Related papers: Formal Theories for Logspace Counting
For fragments L of first-order logic (FO) with counting quantifiers, we consider the definability problem, which asks whether a given L-formula can be equivalently expressed by a formula in some fragment of L without counting, and the more…
A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical…
We rely on the strength of linguistic and philosophical perspectives in constructing a framework that offers a unified explanation for presuppositions and existential commitment. We use a rich ontology and a set of methodological principles…
Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory…
A general procedure is presented to determine, given any suitable representation of the modular group, the characters of all possible Rational Conformal Field Theories whose associated modular representation is the given one. The relevant…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then…
Generalized numbers, arithmetic operators and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of $q$-logarithm/$q$-exponential inverse functions. Some of the…
Category theory offers a mathematical foundation for knowledge representation and database systems. Popular existing approaches model a database instance as a functor into the category of sets and functions, or as a 2-functor into the…
Finite (word) state transducers extend finite state automata by defining a binary relation over finite words, called rational relation. If the rational relation is the graph of a function, this function is said to be rational. The class of…
Descriptive complexity theory is an important area in the study of computational complexity. In this direction, it is possible to describe combinatorial problems exclusively by logical methods, without resorting to the use of complicated…
The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for…
The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…
We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate…
Descriptive complexity theory aims at inferring a problem's computational complexity from the syntactic complexity of its description. A cornerstone of this theory is Fagin's Theorem, by which a graph property is expressible in existential…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
We investigate the complexity of the partial order relation of Young's lattice. The definable relations are characterized by establishing the maximal definability property modulo the single automorphism given by conjugation; consequently,…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…