Master’s Theses Topics in Mathematical Logic This document contains some examples of possible Master’s theses topics in different areas of logic. The purpose of these examples is to give the reader a flavour of the possible theses topics. The specific topic is always agreed on with the supervisor at the start of the project. It is recommended to complete the course Mathematical Logic before starting the Master’s thesis project. Topic specific prerequisities are mentioned separately below. FINITE MODEL THEORY AND LOGIC IN COMPUTER SCIENCE Consistent Query Answering • Topic: Inconsistency in databases arises in many applications such as data integration and data warehousing. A critical question is to determine how to maintain and query such data. In consistent query answering the approach is to keep all inconsistencies in data and return only those query answers that belong to the outcome of the query no matter how consistency is restored in the database. Determining the complexity of this task in various situations is a timely topic in database theory. • Possible supervisors: Miika Hannula • Material: Jef Wijsen. A survey of the data complexity of consis- tent query answering under key constraints. In Christoph Beierle and Carlo Meghini, editors, Foundations of Information and Knowl- edge Systems, pages 62–78, Cham, 2014. Springer International Publishing. • Prerequisities: Finite model theory or Complexity Theory. Decidable fragments of first-order logic • Topic: The validity problem asks, given a formula F, whether F is a theorem. In other words, the question is whether F is true in all models. This is clearly a highly fundamental problem, as solving it amounts to fully automating logical reasoning. The un- decidability of the validity problem of first-order logic was one of the most significant findings of 20th century logic and mathematics in general. While undecidability was bad news, things obviously did not stop there. In the 80s, a complete classification of the de- cidability/undecidability question was obtained for prefix classes of first-order logic. Modern theory on the topic investigates various further fragments where decidability can be achieved. There are many relevant and timely questions related to this active research 1 2 field, ranging from very easy and accessible to more challenging ones. • Possible supervisors: Antti Kuusisto • Material (1) Egon Borger, Erich Grädel, Yuri Gurevich: The Classical De- cision Problem. Perspectives in Mathematical Logic, Springer 1997. This is an authoritative and comprehensive exposition of the topic. However, it is lengthy and also already somewhat outdated. (2) Emanuel Kieronski, Antti Kuusisto: Complexity and Expres- sivity of Uniform One-Dimensional Fragment with Equality. MFCS (1) 2014: 365-376. This is an example of contemporary work. (It is probably best to read the extended open access version on arXiv: https://arxiv.org/pdf/1409.0731.pdf) (3) Antti Kuusisto: On the Uniform One-dimensional Fragment. DL 2016. The introduction of this paper gives a very brief overview of some of the most active current research efforts. Thus the paper can serve as quick intro to the topic in general. • Prerequisities: Complexity Theory. Descriptive complexity of counting • Topic: The complexity of arithmetic computations is a current focal topic in complexity theory the most prominent complexity class be- ing #P of all functions that count accepting paths of nondeterministic polynomial-time Turing machines. This class has interesting com- plete problems like counting the number of satisfying assignments of propositional formulae. The class #P has also a logical characteriza- tion that is a natural generalization of the famous Fagin’s Theorem: Given a first-order formula with a free relational variable, instead of asking if there exists an assignment to this variable that makes the formula true, we now ask to count how many such assignments there are. In this way, the class #P corresponds exactly to first-order logic. The study of the connections between logic and the complexity of counting is a timely topic for a master’s thesis with a short distance to open research questions. • Possible supervisors: Juha Kontinen, Antti Kuusisto • Material: S. Saluja, K. V. Subrahmanyam, and M. N. Thakur. De- scriptive complexity of #P functions. Journal of Computer and Sys- tem Sciences, 50(3):493-505, 1995. • Prerequisities: Finite model theory and Complexity theory. Factoring in polynomial time with a quantum computer • Topic: The most famous quantum algorithm is Shor’s algorithm for polynomial time factoring. • Possible supervisors: Åsa Hirvonen • Material: 3 (1) M. A. Nielsen, I. L. Chuang, Quantum Computation and Quan- tum Information. Cambridge University Press, 2000. (2) P. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Comput. 26, (1997), 1484– 1509. • Prerequisities: Introduction to quantum computation. Formalization of Arrow’s Impossibility Theorem in propositional de- pendence logic • Topic: Arrow’s Impossibility Theorem is the central theorem in social choice theory. It can be understood as an interesting paradox stating that any aggregation function satisfying a few natural and reasonable requirements for democracy must be a dictatorship. This theorem has attracted much attention of logicians and mathemati- cians. There are in the literature a number of formalizations of the theorem in some familiar and classical logics, such as (classical) first-order logic, propositional logic and modal logic. The aim of this thesis project is to analyze Arrow’s Theorem in dependence logic, which is a non-classical logic for reasoning about dependence and independence concepts, thus a more focused tool for analyzing the interactions of the dependency conditions involved in Arrow’s Theorem. In previous work we have formalized Arrow’s Theorem in first-order dependence logic. In this thesis project, you are supposed to formalize the theorem in propositional dependence logic, and also give a formal derivation of the theorem in the deduction system of the logic. • Possible supervisors: Fan Yang • Material: – Arrow, K.J. A Difficulty in the Concept of Social Welfare, Journal of Political Economy, 58: 328–346, 1950, – U. Endriss. Logic and social choice theory. In A. Gupta and J. van Benthem, editors, Logic and Philosophy Today, pages 333–377. College Publications, London, 2011, – E. Pacuit and F. Yang, Dependence and Independence in Social Choice: Arrow’s Theorem, in S. Abramsky, J. Kontinen, H. Vollmer and J. Väänänen, eds, Dependence Logic: Theory and Applications, Progress in Computer Science and Applied Logic, Birkhauser, June 2016, pp 235-260. • Prerequisities: Dependence logic. Implication problems of data dependencies • Topic In database theory the implication problems for various types of database dependencies have been extensively studied starting from Armstrong’s axiomatization for functional dependencies. On the other hand, team semantics is a rich semantical framework for 4 logics whose primitive building blocks correspond to the various de- pendencies studies in databases. Furthermore, implication problems of data dependencies can be viewed as questions whether logical consequence holds between certain types of formulas. The study of the connections between database theory and team semantics offers a wide range of possible theses topics. • Possible supervisors: Juha Kontinen, Miika Hannula • Material: (1) Miika Hannula, Juha Kontinen, Sebastian Link. On the finite and general implication problems of independence atoms and keys. J. Comput. Syst. Sci. 82(5): 856-877, 2016, (2) Miika Hannula, Juha Kontinen. A finite axiomatization of condi- tional independence and inclusion dependencies. Inf. Comput. 249: 121-137, 2016. (3) Prerequisities: Dependence logic. Temporal logic • Topic: Temporal logic extend basic modal logic with various tem- poral operators. Temporal logic is an active area with numerous applications in formal verification. In the thesis it is possible to focus either on the computational or model-theoretical aspects of temporal logics. • Possible supervisors: Juha Kontinen, Antti Kuusisto • Material: See, e.g, Temporal Logic (Stanford Encyclopedia of Phi- losophy). Plato.stanford.edu. • Prerequisities: Finite model theory, complexity theory. MODEL THEORY Hrushovski’s amalgamation construction • Topic: In the thesis one constructs a counterexample to Zilber’s trichotomy conjecture. • Possible supervisors: Tapani Hyttinen • Prerequisities: Model Theory. Model theory of difference fields • Topic: Difference field is a a structure (F;+;×;p) where (F;+;×) is a field and p is an automorphism of (F;+;×). In the thesis one develops the basic model theory for difference fields. • Possible supervisors: Tapani Hyttinen • Prerequisities: Model Theory. Model theory of differentially closed fields 5 • Topic: Differential fields are structures (F;+;×;D) where (F;+;×) is a field and D is a unary function that satisfies the usual properties of derivation i.e. D(a+b) = D(a)+D(b) and D(ab) = D(a)b+aD(b). Examples of these one gets e.g. from polynomial rings with the usual derivation of polynomials. In the thesis one develops the basic model theory for existentially closed (i.e. differentially closed) differential fields. • Possible supervisors: Tapani Hyttinen • Prerequisities: Model Theory. On the strength of metric logics • Topic:Lindström’s theorem states that first order predicate logic is the strongest logic having compactness and downward Löwenheim- Skolem