Related papers: The Cook-Reckhow definition
This thesis studies three topics in quantum computation and information: The approximability of quantum problems, quantum proof systems, and non-classical correlations in quantum systems. In the first area, we demonstrate a polynomial-time…
We explore a definition of complexity based on logic functions, which are widely used as compact descriptions of rules in diverse fields of contemporary science. Detailed numerical analysis shows that (i) logic complexity is effective in…
A general notion of information-related complexity applicable to both natural and man-made systems is proposed. The overall approach is to explicitly consider a rational agent performing a certain task with a quantifiable degree of success.…
The computational complexity of a quantum state quantifies how hard it is to make. `Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we…
The paper "Is Complexity an Illusion?" (Bennett, 2024) provides a formalism for complexity, learning, inference, and generalization, and introduces a formal definition for a "policy". This reply shows that correct policies do not exist for…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
In this paper, we investigate which questions are challenging for retrieval-based Question Answering (QA). We (i) propose retrieval complexity (RC), a novel metric conditioned on the completeness of retrieved documents, which measures the…
We give a new, direct proof of the tetrachotomy classification for the model-checking problem of positive equality-free logic parameterised by the model. The four complexity classes are Logspace, NP-complete, co-NP-complete and…
Theorem proving is one of the oldest applications which require heuristics to prune the search space. Invertible proof procedures has been the major tool. In this paper, we present a novel and powerful heuristic called $nongshim$ which can…
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…
This paper is addressed to logicians not familiar with category theory. It gives a new proof of coherence for symmetric monoidal closed categories, proven by Kelly and Mac Lane in early 1970s. We find this result of great importance for…
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of…
These expanded lecture notes are based on a tutorial on categorical proof theory presented at the summer school associated with the conference "Topology, Algebra, and Categories in Logic 2021-2022." The chapter delves into various…
Induction is typically formalized as a rule or axiom extension of the LK-calculus. While this extension of the sequent calculus is simple and elegant, proof transformation and analysis can be quite difficult. Theories with an induction…
The evolution of complexity has been a central theme for Biology and Artificial Life (Bonner, 1988; Bedau et al., 2000). Complexification has been interpreted in different ways: as a process of diversification between evolving units…
The standard assumptions that underlie many conceptual and quantitative frameworks do not hold for many complex physical, biological, and social systems. Complex systems science clarifies when and why such assumptions fail and provides…
Cyclic proof theory breaks tradition by allowing certain infinite proofs: those that can be represented by a finite graph, while satisfying a soundness condition. We reconcile cyclic proofs with traditional finite proofs: we extend abstract…
We report on our formalization of matrix-interpretation in Isabelle/HOL. Matrices are required to certify termination proofs and we wish to utilize them for complexity proofs, too. For the latter aim, only basic methods have already been…
In this paper we introduce a general framework for defining the depth of a sequence with respect to a class of observers. We show that our general framework captures all depth notions introduced in complexity theory so far. We review most…