Related papers: Computational Complexity on Signed Numbers
We survey a collective achievement of a group of researchers: the PCP Theorems. They give new definitions of the class \np, and imply that computing approximate solutions to many \np-hard problems is itself \np-hard. Techniques developed to…
We present a new manifestation of G\"odel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic…
A generalization of numeration system in which the set N of the natural numbers is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. Here we show that if P belonging to Q[x]…
The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets…
Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the…
We propose a duality between thermodynamics and computational complexity, elevating the difficulty of a computational task to the status of a thermodynamic variable. By introducing a complexity measure C as a novel coordinate, we formulate…
Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was claimed that quantum mechanics based on real numbers can be…
We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting…
The anthropic principle is an inevitable constraint on the space of possible theories. As such it is central to determining the limits of physics. In particular, we contend that what is ultimately possible in physics is determined by…
We demonstrate that techniques of Weihrauch complexity can be used to get easy and elegant proofs of known and new results on initial value problems. Our main result is that solving continuous initial value problems is Weihrauch equivalent…
The P=?NP problem is philosophically solved by showing P is equal to NP in the random access with unit multiply (MRAM) model. It is shown that the MRAM model empirically best models computation hardness. The P=?NP problem is shown to be a…
We propose a new type of quantum computer which is used to prove a spectral representation for a class F of computable sets. When S in F codes the theorems of a formal system, the quantum computer produces through measurement all theorems…
Two recent papers (Renou et al., arXiv:2101.10873, and Chen et al., arXiv:2103.08123) have indicated that complex numbers are necessary for quantum theory. This short note is a comment on their result.
Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern…
We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than…
This paper is about the recent notion of computably probably approximately correct learning, which lies between the statistical learning theory where there is no computational requirement on the learner and efficient PAC where the learner…
When expert systems based on causal probabilistic networks (CPNs) reach a certain size and complexity, the "combinatorial explosion monster" tends to be present. We propose an approximation scheme that identifies rarely occurring cases and…
How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal…