### News

**The first exam will take place in room***5055*instead of AH6.**13.12.2016, regarding exercise 2: Since the weakest liberal precondition (wlp) calculus has not been introduced in the lecture yet, you can ignore part (a), (d), and (e) of task 4.**- The exercise class on 23.12.2016 is cancelled and rescheduled for
**09.01.2017 at 14:15**in room 9U10. - Registration for the written exam on Tuesday February 14, 2017 is
**open**. - The kick-off lecture takes place on
**Monday October 24, 2016.** - This is a condensed course (“
**Blockveranstaltung**“), taking place two lectures/week in the weeks 49, 50, 51, 2, 3 and 5.

### Schedule

Type |
Day |
Time |
Hall |
Start |
Lecturer |

Lecture | Mon | 16:15 – 17:45 | 9U10 | 24.10 | Katoen |

Tue | 10:15 – 11:45 | 9U10 | 06.12 | Katoen | |

Exercise | Fri | 14:15 – 15:45 | 9U10 | 09.12 | Olmedo, Kaminski, Matheja |

**Motivation and background**

Probabilistic programs support *random choices* like “execute program *P* with probability 1/3 and program *Q* with probability 2/3″. Probabilistic programs are typically normal–looking sequential programs describing posterior probability distributions. Describing randomized algorithms has been the classical application of these programs. Applications in biology, machine learning, quantum computing, security, and so on, have led to a rapidly growing interest in probabilistic programs in the last decade. For almost all programming languages, such as C, C#, Prolog, Haskell, Scala, LISP, a probabilistic variant does exist; and even a variant of Microsoft’s Excel has been recently developed, see the probabilistic-programming.org.

Most of these languages feature, in addition to sampling from probability distributions, the ability to *condition* values of variables in a program. Conditioning allows for adding information about observed events into the program that may influence the posterior distribution. It is one of the key features in Bayesian networks that rely on Bayes’ rule as the basis for updating information. It is this feature that distinguishes modern probabilistic programming languages from those in the early days describing randomized algorithms. Randomized quicksort is an example of such program:

What is special about probabilistic programs? Although probabilistic programs are normal–looking programs, reasoning about their correctness is intricate. They do not generate a single output but a *distribution over outputs*. The key property of program termination exemplifies this. Whereas a classical program terminates or not, this is no longer true for probabilistic programs. They can diverge, but this may happen with probability zero. That is, they may have non-terminating runs! In addition, in contrast to classical programs that either do not terminate at all or terminate in finitely many steps, a probabilistic program may take infinitely many steps on average to terminate, even if its termination probability is one.

Establishing correctness of probabilistic programs needs—even more so than ordinary programs—formal reasoning.

This lecture series will introduce the probabilistic programming language *webppl*, a functional language with a simple web interface. Subsequently, we will go into questions such as:

- What do probabilistic programs mean? Exactly.
- What does it mean for a program to not terminate with some likelihood?
- How to analyze that a probabilistic program does what it is supposed to do?
- How to prove that a probabilistic program is (more) efficient?

### Prerequisites

### Exercises

- Exercises can be worked on in groups of at most two students.
- To achieve a certificate to this course or to be admitted to the final exam, at least half of the exercises has to be reasonably dealt.
- The exercise sheets will be issued weekly, starting from December 6, 2017.

### Exam

- Exam admission: at least 50% of all points in the homework exercises need to be achieved.
- First written exam: Tuesday February 14, 2017, 10:00-13:00 (
~~AH6~~5055) - Second written exam: Monday March 20, 2017, 09:00-12:00 (AH3)

### Slides

No. | Date | Subject | Material | Exercise | Solution |
---|---|---|---|---|---|

1 | 24.10 | Introduction | PP_lecture1.pdf | ||

2 | 05.12 | WebPPL | webppl_examples | ex-01 | s-01 |

3 | 06.12 | Weakest preconditions for GCL | ex-02 | s-02 | |

4 | 12.12 | Weakest preconditions for probabilistic GCL | ex-03 | s-03 | |

5 | 13.12 | Operational semantics of pGCL | |||

6 | 19.12 | Axiomatic versus operational pGCL semantics | |||

7 | 20.12 | Conditioning: operationally | webppl_examples | ex-04 | s-04 |

8 | 09.01 | Conditioning: denotationally | removing_observe | ex-05 | s-05 |

9 | 10.01 | Bayesian networks | bayesian-networks | ||

10 | 16.01 | Termination | ex-06 | s-06 | |

11 | 17.01 | Termination | |||

12 | 30.01 | Nondeterminism | |||

13 | 31.01 | Expected run-time analysis | s-07 |

### Further information

- The lecture will be given in
**English**. All course material will be in English. - The students will be awarded
**4****ECTS**credits for the lecture after passing the final exam.

**Literature**

The course is based several research papers and material from several books:

- Annabelle McIver and Carroll Morgan — Abstraction, Refinement and Proof for Probabilistic Systems, Springer, 2005.
- Noah Goodman and Andreas Stuhlmüller (electronic) — The Design and Implementation of Probabilistic Programming Languages, 2016.