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arXiv · 2310.17295

Normal Forms for Elements of ${}^*$-Continuous Kleene Algebras Representing the Context-Free Languages

Mark Hopkins·Hans Leiß
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Abstract

Within the tensor product $K \mathop{\otimes_{\cal R}} C_2'$ of any ${}^*$-continuous Kleene algebra $K$ with the polycyclic ${}^*$-continuous Kleene algebra $C_2'$ over two bracket pairs there is a copy of the fixed-point closure of $K$: the centralizer of $C_2'$ in $K \mathop{\otimes_{\cal R}} C_2'$. Using an automata-theoretic representation of elements of $K\mathop{\otimes_{\cal R}} C_2'$ \`a la Kleene, with the aid of normal form theorems that restrict the occurrences of brackets on paths through the automata, we develop a foundation for a calculus of context-free expressions without variable binders. We also give some results on the bra-ket ${}^*$-continuous Kleene algebra $C_2$, motivate the ``completeness equation'' that distinguishes $C_2$ from $C_2'$, and show that $C_2'$ already validates a relativized form of this equation.

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Mark Hopkins, Hans Leiß. 2023-10-26. Normal Forms for Elements of ${}^*$-Continuous Kleene Algebras Representing the Context-Free Languages. https://doi.org/10.46298/fi.12479

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