Related papers: The Arithmetical Hierarchy: A Realizability-Theore…

Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand…

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic…

Quantum advantage is notoriously hard to find and even harder to prove. For example the class of functions computable with classical physics actually exactly coincides with the class computable quantum-mechanically. It is strongly believed,…

Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…

The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to…

We develop a notion of realizability for Classical Linear Logic based on a concurrent process calculus.

"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…

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 an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…

This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability…

This book is concerned with the various aspects of hierarchical collective behaviour which is manifested by most complex systems in nature. From the many of the possible topics, we plan to present a selection of those that we think are…

We introduce a notion of realizability with ordinal Turing machines based on recognizability rather than computability, i.e., the ability to uniquely identify an object. We show that the arising concept of $r$-realizabilty has the property…

Many proposed applications of neural networks in machine learning, cognitive/brain science, and society hinge on the feasibility of inner interpretability via circuit discovery. This calls for empirical and theoretical explorations of…

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…

This paper examines whether unitary evolution alone is sufficient to explain emergence of the classical world from the perspective of computability theory. Specifically, it looks at the problem of how the choice related to the measurement…

One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…

Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…

Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…

To deal with global problems we suggest to consider complex systems not in space and time, but in a possible deeper reality, i.e., the hierarchical network of prime integer relations. Encoded by arithmetic through the self-organization…

I survey recent progress on a classic and challenging problem in social choice: the fair division of indivisible items. I discuss how a computational perspective has provided interesting insights into and understanding of how to divide…