## Projects for Students

We offer projects suitable for

• paid work as “studentische Hilfskraft” (HiWi) with flexible conditions;
• bachelor theses;
• master theses.

The following projects are currently available.

### Verification of approximation algorithms for graphs

Many basic graph theoretic problems are either NP-hard or are not known to be solved in better than impractical polynomial times. This makes solving those problems prohibitive if not impossible for real-world graphs. Approximation algorithms circumvent that by using less resources than exact algorithms, at the expense of providing only approximate solutions. In this project the student would formally verify that 1) the approximate solutions of those algorithms meet a certain quality guarantee and 2) the upper bounds on their runtimes are correct. Particularly interesting algorithms are for computing lower-bounds on (directed) graph diameters and approximate solutions to the All-Pairs-Shortest-Paths problem. Some of those algorithms are deterministic, and others have elements of randomness in them. A randomised algorithm would be more interesting from a verification perspective.

### Verified state space based classical planning

State space search is a successful technique to solve classical AI planning problems, i.e. deterministic problems with finite states. The main task of this project is to (partially) build a formally verified A* based planning algorithm. Tasks in this project include

• formalising existing heuristics for classical planning in Isabelle/HOL and verifying that they are admissible. There are many such heuristics, and they are reviewed in sections 2.3 and 2.7.9 in a book by Gallab et al. [1]. A first step is to verify abstraction based heuristics like pattern databases and/or merge-and-shrink.
• integrating an existing Isabelle formalisation of A* with 1) already formalised semantics of planning problems to obtain a version of A* that operates on planning state spaces and 2) planning heuristics that are verified.
• refining the integrated A* to executable code.

### Verified planning under uncertainty (i.e. (PO)MDP planning)

AI planning is the discipline aiming at building agents (i.e. computer programs) that act to achieve a certain goal, given a declarative description of the environment and the actions available to it. In practical applications, like autonomous vehicles driving, it is not possible to model the exact effects of actions. Planning in settings with uncertainty has been explored in the areas of AI, control, and operations research. In those settings the environment is commonly modelled as a factored Markov decision process (MDP) or a partially observable MDP. Tasks in this project include

• extending the existing Isabelle formalisation of MDPs to include the notion of agent rewards
• formalising the notion of (optimal) strategies, verifying algorithms that solve these problems, like policy/value iteration
• refining the verified algorithm to executable code using the existing Isabelle refinement framework.

### Group Theory

The purpose of this project is to add various missing bits of group theory to HOL-Algebra, with a particular focus on finite groups. Topics of interest include:

• Decomposition of a group into an interior direct product
• Indexed direct products
• The fundamental theorem of finite(ly generated) abelian groups
• Finite simple groups

### Analytic Number Theory

There is an ongoing effort to formalise Tom Apostol's classic textbook on Analytic Number Theory. Most of the core content has already been formalised, but there are some chapters that have been left out completely, e.g. Number Partitions.

The goal of the practical course work would be to formalise as much as possible of this material.

### Primality Testing

There are many different tests to determine if a natural number is prime. Most of these are *probabilistic* and always recognise a prime number as prime, but, with some probability, also mistakenly classify composites as primes. Some of these tests (Fermat, Miller–Rabin, Solovay–Strassen) are already formalised in Isabelle.

In this project, more primality tests are to be formalised, e.g.:

• Frobenius test (randomised)
• Lucas and Fibonacci pseudoprimes (randomised)
• Lucas–Lehmer test for Mersenne primes (deterministic)
• AKS test (deterministic)

### Specialised Mathematical Tactics and Simprocs

The goal is to implement various additional tactics and/or simplification procedures ('simprocs') to simplify/expand certain mathematical terms that are difficult to simplify using simp rules (e.g.: proving/disproving/rewriting statements like "4/3 ∈ ℤ", k-th integer root and k-th power testing ,...). Some of these can be done using externally-computed certificates (e.g.: primality: Pratt certificate, prime power: prime factor (with Pratt certificate) and multiplicity, compositeness: two non-trivial factors, ...)

Proofs of concepts for some of these already exist (e.g. in the Pratt_Certificate and Bertrands_Postulate AFP entries). The goal would be to make a bigger collection of fast and robust tactics/simprocs.

### Automatic Laurent Series Expansions

Sufficiently nice complex-valued functions admit Laurent series expansions, which can be computed symbolically. This is an important feature in Computer Algebra Systems like Mathematica that can be used to e.g. compute limits and complex residues. In this Master's thesis project, a procedure is developed that computes such Laurent series inside the proof assistant Isabelle/HOL to aid the user by automatically proving limits, residues, and reasoning about poles.

### Ein Testrahmen für funktionale Datenstrukturen

Es soll ein Testrahmen implementiert und ausgewertet werden, um die Effizienz funkionaler Datenstrukturen messen zu können. Dabei soll das Lastprofil, also der Mix der verschiedenen Operationen, flexibel spezifizierbar sein. Sodann sollen eine Reihe effizienter funktionaler Datenstrukturen für Suchbäume und Prioritätswarteschlangen implementiert und mit Hilfe dieses Testrahmens auf ihre Effizienz hin untersucht werden.

### Efficient Algorithms and Data Structures in Isabelle/HOL

The goal of this project is to formalize some algorithm (e.g. the blossom algorithm for maximal cardinality matching, algorithm for closest pair of points,...) or efficient data structure in Isabelle/HOL and use refinement techniques to refine the algorithm down to efficiently executable code.

### Numerical Proof Procedures in Isabelle/HOL

The Isabelle theorem prover can prove variable-free arithmetic statements automatically, for example arctan 1.5 < 1. The proof procedure uses Taylor polynomials and interval arithmetic. It should be extended to support more operations or make use of IEEE floating point numbers.

### Efficient Functional Data Structures

Mittels Codegenerierung können aus Spezifikationen unmittelbar Prototypen in einer funktionalen Programmiersprache erzeugt werden, z.B. in ML, in Haskell, oder in (einer Teilmenge von) Scala. Um die Effizienz dieser Prototypen zu steigern, sind effiziente Datenstrukturen erforderlich. Bekannte Beispiele hierfür sind balancierende Suchbäume (z.B. 2-3-Bäume); es gibt aber noch viel mehr Kandidaten. Eine Auswahl hiervon soll als Isabelle-Theorie implementiert und gegen eine naive Referenzimplementierung verifiziert werden.