Kleene star in the context of Free monoid

In computer science, a production or production rule is a rewrite rule that replaces some symbols with other symbols. A finite set of productions ${\displaystyle P}$ is the main component in the specification of a formal grammar (specifically a generative grammar).

In such grammars, a set of productions is a special case of relation on the set of strings ${\displaystyle V^{*}}$ (where ${\displaystyle {}^{*}}$ is the Kleene star operator) over a finite set of symbols ${\displaystyle V}$ called a vocabulary that defines which non-empty strings can be substituted with others.The set of productions is thus a special kind subset

Stephen Cole Kleene (/ˈkleɪni/ KLAY-nee; January 5, 1909 – January 25, 1994) was an American mathematician and logician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.