University of Oxford Department of Computer Science

Please feel free to get in touch, if you would like to discuss any of the topics listed below or any other project ideas related to automata, lambda calculi, programming languages or verification. The project topics listed below are not set in stone and can be tailored to individual interests. I have grouped them to indicate (roughly) their relationship to taught courses.

**Models of Computation**: DPDA Equivalence, Lambda Equations, Security Protocols, Syntactic Control of Interference**Lambda Calculus**: Contextual Equivalence, Lambda Equations, Logical Relations, Syntactic Control of Interference**Programming Languages**: Contextual Equivalence, Logical Relations, Syntactic Control of Interference**Verification**: Logical Relations, Security Protocols, Syntactic Control of Interference

βη-equivalence is the main notion of equivalence studied in courses on the lambda calculus.
However, from the point of view of programming languages, another notion of equivalence is more relevant:
*contextual equivalence*.
Intuitively, two terms are contextually equivalent if they behave in the same way in any context.
The purpose of the project is to design and implement a contextual equivalence checker for the simply-typed lambda calculus
by combining ideas from various research papers as well as adding your own.

Enthusiasm for the lambda calculus and strong proof skills are essential.

- H. Barendregt, W. Dekkers, R. Statman. Lambda Calculus with Types. Cambridge University Press, 2013.
- J. Laird. Bistability: An Extensional Characterization of Sequentiality. Proceedings of CSL, Springer, 2003.
- R. Loader. An algorithm for the minimal model. 1997.
- V. Padovani. Decidability of All Minimal Models (Revised Version - 2012). 2012.

Two pushdown automata (PDA) are equivalent if they accept the same language. The problem of deciding whether two given PDA are equivalent is known to be undecidable. However, it becomes decidable when the two automata are deterministic. It took 30 years to find an algorithm that, given two deterministic PDA, will be able to determine whether or not they are equivalent. The first one was given in 1997 by Geraud Senizergues, who was subsequently awarded the Goedel Prize for the result. Senizergues' original algorithm has since been considerably simplified by other researchers.

The primary objective of this project will be to understand the latest simplification due to Jancar (cited below) and implement the algorithm in several special cases to understand its complexity. The complexity of the general case is still an open problem, which would be more suitable for a DPhil project.

This is a challenging project focussed on a cutting-edge problem in theoretical computer science. An excellent grasp of automata theory and enthusiasm for exploring the latest techniques in the area are essential.

- Petr Jancar. Decidability of DPDA Language Equivalence via First-Order Grammars, 2018.

This project contains solving equations of the form F X = G (for given F, G) with respect to βη-equality. Such equations can be viewed as matching problems (up to βη). In restricted cases (types of low order), it is known how to solve them using tree automata. The purpose of the project is review existing decision procedures for the problem, optimise/simplify them as much as possible, and implement a solver.

Enthusiasm for the lambda calculus and automata.

- H. Barendregt, W. Dekkers, R. Statman. Lambda Calculus with Types. Cambridge University Press, 2013.
- H. Comon, Y. Jurski. Higher-Order Matching and Tree Automata. Proceedings of CSL, Springer, 1997.

Logical relations are an established technique for proving properties of lambda terms and, more broadly, programming languages involving higher-order types (not necessarily purely functional).
Among others, they can be used to show that the simply-typed lambda calculus is strongly
normalising or to support equivalence proofs for programs, such as
those needed to validate compiler transformations. Two programs are *contextually equivalent*
if and only if they can be used interchangeably without affecting the result.
The aim of the project is to explore the logical relations technique and automate some of its aspects.

This project is related to the lambda calculus and principles of programming languages. It will attempt to transfer techniques from computer-aided formal verification to the setting of higher-order programming languages.

- Lau Skorstengaard. An Introduction to Logical Relations (Proving Program Properties Using Logical Relations). Notes from Amal Ahmed's summer school course, 2015.
- A. M. Pitts and I.D.B. Stark. Operational Reasoning for Functions with Local State. In A.D. Gordon and A. M. Pitts (Eds), Higher Order Operational Techniques in Semantics, Publications of the Newton Institute (Cambridge University Press, 1998), pp 227-273.
- Derek Dreyer, Georg Neis, Lars Birkedal. The impact of higher-order state and control effects on local relational reasoning. Journal of Functional Programming 22(4-5): 477-528 (2012).

The aim of the project is to understand the state of the art in verifying security protocols, and to try to deploy automata-theoretic methods in the area. Standard automata theory is based on finite alphabets, which is not quite suitable in this context, so the project will explore automata models over infinite alphabets.

Good knowledge of automata theory and a liking for algorithm design.

- Rohit Chadha, Vincent Cheval, Stefan Ciobaca, and Steve Kremer. Automated verification of equivalence properties of cryptographic protocols. ACM Transactions on Computational Logic, 17(4), November 2016. Listed in ACM Computing Reviews' 21st Annual Best of Computing list of notable books and articles for 2016.
- F. Neven, T. Schwentick and V. Vianu. Finite state machines for strings over infinite alphabets. ACM Transactions on Computational Logic 5(3): 403-435 (2004)

The Syntactic Control of Interference (SCI) is a typing system for higher-order programs with state that eliminates covert interference by preventing sharing of variables between functions and their arguments. Reasoning about such programs is easier and decidable. The technical goal of the project is to understand how SCI programs are modelled using multi-tape deterministic finite state automata and to identify fragments that can be analyzed in an efficient way, possibly using simpler automata-theoretic formalisms.

Good knowledge of automata theory and a liking for type systems and algorithm design.

- James Laird. Decidability and syntactic control of interference. Theoretical Computer Science 394(1-2): 64-83 (2008)