Elena Calude

Physical versus computational complementarity. I

The dichotomy between endophysical/intrinsic and exophysical/extrinsic perception concerns how a model—mathematical, logical, computational—universe is perceived, from inside or from outside. This paper, the first in a proposed series, discusses some limitations and tradeoffs between endophysical/intrinsic and exophysical/extrinsic perceptions in both physical and computational contexts. We build our work on E. F. Moores Gedanken-experiments in which the universe is modeled by a finite deterministic automaton. A new type of computational complementarity, which mimics the state of quantum entanglement, is introduced and contrasted with Moores computational complementarity. Computer simulations of both types of computational complementarity are developed for fourstate Moore automata.

Finite nondeterministic automata: simulation and minimality

Abstract Motivated by recent applications of finite automata to theoretical physics, we study the minimization problem for nondeterministic automata (with outputs, but no initial states). We use Ehrenfeucht–Fraisse-like games to model automata responses and simulations. The minimal automaton is constructed and, in contrast with the classical case, proved to be unique up to an isomorphism. Finally, we investigate the partial ordering induced by automata simulations. For example, we prove that, with respect to this ordering, the class of deterministic automata forms an ideal in the class of all automata.

Deterministic automata simulation, universality and minimality

Abstract Finite automata have been recently used as alternative, discrete models in theoretical physics, especially in problems related to the dichotomy between endophysical/intrinsic and exophysical/ extrinsic perception (see, for instance [3, 6, 18–21]). These studies deal with Moore experiments; the main result states that it is impossible to determine the initial state of an automaton, and, consequently, a discrete model of Heisenberg uncertainty has been suggested. For this aim the classical theory of finite automata — which considers automata with initial states — is not adequate, and a new approach is necessary. A study of finite deterministic automata without initial states is exactly the aim of this paper. We will define and investigate the complexity of various types of simulations between automata. Minimal automata will be constructed and proven to be unique up to an isomorphism. We will build our results on an extension of Myhill-Nerode technique; all constructions will make use of “automata responses” to simple experiments only, i.e., no information about the internal machinery will be considered available.

Guest Column: Adiabatic Quantum Computing Challenges

The paper presents a brief introduction to quantum computing with focus on the adiabatic model which is illustrated with the commercial D-Wave computer. We also include new theory and experimental work done on the D-Wave computer. Finally we discuss a hybrid method of combining classical and quantum computing and a few open problems.

Inductive Complexity Measures for Mathematical Problems

An algorithmic uniform method to measure the complexity of finitely refutable statements [6, 7, 9] was used to classify famous/interesting mathematical statements like Fermats last theorem, the four colour theorem, and the Riemann hypothesis [8, 15, 16]. Working with inductive Turing machines of various orders [1] instead of classical computations, we propose a class of inductive complexity measures and inductive complexity classes for mathematical statements which generalise the previous method. In particular, the new method is capable to classify Π2–statements. As illustrations, we evaluate the inductive complexity of the Collatz and twin prime conjectures — statements which cannot be evaluated with the original method.

Inductive complexity of p versus NP problem

Using the complexity measure developed in [7,3,4] and the extensions obtained by using inductive register machines of various orders in [1,2], we determine an upper bound on the inductive complexity of second order of the P versus NP problem. From this point of view, the P versus NP problem is more complex than the Riemann hypothesis.

Fermat's last theorem and chaoticity

Proving that a dynamical system is chaotic is a central problem in chaos theory (Hirsch in Chaos, fractals and dynamics, 1985]. In this note we apply the computational method developed in (Calude and Calude in Complex Syst 18:267–285, 2009; Calude and Calude in Complex Syst 18:387–401, 2010; Calude et al in J Multi Valued Log Soft Comput 12:285–307, 2006) to show that Fermat’s last theorem is in the lowest complexity class

Computational complementarity for probabilistic automata

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