Related papers: Getting around the Halting Problem

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $\Gamma$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/\Gamma$ is divergent if eventually it leaves every compact subset. In…

We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem…

Counter automata are more powerful versions of finite-state automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of $\Z^n$ is accepted by a…

A bottleneck plane perfect matching of a set of $n$ points in $\mathbb{R}^2$ is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as {\em bottleneck}. The…

We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…

The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive…

For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the…

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The problem: given a propositional logic program P and a non-negative integer k, decide whether P has a stable…

Integer Linear Programming with $n$ binary variables and $m$ many $0/1$-constraints can be solved in time $2^{\tilde O(m^2)} \text{poly}(n)$ and it is open whether the dependence on $m$ is optimal. Several seemingly unrelated problems,…

Guidelines and consistency rules of UML are used to control the degrees of freedom provided by the language to prevent faults. Guidelines are used in specific domains (e.g., avionics) to recommend the proper use of technologies. Consistency…

On non-K\"ahler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of…

There have been many attempts to solve the P versus NP problem. However, with a new proof method, P not equal NP can be proved. A time limit is set for an arbitrary Turing machine and an input word is rejected on a timeout. The time limit…

There are many techniques and tools to prove termination of C programs, but up to now these tools were not very powerful for fully automated termination proofs of programs whose termination depends on recursive data structures like lists.…

Transformer large language models (LLMs) have sparked admiration for their exceptional performance on tasks that demand intricate multi-step reasoning. Yet, these models simultaneously show failures on surprisingly trivial problems. This…

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…

Chinese Remainder Theorem (CRT) is a powerful approach to solve ambiguity resolution related problems such as undersampling frequency estimation and phase unwrapping which are widely applied in localization. Recently, the deterministic…

A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing's for the unsolvability of the halting problem, which is permitted in classical logic, shows that…

A pattern $\alpha$ is a string of variables and terminal letters. We say that $\alpha$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $\alpha$ by terminal words. The matching…

Robin's Inequality posits $G(n)<e^{\gamma}$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^{\gamma}\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By…

We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous…