Ilias Garnier

Certified Complexity (CerCo)

We provide an overview of the FET-Open Project CerCo (‘Certified Complexity’). Our main achievement is the development of a technique for analysing non-functional properties of programs (time, space) at the source level with little or no loss of accuracy and a small trusted code base. The core component is a C compiler, verified in Matita, that produces an instrumented copy of the source code in addition to generating object code. This instrumentation exposes, and tracks precisely, the actual (non-asymptotic) computational cost of the input program at the source level. Untrusted invariant generators and trusted theorem provers may then be used to compute and certify the parametric execution time of the code.

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.

Dirichlet is Natural

Giry and Lawveres categorical treatment of probabilities, based on the probabilistic monad G, offer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process. This family is widely used in non-parametric Bayesian learning.Given a Polish space X, we build a family of higher-order probabilities in G(G(X)) indexed by M*(X) the set of non-zero finite measures over X. The construction relies on two ingredients. First, we develop a method to map a zero-dimensional Polish space X to a projective system of finite approximations, the limit of which is a zero-dimensional compactification of X. Second, we use a functorial version of Bochners probability extension theorem adapted to Polish spaces, where consistent systems of probabilities over a projective system give rise to an actual probability on the limit. These ingredients are combined with known combinatorial properties of Dirichlet processes on finite spaces to obtain the Dirichlet family DX on X. We prove that the family DX is a natural transformation from the monad M* to G?G over Polish spaces, which in particular is continuous in its parameters. This is an improvement on extant constructions of DX 17,26.

Computing Continuous-Time Markov Chains as Transformers of Unbounded Observables

The paper studies continuous-time Markov chains CTMCs as transformers of real-valued functions on their state space, considered as generalised predicates and called observables. Markov chains are assumed to take values in a countable state space

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

\mathbf {S}

Stochastic mechanics of graph rewriting

; observables

Borel Kernels and their Approximation, Categorically.

f: \mathbf {S} \rightarrow {\mathbb {R}}