Related papers: Classifying the complexity of models of arithmetic
Throughout the course of mathematical history, generalizations of previously understood concepts and structures have led to the fruitful development of the hierarchy of number systems, non-euclidean geometry, and many other epochal phases…
Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the…
Finitely generated Z-modules have canonical decompositions. When such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith…
We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center…
Modal logics are widely used in computer science. The complexity of modal satisfiability problems has been investigated since the 1970s, usually proving results on a case-by-case basis. We prove a very general classification for a wide…
We define instantiational and algorithmic completeness for a formal language. We show that, in the presence of Church's Thesis, an alternative interpretation of Goedelian incompleteness is that Peano Arithmetic is instantiationally…
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and…
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…
Among the approximation methods for the verification of counter systems, one of them consists in model-checking their flat unfoldings. Unfortunately, the complexity characterization of model-checking problems for such operational models is…
SHAP explanations are a popular feature-attribution mechanism for explainable AI. They use game-theoretic notions to measure the influence of individual features on the prediction of a machine learning model. Despite a lot of recent…
The linear ordering problem (LOP), which consists in ordering M objects from their pairwise comparisons, is commonly applied in many areas of research. While efforts have been made to devise efficient LOP algorithms, verification of whether…
Work in progress concerning alternative formalizations of arithmetic.
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually…
We demonstrate that any $\Pi_\alpha$ sentence of the infinitary logic $L_{\omega_1 \omega}$ extending the theory of linear orderings has a model with a $\Pi_{\alpha+4}$ Scott sentence and hence of Scott rank at most $\alpha+3$. In other…
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…
We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that…
In this paper, we analyze the complexity of functional programs written in the interaction-net computation model, an asynchronous, parallel and confluent model that generalizes linear-logic proof nets. Employing user-defined sized and…