Fredrik Dahlqvist

Giry and the Machine

Abstract We present a general method – the Machine – to analyse and characterise in finitary terms natural transformations between well-known functors in the category Pol of Polish spaces. The method relies on a detailed analysis of the structure of Pol and a small set of categorical conditions on the domain and codomain functors. We apply the Machine to transformations from the Giry and positive measures functors to combinations of the Vietoris, multiset, Giry and positive measures functors. The multiset functor is shown to be defined in Pol and its properties established. We also show that for some combinations of these functors, there cannot exist more than one natural transformation between the functors, in particular the Giry monad has no natural transformations to itself apart from the identity. Finally we show how the Dirichlet and Poisson processes can be constructed with the Machine.

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-via-canonicity. Specifically, we use the topological theory of canonical (in) equations in distributive lattice expansions to show that distributive substructural logics are strongly complete with respect to their relational semantics. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions.

Robustly Parameterised Higher-Order Probabilistic Models

We present a method for constructing robustly parameterised families of higher-order probabilistic models. Parameter spaces and models are represented by certain classes of functors in the category of Polish spaces. Maps from parameter spaces to models (parameterisations) are continuous and natural transformations between such functors. Naturality ensures that parameterised models are invariant by change of granularity -- ie that parameterisations are intrinsic. Continuity ensures that models are robust with respect to their parameterisation. Our method allows one to build models from a set of basic functors among which the Giry probabilistic functor, spaces of cadlag trajectories (in continuous and discrete time), multisets and compact powersets. These functors can be combined by guarded composition, product and coproduct. Parameter spaces range over the polynomial closure of Giry-like functors. Thus we obtain a class of robust parameterised models which includes the Dirichlet process, various point processes (random sequences with values in Polish spaces) and other classical objects of probability theory. By extending techniques developed in prior work, we show how to reduce the questions of existence, uniqueness, naturality, and continuity of a parameterised model to combinatorial questions only involving finite spaces.

Bayesian Inversion by Omega-Complete Cone Duality (Invited Paper)

The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. In this paper, we investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings, or simply kernels) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators: For the direct option, we introduce a typed version of the category of Markov kernels and use the so-called ‘disintegration of measures’. Here, one has to specialise to measurable spaces borne from a simple class of topological spaces -e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development in Ref. [4]. For the operator option, we use a cone version of the category of Markov operators (kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being ω-chain-continuous.1 Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of L1and L∞-cones [3]. Inversion, seen through the operator prism, is just adjunction.2 No topological assumption is needed. We show that both categories (Markov kernels and ω-chain-continuous Markov operators) are related by a family of contravariant functors Tp for 1 ≤ p ≤ ∞. The Tp’s are Kleisli extensions of (duals of) conditional expectation functors introduced in Ref. [3]. With this bridge in place, we can prove that both notions of inversion agree when both defined: if f is a kernel, and f† its direct inverse, then T∞(f) = T1(f). 1998 ACM Subject Classification Semantics of programming languages

Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.

Coalgebraic Completeness-via-Canonicity

We present the technique of completeness-via-canonicity in a coalgebraic setting and apply it to both positive and boolean coalgebraic logics with relational semantics.