Master's Theses Topics in Mathematical Logic

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.

FINITEMODELTHEORYANDLOGICINCOMPUTERSCIENCE 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 consistent 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 Φ, whether Φ is a theorem. In other words, the question is whether Φ is true in all models. This is clearly a highly fundamental problem, as solving it amounts to fully automating logical reasoning. The undecidability 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 decidability/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 complete 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 dependence 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 dependencies 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 temporal 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,+,×,π) where (F,+,×) is a field and π 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 property. Corresponding maximality theorems have been proven for continuous logic (and other similar logics). Possible topics range from metric Lindström’s theorem (references below) to infinitary metric logics. • Possible supervisors:Åsa Hirvonen • Material: (1) J. Iovino, On the maximality of logics with approximations. Journal of Symbolic Logic, 66 (2001), 1909–1918. (2) X. Caicedo, Maximality of continuous logic. Beyond first order model theory, 105–130, CRC Press, Boca Raton, FL, 2017. (3) X. Caicedo and J. Iovino, Omitting uncountable types, and the strength of [0,1]-valued logics. Ann. Pure Appl. Logic 165 (2014), no. 6, 1169–1200. • Prerequisities: Model theory, Introduction to continuous logic.

Totally categorical theories are not ﬁnitely axiomatizable • Topic: In the thesis one proves the claim in the title. The claim was a theorem by B. Zilber and it was the starting point of geometric model theory. • Possible supervisors: Tapani Hyttinen • Prerequisities: Model Theory.

SET-THEORY

Cantor’s theorem and Borel sets • Topic: The topic is Cantor’s theorem and Borel sets. Cantor’s theorem says that if you are given a sequence of real numbers then there is a real which is not in the sequence. It is possible to prove that the new real cannot be obtained by a Borel function form the sequence. The proof (due to H. Friedman) uses absoluteness, forcing, and the theory of Borel sets but is not difﬁcult. The thesis presents this result. 6

• Possible supervisors: Juliette Kennedy, Jouko Väänänen, Åsa Hir- vonen, Tapani Hyttinen • Material: (1) T. Jech. Set Theory. Springer. (2) A. Kanamori. The Higher Inﬁnite. Springer. • Prerequisities: Axiomatic Set Theory.

Cardinal invariants • Topic: Some of the topological and measure theoretic properties of the real line can be described using cardinal characteristics. E.g. add(M) is the least size of a set of meager sets whose union is not meager and co f (N) is the least size of a set X of null sets such that every null set is a subset of some member of X. One can show that add(M) ≤ co f (N) and that add(M) < co f (N) is independent of ZFC + ¬CH. In the thesis one would study some of these invariants (one can choose the invariants rather freely but the difﬁculty depends on the choice). • Possible supervisors: Tapani Hyttinen • Prerequisities: Axiomatic Set-Theory.

Diamond • Topic: Diamond is a combinatorial principle which is in principle unprovable in ZFC even if CH is assumed. However, above ω1 it can be proved from GCH in ZFC (Shelah). There is also an earlier proof by Gregory from GCH. In the thesis these two proofs are presented. • Possible supervisors: Juliette Kennedy, Jouko Väänänen, Åsa Hir- vonen, Tapani Hyttinen • Material: (1) T. Jech. Set Theory. Springer. (2) A. Kanamori. The Higher Inﬁnite. Springer. • Prerequisities: Axiomatic Set Theory.

Extended Constructibility • Topic:The constructible hierarchy (denoted L) is built over frst order logic, but other logics can be used in the construction of L. Various strong logics (so-called) have been used, which extend frst order logic, and new inner models of set theory have been obtained. The task here is to describe the inner model constructed from the quan- tifer Qω x,yφ(x,y) meaning that "φ(x,y) defines a linear order of cofinality ω," and prove in detail that the inner model constructed from this cofinality quantifier differs from L. • Possible supervisors: Juliette Kennedy, Jouko Väänänen, Åsa Hir- vonen, Tapani Hyttinen • Material: 7

(1) J. Kennedy, M. Magidor, J. Väänänen. Inner Models from Extended Logics. Arxiv. (2) T. Jech. Set Theory. Springer. (3) A. Kanamori. The Higher Inﬁnite. Springer. • Prerequisities: Axiomatic Set Theory.

GCH and large cardinals • Topic: The Continuum Hypothesis is independent of large cardinals. However, GCH has an interesting relationship with large cardinals. Scott proved that if GCH holds below a measurable cardinal, it holds at the measurable cardinal. Solovay proved that if GCH holds below a singular cardinal above a strongly compact cardinal, if hold at the singular cardinal itself. The thesis presents these two results. • Possible supervisors: Juliette Kennedy, Jouko Väänänen, Åsa Hir- vonen, Tapani Hyttinen • Material: (1) T. Jech. Set Theory. Springer. (2) A. Kanamori. The Higher Inﬁnite. Springer. • Prerequisities: Axiomatic Set Theory.

Infinite combinatorics • Topic: This is a topic in infinite combinatorics. Weak compactness has various characterisations. In his "Weakly Compact Cardinals: A Combinatorial Proof," Shelah gives a direct proof that the weak compactness of κ is equivalent to a combinatorial statement involving families of functions on ordinals below κ. In the thesis this proofs is presented. • Possible supervisors: Juliette Kennedy, Jouko Väänänen, Åsa Hir- vonen • Material: (1) T. Jech. Set Theory. Springer. (2) A. Kanamori. The Higher Infinite. Springer. (3) Weakly Compact Cardinals: A Combinatorial Proof, Shelah. (The Journal of Symbolic Logic, Vol. 44, No. 4 (Dec., 1979), pp. 559-562) • Prerequisities: Axiomatic Set Theory.

Independence of Whitehead problem • Topic: In the thesis one would prove that Whitehead problem is independent from ZFC. This problem was considered as an important problem in group theory. • Possible supervisors: Tapani Hyttinen • Prerequisities: Axiomatic Set-Theory.

Measurability and large cardinals 8

• Topic: Existence of large cardinals imply measurability of compli- cated sets. In the thesis one would prove one of such results. • Possible supervisors: Tapani Hyttinen • Prerequisities: Axiomatic Set-Theory.