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The following are some proposals for Master thesis topics which
I
am willing to supervise. **It is not necessary to address all
goals listed for a given topic.** Only one or some may be
sufficient for a master or a bachelor thesis, depending on which
ones and in how much depth they are treated. Contact me to talk
about the details: lukasz.czajka at tu-dortmund.de (can be in
German if you like).

Some paper links below require a subscription and are therefore accessible only through the university network.

Choose some of your favourite functional data structures (e.g. from [1]) and formalize them in Coq or Isabelle/HOL.

- C. Okasaki: Purely functional data structures

- Functional programming
- Coq or Isabelle/HOL

Hammers [2] are automated reasoning tools for proof assistants [1], which combine machine-learning with automated theorem proving. A typical use is to prove relatively simple goals using available lemmas. The problem is to find appropriate lemmas in a large collection of all accessible lemmas and combine them to prove the goal.

The machine-learning component of a hammer, called premise selection, tries to solve the following problem: given a large library of lemma statements together with their proofs, and a goal statement G, predict which lemmas are likely useful for proving G. For machine-learning purposes, proofs are usually reduced to a set of dependencies. A dependency of a lemma L is any lemma which is used in a proof of L. Hence, given a large dataset of lemma statements with their dependencies, we want to predict an over-approximation of the set of dependencies of a new statement.

CoqHammer [3,4] is a hammer tool for Coq -- a proof assistant based on dependent type theory. The goal of a thesis would be to improve premise selection in CoqHammer by adapting the work done for other proof assistants.

- Design a better set of features, basing on [5].
- Implement more machine-learning methods and compare them [2, Section 2].
- Investigate adapting a deep learning approach [6].

- H. Geuvers: Proof Assistants: History, Ideas and Future
- J. Blanchette, C. Kaliszyk, L. Paulson, J. Urban: Hammering towards QED
- CoqHammer
- Ł. Czajka, C. Kaliszyk: Hammer for Coq: Automation for Dependent Type Theory
- C. Kaliszyk, J. Urban, J. Vyskocil: Efficient Semantic Features for Automated Reasoning over Large Theories
- A. Alemi, F. Chollet, N. Elen, G. Irving, C. Szegedy, J. Urban: DeepMath - Deep Sequence Models for Premise Selection

- Machine learning
- Functional programming
- Coq and type theory (only superficial knowledge of Coq and type theory is necessary as a prerequisite, but deeper knowledge will help)

Proof reconstruction in CoqHammer [1,2] (see also the previous topic) uses proof-search tactics written in LTac (Coq's tactic language) to prove statements basing on the information returned by automated theorem provers (ATPs). Currently, the only information used is a list of dependencies. The goal of a thesis would be to adapt CoqHammer reconstruction tactics to take into account more information available such as the inversion, injection and discrimination axioms, typings, and the names of (co)inductive types used in ATP proofs. This will also require OCaml-side plugin programming, and possibly a modification of the CoqHammer translation component to include more axioms in the ATP input files (the presence of which axioms can later be used to guide proof search during proof reconstruction).

- CoqHammer
- Ł. Czajka, C. Kaliszyk: Hammer for Coq: Automation for Dependent Type Theory

- Coq and LTac
- Functional programming

The goal of a thesis would be to generalise slightly some results from the literature [1,2,3] on the decidability of certain fragments of intuitionistic first-order logic and implement decision procedures for these fragments. Also implement an extension of the Intuition prover [4], which searches for proofs in minimal first-order logic (i.e. the universal-implicational fragment of intuitionistic logic), to handle all connectives.

- Extend the result from [1] on EXPSPACE-completeness of the negative fragment of minimal first-order logic without function symbols to handle all connectives and function symbols with the restriction that the size of each individual term occurring in a proof must be bounded by a constant. Analogously, extend the result from [1] on the co-NEXPTIME-completeness of the arity-bounded negative fragment of minimal first-order logic without function symbols.
- Adapt the results of [7] to show EXPTIME-completeness of the Horn fragment (i.e. derivability of an atom from a set of Horn clauses) with restrictions as in the previous point.
- Implement the above decision procedures.
- Extend Intuition [4] to handle all connectives, basing on [5]. The paper [5] and the current procedure of Intuition are based on an extension of the automata-theoretic algorithm from [6].
- Implement a decision procedure for the positive fragment [2,3]. This goal is hard and could constitute a Master thesis by itself if done in sufficient depth.

- A. Schubert, P. Urzyczyn, K. Zdanowski: On the Mints Hierarchy in First-Order Intuitionistic Logic
- G. Mints: Solvability of the problem of deducibility in LJ for a class of formulas not containing negative occurrences of quantifiers
- G. Dowek, Y. Jiang: Eigenvariables, bracketing and the decidability of positive minimal intuitionistic logic
- Intuition prover
- M. Zielenkiewicz, A. Schubert: Automata Theory Approach to Predicate Intuitionistic Logic
- A. Schubert, W. Dekkers, H. Barendregt: Automata Theoretic Account of Proof Search
- B. Düdder, M. Moritz, J. Rehof, P. Urzyczyn: Bounded Combinatory Logic

- Logic and type theory

The goal of a thesis would be to compare and/or implement the proof search procedures for Pure Type Systems (or the Lambda-Cube) from the papers [1,2,3]. Covering [3] is optional.

- G. Dowek: A Complete Proof Synthesis Method for the Cube of Type Systems
- S. Lengrand, R. Dyckhoff, J. McKinna: A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
- F. Gutierres, B. Ruiz: Cut Elimination in a Class of Sequent Calculi for Pure Type Systems

- Logic and type theory

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