# Post doc

# DafPUP - Domain Approximations for Programs with Uncertain Probabilities.

**Axis :**SciLex - Tâche 2

**Coordinateurs :**Jean Goubault-Larrecq

**Candidate :**Jia, Xiaodong

**Institutions :**

**Administrator laboratory :**LSV

**Adossé à l'action DigiCosme :**

**GT SHy**

**Durée & Dates de la mission :**1 an - avril 2018 - mars 2019

**Context :**

The starting point of this proposal has its origins in the paper static analysis of programs with imprecise inputs, by Adjé et al., where a static analysis framework is proposed for numerical programs with inputs given as imprecise probabilities.

Such programs form an important family of critical mission programs, including controllers in planes and trains. The imprecise probabilistic inputs are typically given by sensors, whose value is probabilistic, but obeys partially known probability distributions. For example, it might be known that a temperature sensor returns a Gaussian distributed value whose average is the outside temperature T, with standard deviation +/- 0.5 degrees C, but the outside temperature T itself is only known to be between -55 degrees and -45 degrees when the plane reaches its cruise altitude.

The above paper rests on a formal concrete semantics based on a measure-theoretic approach, where values of sensors, hence also of program variables, are certain convex sets of probability distributions encoded as so-called measure-theoretic previsions. One would like to have a domain-theoretic explanation of those mathematical objects. That would open the way to natural methods of approximation.

**Objectives :**

Perhaps the best way to state the objective of this project is in the form of questions:

- Can one find a measure-theoretic semantics of uncertain probabilities as a form of limit of domain-theoretic semantics of uncertain probabilities?
- Can one find measurable maps from X to R as some form of limit of lower semicontinuous maps, perhaps involving computational models in the style of Lawson, Martin, and others?

See the proposal for more details.

**Scientific production :**

Xiaodong Jia has arrived, and has been working on the project since May 1st, 2018.