Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page

Linear representations of regular rings and complemented modular lattices with involution

Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between classes of spaces and classes of representables is analyzed; for a class of spaces which is closed under ultraproducts and non-degenerate finite dimensional subspaces, the latter are shown to be closed under complemented [regular] subalgebras, homomorphic images, and ultraproducts and being generated by those members which are associated with finite dimensional spaces. Under natural restrictions, this is refined to a $1$-$1$-correspondence between the two types of classes

One-Variable Fragments of First-Order Many-Valued Logics

In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences

Levels of discontinuity, limit-computability, and jump operators

We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $\Delta^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem

Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras

Satisfiability of Boolean circuits is NP-complete in general but becomes polynomial time when restricted for example either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is connected with solving equations over finite algebras. This in turn is one of the oldest and well-known mathematical problems which for centuries was the driving force of research in algebra. Let us only mention Galois theory, Gaussian elimination or Diophantine Equations. The last problem has been shown to be undecidable, however in finite realms such problems are obviously decidable in nondeterministic polynomial time. A project of characterizing finite algebras m A with polynomial time algorithms deciding satisfiability of circuits over m A has been undertaken in [Pawel M. Idziak and Jacek Krzaczkowski, 2018]. Unfortunately that paper leaves a gap for nilpotent but not supernilpotent algebras. In this paper we discuss possible attacks on filling this gap