebraic theory of fuzzy transition systems. Main conclusions include: the functor FA for deterministic fuzzy transition systems and the functor (P∘F)A for nondeterministic fuzzy transition systems preserve weak pullbacks, and the functor FA has a final coalgebra under some restricted conditions. Moreover, we show how to get a concrete (fuzzy) bisimulation from a coalgebraic bisimulation.

Unifying Exact Completions

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category of exact categories to existential elementary doctrines has a left biadjoint that can be obtained as a composite of two others. Finally, we conclude how this notion encompasses both that of the exact completion of a regular category as well as that of the exact completion of a category with binary products, a weak terminal object and weak pullbacks.

Relation lifting, with an application to the many-valued cover modality

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad" on categories, one is the preservation by T of "exactness" of certain squares. Both characterisations are generalisations of the "classical" results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks. The results presented in this paper enable us to compute predicate liftings of endofunctors of, for example, generalised (ultra)metric spaces. We illustrate this by studying the coalgebraic cover modality in this setting.

Completeness for the coalgebraic cover modality

We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation. As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ Pattinson's stratification method, showing that our derivation system can be stratified in countably many layers, corresponding to the modal depth of the formulas involved. In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion of relation lifting, induced by an arbitrary but fixed set functor. We introduce a category of Boolean algebra presentations, and establish an adjunction between it and the category of Boolean algebras. Given the fact that our derivation system involves only formulas of depth one, it can be encoded as a endo-functor on Boolean algebras. We show that this functor is finitary and preserves embeddings, and we prove that the Lindenbaum-Tarski algebra of our logic can be identified with the initial algebra for this functor.

On monotone modalities and adjointness

We fix a logical connection (Stone ˧ Pred : Setop → BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T: Set → Set that is finitary and standard and preserves weak pullbacks and finite sets. We prove that for any such T, the cover modality nabla is a left (and its dual delta is a right) adjoint relative to ω. We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to ω.

Neighbourhood Structures: Bisimilarity and Basic Model Theory

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2^2. We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2^2-bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2^2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 2^2-bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2^2-bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem's characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.

Vietoris Bisimulations

Building on the fact that descriptive frames are coalgebras for the Vietoris functor on the category of Stone spaces, we introduce and study the concept of a Vietoris bisimulation between two descriptive modal models, together with the associated notion of bisimilarity. We prove that our notion of bisimilarity, which is defined in terms of relation lifting, coincides with Kripke bisimilarity (with respect to the underlying Kripke models), with behavioural equivalence, and with modal equivalence, but not with Aczel–Mendler bisimilarity. As a corollary, we obtain that the Vietoris functor does not preserve weak pullbacks. Comparing Vietoris bisimulations between descriptive models to Kripke bisimulations on the underlying Kripke models, we prove that the closure of such a Kripke bisimulation is a Vietoris bisimulation. As a corollary, we show that the collection of Vietoris bisimulations between two descriptive models forms a complete lattice. Finally, we provide a game-theoretic characterization of Vietoris bisimilarity.

Coalgebraic Automata Theory: Basic Results

We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.

Automata and fixed point logic: A coalgebraic perspective

This paper generalizes existing connections between automata and logic to a coalgebraic abstraction level. Let F: Set to Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally, we show that the two approaches are equivalent in expressive power. We prove that any coalgebraic fixed point formula can be transformed into an F-automaton that accepts precisely those pointed F-coalgebras in which the formula holds. And conversely, we prove that any F-automaton can be converted into an equivalent fixed point formula that characterizes the pointed F-coalgebras accepted by the automaton.

A Coalgebraic Perspective on Monotone Modal Logic

The paper has two main parts: First we make the connection between monotone modal logic and the general theory of coalgebras precise by defining functors UpP:Set→Set and UpV:Stone→Stone such that UpP- and UpV-coalgebras correspond to monotone neighbourhood frames and descriptive general monotone frames, respectively. Then we investigate the relationship between the coalgebraic notions of equivalence and monotone bisimulation. In particular, we show that the UpP-functor does not preserve weak pullbacks, and we prove interpolation for a number of monotone modal logics using results on UpP-bisimulations.

Automata and Fixed Point Logics for Coalgebras

It is the aim of this paper to generalize existing connections between automata and logic to a more general, coalgebraic level.Let F:Set→Set be a standard functor that preserves weak pullbacks. We introduce the notion of an F-automaton, a device that operates on pointed F-coalgebras; the criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game.We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an F-automaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed F-coalgebras in which p holds.

Towards coalgebraic behaviourism

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra.In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F-coalgebras coincides with having the same observable behaviour. We show that such functors have the final coalgebra of a rather simple nature, and preserve some weak pullbacks. We also show that this is the case if and only if F-bisimilarity corresponds to logical equivalence in the finitary fragment of the coalgebraic logic.In the second part of the paper, we present a construction of a “final” coalgebra that captures the observable behaviour of F-coalgebras. We keep the word “final” quoted since the object we are going to construct need not belong to the original category. The construction is carried out for arbitrary Set-endofunctor F, throughout the construction we remain in Set, but the price to pay is the introduction of new morphisms. The paper concludes with a hint to a possible application to modelling weak bisimilarity for coalgebras.

Operators on classes of coalgebras

A class K of T-coalgebras is called a covariety if K=SHΣ( K) . SHΣ is just one of several class operators which can be formed by composing H, S and Σ. In this paper, we show that starting from H, S and Σ one can form exactly 13 different class operators (including the operator I of taking the isomorphic copies). We first describe the partially ordered monoid generated by these three operators for the class of all functors Set→ Set , then for the class of all functors preserving weak pullbacks along injective mappings, and finally, for particular functors from a rather large class which includes all non-constant polynomial functors.

Functors for Coalgebras

Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor \( {\cal F} \) which associates a set X with the set \( {\cal F} \)(X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as \( {\cal F} \)-coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated \( {\cal F} \)-coalgebras need not exist.

Relators and Metric Bisimulations: (Extended Abstract)

Coalgebras of set functors preserving weak pullbacks are particularly well-behaved. Invoking a result by Carboni, Kelly, and Wood (1990), we show that this can be explained by the fact that such functors can be uniquely extended to a relator. This insight next suggests a definition of metric bisimulation.

Toposes of Coalgebras and Hidden Algebras

There is a long tradition in computer science of modelling finite data types such as stacks and natural numbers by algebras. More recently coalgebras, which are dual to algebras, have been used to model infinite data types and in operational semantics.In this report we consider categories of coalgebras from the perspective of topos theory. A mild generalisation of a well known theorem about toposes of coalgebras of a comonad is given. This result is used to prove that if B is a pullback preserving functor on a topos ∊, and if the forgetful functor UB : ∊B → ∊ has a right adjoint then ∊B, the category of B-coalgebras, is itself a topos.We also show that, if B is a bounded functor on Set preserving weak pullbacks, then the category SetB of B-coalgebras has a subobject classifier and is a reflective full subcategory of a Grothendieck topos. We mention an elementary construction of this topos, pointed out to us by [12], as the reflection of SetB in the category of effective regular categories.We are particularly interested in hidden algebra, a formalism in the algebraic specification tradition with close links to coalgebra, capturing notions of state and behavioural equivalence. For a hidden destructor signature Σ we exhibit the topos structure of the category of hidden Σ-algebras.