Related papers: A Turing Incomputable Coloring Function

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Any graph which is an induced subgraph of a graph in this sequence is called a Burling graph. These graphs have attracted some attention because they…

The Turing machine (TM) and the Church thesis have formalized the concept of computable number, this allowed to display non-computable numbers. This paper defines the concept of number "approachable" by a TM and shows that some (if not all)…

Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive…

We define a Graphics Turing Test to measure graphics performance in a similar manner to the definition of the traditional Turing Test. To pass the test one needs to reach a computational scale, the Graphics Turing Scale, for which Computer…

In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for…

We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.

Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only…

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…

Computational color constancy refers to the problem of computing the illuminant color so that the images of a scene under varying illumination can be normalized to an image under the canonical illumination. In this paper, we adopt a deep…

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique…

Networks of coupled nonlinear optical resonators have emerged as an important class of systems in ultrafast optical science, enabling richer and more complex nonlinear dynamics compared to their single-resonator or travelling-wave…

In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with…

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements…

Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.

We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or…

Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored…

We show in this article that uncomputability is also a relative property of subrecursive classes built on a recursive relative incompressible function, which acts as a higher-order "yardstick" of irreducible information for the respective…

Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring…

This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named…

Set-coloring a graph means giving each vertex a subset of a fixed color set so that no two adjacent subsets have the same cardinality. When the graph is complete one gets a new distribution problem with an interesting generating function.…