Related papers: Betwixt Turing and Kleene
The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to…
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
Code clones are pairs of code snippets that implement similar functionality. Clone detection is a fundamental branch of automatic source code comprehension, having many applications in refactoring recommendation, plagiarism detection, and…
This is a comprehensive review on fault-tolerant topological quantum computation with the surface codes. The basic concepts and useful tools underlying fault-tolerant quantum computation, such as universal quantum computation, stabilizer…
The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The…
Chain-of-Thought (CoT) and Looped Transformers have been shown to empirically improve performance on reasoning tasks and to theoretically enhance expressivity by recursively increasing the number of computational steps. However, their…
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons…
The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see…
Finite Turing computation has a fundamental symmetry between inputs, outputs, programs, time, and storage space. Standard models of transfinite computational break this symmetry; we consider ways to recover it and study the resulting model…
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions…
Computation codes in network information theory are designed for the scenarios where the decoder is not interested in recovering the information sources themselves, but only a function thereof. K\"orner and Marton showed for distributed…
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally…
This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…
We give an effective procedure that produces a natural number in its output from any natural number in its input, that is, it computes a total function. The elementary operations of the procedure are Turing-computable. The procedure has a…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
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…
A novel computing model, called \emph{Probe Machine}, is proposed in this paper. Different from Turing Machine, Probe Machine is a fully-parallel computing model in the sense that it can simultaneously process multiple pairs of data, rather…
We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to some equivalence relation (this relation may be equality), namely, each of these theories is consistent with the formula, whose sense…