Related papers: Computational Complexity on Signed Numbers
While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural…
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…
The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is sufficient evidence to establish the…
In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some…
We study the computational complexity of certain integrable quantum theories in 1+1 dimensions. We formalize a model of quantum computation based on these theories. In this model, distinguishable particles start out with known momenta and…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
NP complete problem is one of the most challenging issues. The question of whether all problems in NP are also in P is generally considered one of the most important open questions in mathematics and theoretical computer science as it has…
By Fagin's Theorem, NP contains precisely those problems that can be described by formulas starting with an existential second-order quantifier, followed by only first-order quantifiers (ESO formulas). Subsequent research refined this…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by being based on two…
The aim of this work is to show that contemporary mathematics, including Peano arithmetic, is inconsistent, to construct firm foundations for mathematics, and to begin building on these foundations.
Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around.…