PhD First-class Coinduction in Proof Assistants

Jobdescription

Essential to writing and reasoning about software that operates continuously is a practically usable language for programs and proofs of coinductive types. Proof assistants currently rely on either syntactic guardedness conditions on programs, sized types, or guarded recursion. However, the syntactic guardedness condition is very restrictive and brittle, while both sized types and guarded recursion require significant additional annotations to the program.

As a PhD student in this project, you will work towards developing a new system for defining and reasoning about coinductive processes in proof assistants. The key challenge is to develop a system that is both modular (so productivity information is propagated through the type system) and user-friendly (so few or no manual annotations are needed).

This project combines and builds on existing approaches proposed in the literature:

Sized types and guarded type theory, in particular their implementation in Agda (see A. Abel and B. Pientka: “Wellfounded recursion with copatterns: a unified approach to termination and productivity”, ICFP’13).

Coinductive reasoning in (higher) observational type theory and cubical type theory (see R. Atkey and C. McBride: “Productive coprogramming with guarded recursion”, ICFP’13).

In order to ensure soundness and usability, this project will require a combination of theoretical analysis of a type theory with first-class reasoning about coinductive processes and the development of a prototype implementation of this type theory as an extension to Agda.

This PhD position is part of the NWO XL project “Cyclic Structures in Programs and Proofs”, which has positions in Groningen, Leiden, Twente, and Nijmegen. As a PhD student in this project, you will collaborate with other researchers working within this project by attending the NetTCS workshop ( and through individual collaborations.For other vacancies:

Job requirements

We seek strong, highly motivated applicants who:

Have (or are close to completing) an MSc in Computer Science, Logic, Mathematics, or a related field.

Have background or research experience in any of the areas above (preferred but not required).

Have strong communication skills (oral and written) in English.

TU Delft (Delft University of Technology)

Delft University of Technology is built on strong foundations. As creators of the world-famous Dutch waterworks and pioneers in biotech, TU Delft is a top international university combining science, engineering and design. It delivers world class results in education, research and innovation to address challenges in the areas of energy, climate, mobility, health and digital society. For generations, our engineers have proven to be entrepreneurial problem-solvers, both in business and in a social context.

At TU Delft we embrace diversity as one of our core values and we actively engage to be a university where you feel at home and can flourish. We value different perspectives and qualities. We believe this makes our work more innovative, the TU Delft community more vibrant and the world more just. Together, we imagine, invent and create solutions using technology to have a positive impact on a global scale. That is why we invite you to apply. Your application will receive fair consideration.

Challenge. Change. Impact!

Faculty Electrical Engineering, Mathematics and Computer Science

The Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) brings together three scientific disciplines. Combined, they reinforce each other and are the driving force behind the technology we all use in our daily lives. Technology such as the electricity grid, which our faculty is helping to make completely sustainable and future-proof. At the same time, we are developing the chips and sensors of the future, whilst also setting the foundations for the software technologies to run on this new generation of equipment – which of course includes AI. Meanwhile we are pushing the limits of applied mathematics, for example mapping out disease processes using single cell data, and using mathematics to simulate gigantic ash plumes after a volcanic eruption. In other words: there is plenty of room at the faculty for ground-breaking research. We educate innovative engineers and have excellent labs and facilities that underline our strong international position. In total, more than 1000 employees and 4,000 students work and study in this innovative environment.

Click here to go to the website of the Faculty of Electrical Engineering, Mathematics and Computer Science.

Conditions of employment

Doctoral candidates will be offered a 4-year period of employment in principle, but in the form of 2 employment contracts. An initial 1,5 year contract with an official go/no go progress assessment within 15 months. Followed by an additional contract for the remaining 2,5 years assuming everything goes well and performance requirements are met.

Salary and benefits are in accordance with the Collective Labour Agreement for Dutch Universities, increasing from € 2901 per month in the first year to € 3707 in the fourth year. As a PhD candidate you will be enrolled in the TU Delft Graduate School. The TU Delft Graduate School provides an inspiring research environment with an excellent team of supervisors, academic staff and a mentor. The Doctoral Education Programme is aimed at developing your transferable, discipline-related and research skills.

The TU Delft offers a customisable compensation package, discounts on health insurance, and a monthly work costs contribution. Flexible work schedules can be arranged.

Will you need to relocate to the Netherlands for this job? TU Delft is committed to make your move as smooth as possible! The HR unit, Coming to Delft Service, offers information on their website to help you prepare your relocation. In addition, Coming to Delft Service organises events to help you settle in the Netherlands, and expand your (social) network in Delft. A Dual Career Programme is available, to support your accompanying partner with their job search in the Netherlands.

Additional information

If you would like more information about this vacancy or the selection procedure, please contact Jesper Cockx, via .

Application procedure

Are you interested in this vacancy? Please apply no later than June 29, 2025, via the application button and upload the following documents, al in PDF format:

CV.

A motivation letter (max 1 page A4), in which you motivate why you would be a good fit for our consortium and the selected position(s).

Your MSc thesis, if available; otherwise, another sample of research, such as your BSc thesis.

Grade transcripts of BSc and MSc studies so far.

Moreover, you are asked to provide names and contacts of two academic references.

You can address your application to Jesper Cockx.

Doing a PhD at TU Delft requires English proficiency at a certain level to ensure that the candidate is able to communicate and interact well, participate in English-taught Doctoral Education courses, and write scientific articles and a final thesis. For more details please check the Graduate Schools Admission Requirements.

Please note:

You can apply online. We will not process applications sent by email and/or post.

A pre-employment screening can be part of the selection procedure.

For the final candidates, a knowledge security check will be part of the application procedure. For more information on this check, please consult Chapter 8 of the National Knowledge Security Guidelines. We carry out this check on the basis of legitimate interest.

Please do not contact us for unsolicited services.