Related papers: Self-referential theories
Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory $T$ has enough structure, then the category $T\text{-}\mathbf{Mod}$ of its models carries the structure of a model…
In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of…
A family of solvable self-dual Lie algebras is presented. There exist a few methods for the construction of non-reductive self-dual Lie algebras: an orthogonal direct product, a double-extension of an Abelian algebra, and a Wigner…
We present a logical framework that enables us to define a formal theory of computational trust in which this notion is analysed in terms of epistemic attitudes towards the possible objects of trust and in relation to existing evidence in…
A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…
We formalize the general principle of significance with respect to binary relations which is a universal tool for description and analysis of various situations in and apart from mathematics. We derive the basic properties and focus on a…
In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds' notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These…
Hypothesis testing and other statistical inference procedures are most efficient when a reliable low-dimensional parametric family can be specified. We propose a method that learns such a family when one exists but its form is not known a…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
Machine learning plays a role in many deployed decision systems, often in ways that are difficult or impossible to understand by human stakeholders. Explaining, in a human-understandable way, the relationship between the input and output of…
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…
At the heart of intuitionistic type theory lies an intuitive semantics called the "meaning explanations"; crucially, when meaning explanations are taken as definitive for type theory, the core notion is no longer "proof" but "verification".…
We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that…
We investigate abstract model theoretic properties which holds for models in which a truth or satisfaction predicate for a sublanguage of the signature is definable. We analyse in which cases those properties in fact ensure the definability…
Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area…
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the…
Many real-world complex networks contain a significant amount of structural redundancy, in which multiple vertices play identical topological roles. Such redundancy arises naturally from the simple growth processes which form and shape many…