Generalized Turing machines (GTMs) are a variant of non-halting Turing machines, by computational power similar to machines with the oracle for the halting problem. GTMs allow a definition of a kind of descriptive (Kolmogorov) complexity that is uniform for finite and infinite sequences. There are several natural modifications of the definition (as there are several monotone complexities). This...

Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies “I don’t know” on hard instances, another variant fails to halt, and yet another replies incorrectly “yes” or “no”. Also the halting problem has three variants. A tester may test halting on the empty inpu...

William I. Gasarch Department of Computer Science Institute for Advanced Study University of Maryland College Park, MD 20742 ABSTRACT A recursive graph is a graph whose edge set and vertex set are both recursive. Although the chromatic number of a recursive graph G (denoted χ(G)) cannot be determined recursively, it can be determined if queries to the halting set are allowed. We show that the p...

The genetic programming (GP) paradigm was designed to evolve functions that are progressively better approximations to some target function. The introduction of memory into GP has opened the Pandora's box which is algorithms. It has been shown that the combination of GP and Indexed Memory can be used to evolve any target algorithm. What has not been shown is the practicality of doing so. This p...

We examine the converse of Church-Turing thesis and establish the existence of uncountable number of accelerated Turing machines. This leads to the case that these machines are unaffected by Gödel’s incompleteness

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post’s Correspondence Problem via several new encoding techniques...

Using the counterfactual effect, we demonstrate that with better than 50% chance we can determine whether an arbitrary universal Turing machine will halt on an arbitrarily given program. As an application we indicate a probabilistic method for computing the busy beaver function— a classical uncomputable function. These results suggest a possibility of going beyond the Turing barrier.

Proof: If a deterministic Turing machine T decides language BHP by computing the first n steps of each of the possible computation paths of the nondeterministic Turing machine Mi applied to x, then the worst-case running-time of T must be exponential, since Mi applied to x may have a choice of an exponential number of computation paths. So if T is to decide language BHP and still run in polynom...

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post’s Correspondence Problem via several new encoding techniques...