Algebraic Aspects of Families of Fuzzy Languages

We study operations on fuzzy languages such as union, concatenation, Kleene $\star$, intersection with regular fuzzy languages, and several kinds of (iterated) fuzzy substitution. Then we consider families of fuzzy languages, closed under a fixed collection of these operations, which results in the concept of full Abstract Family of Fuzzy Languages or full AFFL. This algebraic structure is the fuzzy counterpart of the notion of full Abstract Family of Languages that has been encountered frequently in investigating families of crisp (i.e., non-fuzzy) languages. Some simpler and more complicated algebraic structures (such as full substitution-closed AFFL, full super-AFFL, full hyper-AFFL) will be considered as well.\ud In the second part of the paper we focus our attention to full AFFL's closed under iterated parallel fuzzy substitution, where the iterating process is prescribed by given crisp control languages. Proceeding inductively over the family of these control languages, yields an infinite sequence of full AFFL-structures with increasingly stronger closure properties

Controlled Fuzzy Parallel Rewriting

We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages

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

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

String-based Multi-adjoint Lattices for Tracing Fuzzy Logic Computations

Classically, most programming languages use in a predefined way thenotion of “string” as an standard data structure for a comfortable management of arbitrary sequences of characters. However, in this paper we assign a different role to this concept: here we are concerned with fuzzy logic programming, a somehow recent paradigm trying to introduce fuzzy logic into logic programming. In this setting, the mathematical concept of multi-adjoint lattice has been successfully exploited into the so-called Multi-adjoint Logic Programming approach, MALP in brief, for modeling flexible notions of truth-degrees beyond the simpler case of true and false. Our main goal points out not only our formal proof verifying that stringbased lattices accomplish with the so-called multi-adjoint property (as well as its Cartesian product with similar structures), but also its correspondence with interesting debugging tasks into the FLOPER system (from “Fuzzy LOgic Programming Environment for Research”) developed in our research group