Turing machines based on unsharp quantum logic

In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic Turing machines (EDTMs). We discuss different E-valued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a classical initial state and ENTMs with a classical final state have the same power as ENTMs with quantum initial and final states. In particular, the latter can be simulated by ENTMs with classical transitions under a certain condition. Using these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs are more powerful than EDTMs. This is a notable difference from the classical Turing machines.Comment: In Proceedings QPL 2011, arXiv:1210.029

Bird's-eye view on Noise-Based Logic

Noise-based logic is a practically deterministic logic scheme inspired by the randomness of neural spikes and uses a system of uncorrelated stochastic processes and their superposition to represent the logic state. We briefly discuss various questions such as (i) What does practical determinism mean? (ii) Is noise-based logic a Turing machine? (iii) Is there hope to beat (the dreams of) quantum computation by a classical physical noise-based processor, and what are the minimum hardware requirements for that? Finally, (iv) we address the problem of random number generators and show that the common belief that quantum number generators are superior to classical (thermal) noise-based generators is nothing but a myth.Comment: paper in pres

Fuzzy Automata: A Quantitative Review

Classical automata theory cannot deal with the system uncertainty. To deal with the system uncertainty the concept of fuzzy finite automata was proposed. Fuzzy automata can be used in diverse applications such as fault detection, pattern matching, measuring the fuzziness between strings, description of natural languages, neural network, lexical analysis, image processing, scheduling problem and many more. In this paper, a methodical literature review is carried out on various research works in the field of Fuzzy automata and explained the challenging issues in the field of fuzzy automata

A probabilistic model of computing with words

AbstractComputing in the traditional sense involves inputs with strings of numbers and symbols rather than words, where words mean probability distributions over input alphabet, and are different from the words in classical formal languages and automata theory. In this paper our goal is to deal with probabilistic finite automata (PFAs), probabilistic Turing machines (PTMs), and probabilistic context-free grammars (PCFGs) by inputting strings of words (probability distributions). Specifically, (i) we verify that PFAs computing strings of words can be implemented by means of calculating strings of symbols (Theorem 1); (ii) we elaborate on PTMs with input strings of words, and particularly demonstrate by describing Example 2 that PTMs computing strings of words may not be directly performed through only computing strings of symbols, i.e., Theorem 1 may not hold for PTMs; (iii) we study PCFGs and thus PRGs with input strings of words, and prove that Theorem 1 does hold for PCFRs and PRGs (Theorem 2); a characterization of PRGs in terms of PFAs, and the equivalence between PCFGs and their Chomsky and Greibach normal forms, in the sense that the inputs are strings of words, are also presented. Finally, the main results obtained are summarized, and a number of related issues for further study are raised

A Theory of Computation Based on Quantum Logic (I)

The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic

Discrete and fuzzy dynamical genetic programming in the XCSF learning classifier system

A number of representation schemes have been presented for use within learning classifier systems, ranging from binary encodings to neural networks. This paper presents results from an investigation into using discrete and fuzzy dynamical system representations within the XCSF learning classifier system. In particular, asynchronous random Boolean networks are used to represent the traditional condition-action production system rules in the discrete case and asynchronous fuzzy logic networks in the continuous-valued case. It is shown possible to use self-adaptive, open-ended evolution to design an ensemble of such dynamical systems within XCSF to solve a number of well-known test problems

A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used