This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an ε-free automaton to prove an equivalent to Kleene’s representation theorem (implementing Glushkov’s instead of Thom...

The class of 2-automatic paperfolding sequences corresponds to the class of ultimately periodic sequences of unfolding instructions. We rst show that a paper-folding sequence is automatic ii it is 2-automatic. Then we provide families of minimal nite-state automata, minimal uniform tag sequences and minimal substitutions describing automatic paperfolding sequences, as well as a family of algebr...

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and fo...

The Krohn-Rhodes complexity theory for pure (without lin-earity) automata is well-known. This theory uses an operation of wreath product as a decomposition tool. The main goal of the paper is to introduce the notion of complexity of linear automata. This notion is ultimately related with decompositions of linear automata. The study of these decom-positions is the second objective of the paper. ...

The set of synchronizing words a given n-state automaton forms regular language recognizable by an with \(2^n - n\) states. size recognizing for the is linked to computational problems related synchronization and length words. Hence, it natural investigate automata extremal this property, i.e., such that minimal deterministic has sync-maximal permutation groups have been introduced in [S. Hoffm...

Quantum-dot Cellular Automata (QCA) is an emerging and promising technology that provides significant improvements over CMOS. Recently QCA has been advocated as an applicant for implementing reversible circuits. However QCA, like other Nanotechnologies, suffers from a high fault rate. The main purpose of this paper is to develop a fault tolerant model of QCA circuits by redundancy in hardware a...

Myhill-Nerode Theorem is regarded as a basic theorem in the theories of languages and automata and is used to prove the equivalence between automata and their languages. The significance of this theorem has stimulated researchers to develop that on different automata thus leading to optimizing computational models. In this article, we aim at developing the concept of congruence in general fuzzy...