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Given a countable set of sites and a collection of flip rates at each site, we give a sufficient condition on the long-range dependancies of the flip rates ensuring the well-definedness of the corresponding spin system. This hypothesis has…
We describe a method to axiomatize computations in deterministic Turing machines. When applied to computations in non-deterministic Turing machines, this method may produce contradictory (and therefore trivial) theories, considering…
The existence of universal quantum computers has been theoretically well established. However, building up a real quantum computer system not only relies on the theory of universality, but also needs methods to satisfy requirements on other…
We extend the capabilities of neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine or Von Neumann architecture but is…
The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes,…
The fundamental impasses and ruptures in various domains of the canonical, unitary science, or the 'end of science', become the more and more evident. The natural unity of being is recovered within a universal nonperturbative method leading…
The fixed point construction is a method for designing tile sets and cellular automata with highly nontrivial dynamical and computational properties. It produces an infinite hierarchy of systems where each layer simulates the next one. The…
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an…
We study the interaction of structural subtyping with parametric polymorphism and recursively defined type constructors. Although structural subtyping is undecidable in this setting, we describe a notion of parametricity for type…
In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
Complex spatiotemporal patterns in nature significantly challenge reductionism-based modern science. The lack of a paradigm beyond reductionism hinders our understanding of the emergence of complexity. The diversity of countless patterns…
The scaling theory of irreversible aggregation is discussed in some detail. First, we review the general theory in the simplest case of binary reactions. We then extend consideration to ternary reactions, multispecies aggregation,…
This paper delves into the intersection of computational theory and music, examining the concept of undecidability and its significant, yet overlooked, implications within the realm of modern music composition and production. It posits that…
The purpose of this thesis is to make an analysis of the concept of Hypercomputation and of some hypermachines. This thesis is separated in three main parts. We start in the first chapter with an analysis of the concept of Classical…
Based on the propositional description of even Goldbach conjecture, in order to verify the truth of even Goldbach conjecture, we will deeply discuss this question and present a new computing model of $G{{N}_{e}}TM$ Turing Machine. This…
This paper demonstrates the relativity of Computability and Nondeterministic; the nondeterministic is just Turing's undecidable Decision rather than the Nondeterministic Polynomial time. Based on analysis about TM, UM, DTM, NTM, Turing…