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• W3140591929 abstract "This paper presents a short description of the PhD thesis of the author [4]. For shortness reasons, almost all technical part and proofs are omitted. Since the goal is to accessibly present the main results and give some intuition of the proof techniques, some of the details and subtleties are removed on purpose. The thesis is devoted to the study of systems of equations φi(X1, . . . , Xn) = ψi(X1, . . . , Xn) for i = 1, . . .m , in which the variables X1, . . . , Xn represent sets of natural numbers. The allowed operations are union, intersection and addition, which is defined as X + Y = {x+ y | x ∈ X, y ∈ Y } . Such systems can be equally interpreted as systems of language equations over a single-letter alphabet and operations of union, intersection and concatenation. The study begins with considering the subclass of systems of equations over sets of numbers, consisting of systems of the resolved form, i.e., Xi = φi(X1, . . . , Xn) for i = 1, . . . , n . The counterparts of such systems among the language equations are the resolved systems of language equations over a single-letter alphabet. These can be also viewed as appropriate grammars: when the allowed operations are union and concatenation, they correspond to context free grammars; when also intersection is allowed, to the conjunctive grammars. It is shown that the resolved systems of equations over sets of natural numbers can have non-ultimately periodic sets as the least solutions. Equivalently, conjunctive grammars over a single-letter alphabet can generate non-regular languages, as opposed to context-free grammars. This claim is firstly demonstrated by giving a simple system, which has the least solution with {4 : n ∈ N} as its first component. This system exploits the properties of the base-4 positional notation of sets, in particular, the aforementioned set should be viewed as the set of numbers with base-4 notation 10, for some ` ≥ 0. Then, this example is generalised. For every set S of numbers, such that base-k positional notations of numbers from S are recognised by a certain type of a realtime cellular automaton, an explicit construction of a resolved system with S as the first component of the least solution is given. Next the systems with no restrictions on the form of the equations imposed are investigated. They are shown to be computationally universal: the class of unique solutions of such systems coincides with the class of recursive sets. Similar characterisations are shown for the class of least (greatest) solutions: this class coincides with the class of recursively enumerable sets (co-recursively enumerable sets, respectively; these sets are called r.e. and co-r.e. from now on). These results hold even when only union and addition (or only intersection and addition) are allowed in the system." @default.
• W3140591929 created "2021-04-13" @default.
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• W3140591929 date "2011-01-01" @default.
• W3140591929 modified "2023-10-11" @default.
• W3140591929 title "CONJUNCTIVE GRAMMARS AND EQUATIONS OVER SETS OF NATURAL NUMBERS" @default.
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