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The purpose of this paper is to present an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical theory must occur in a context of noncommutative…
This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control…
We define several canonical problems related to contrastive explanations, each answering a question of the form ''Why P but not Q?''. The problems compute causes for both P and Q, explicitly comparing their differences. We investigate the…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
The mathematical notion of incompleteness (eg of rational numbers, Turing-computable functions, and arithmetic proof) does not play a key role in conventional physics. Here, a reformulation of the kinematics of quantum theory is attempted,…
Incompleteness theorems of Godel, Turing, Chaitin, and Algorithmic Information Theory have profound epistemological implications. Incompleteness limits our ability to ever understand every observable phenomenon in the universe.…
Minimization of cortical prediction errors has been considered a key computational goal of the cerebral cortex underlying perception, action and learning. However, it is still unclear how the cortex should form and use information about…
We introduce a probabilistic model of early visual processing, beginning with the interaction between a light wavefront and the retina. We argue that perception originates not with deterministic transduction, but with probabilistic…
Cognitive theories for reasoning are about understanding how humans come to conclusions from a set of premises. Starting from hypothetical thoughts, we are interested which are the implications behind basic everyday language and how do we…
The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure…
An overview of the conceptuality interpretation of quantum mechanics is presented, along with an explanation of how it sheds light on key quantum and relativistic phenomena. In particular, we show how the interpretation clarifies…
The $P$ versus $NP$ problem is still unsolved. But there are several oracles with $P$ unequal $NP$ relative to them. Here we will prove, that $P\not=NP$ relative to a $P$-complete oracle. In this paper, we use padding arguments as the proof…
Quantum information science currently poses a troubling contradiction. It can be summarized as: (1) To factor efficiently, quantum computers must perform exponentially precise energy estimation. (2) Exponentially precise energy estimation…
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We…
Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…
We present a general theory of quantum information processing devices, that can be applied to human decision makers, to atomic multimode registers, or to molecular high-spin registers. Our quantum decision theory is a generalization of the…
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…
Beginning with Turing's seminal work in 1950, artificial intelligence proposes that consciousness can be simulated by a Turing machine. This implies a potential theory of everything where the universe is a simulation on a computer, which…
Generalised Probabilistic Theories (GPTs) provide a unifying framework encompassing classical theories, quantum theories, as well as hypothetical alternatives. We investigate the problem of extending a system with a finite set of…