Study On The Properties Of Lattice-valued Languages And Their Cuts

The properties of automata are important subjects in automatatheory. In references [6,7,8], latticed-valued automata have been constructed in themore general algebraic struture: latticed oreded monoid, and the properties of thelatticed-valued automata and their languages are studied. From these references, weknow that automata with truth-values in a lattice-ordered monoid are more powerfulthan fuzzy automata and classical automata. So, the study on the properties oflatticed-valued automata and their languages are very important.On the basis of references [6,7,8,17,19], we mainly work over the algebriac prop-erties and approximation properties of latticed-valued automata and their languagesin this paper, and we aslo study the the algebraic properties and approximationproperties of the cuts of latticed-valued regular languages.The algebraic properties of automata and their languages are considered as theclosure properties of algebraic operations on automata and their languages, such asintersection, union, quotient, reversal, Kleene closure, homomorphism and so on.The articles [6,7,8] show that automata and their languages are closed under alge-braic operations in the latticed oreded monoid with some conditions imposed. Andin [17,19], the notion of approximation is gained between languages or latticed-valuedlanguages, and some conditions are given in which latticed-valued automata(NLFA)can beε-approxomated by deterministic latticed-valued automata(DNLFA).In this thesis, we continue to study the algebriac properties and approxima-tion properties of lattice-valued automata and their languages. First, the notionsof quotient and admissible set are difined, and we show that latticed-valued regularlanguages are closed under quotient, a latticed-valued laguage f is latticed-valuedregular if and only if f has an admissible set. Then we obtain some sufficient or nec-essary conditions in which latticed-valued regular languages can beε-approximatedby deterministic lattced-valued regular languages. From the above discussion, weinvestigate the properties on the cuts of latticed-valued laguages, give some condi-tions about the closure properties of some algebraic operations, for example, theunion, intersection, complement on the cuts of the latticed-valued regular languages are closed in general group, the union, intersection, cocatenation between the cuts oflatticed-valued regular languages and regular languages is also contained in the cutsof latticed-valued regular languages, the cuts of latticed-valued regular languagesare closed under homomorphism, quotient, reversal, but these may not be true forintersection and complemet. Then we debate the approximation properties betweenthe the cuts of latticed-valued languages or latticed-valued regular languages andregular languages, give some conclusions, and show thatε-approximation betweenlatticed-valued languages andε-approximation between the cuts of the latticed lan-guages are related in some extent. For example, if a cut of a latticed-valued regularlanguage can induce aε-cover, then it can beε-approximated by a regular language.