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 $K\mathop{\otimes_{\cal R}} C_2'$ à 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. final version. 42 pages, 4 figures. References sorted alphabetically

Abstract The family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$ . For arbitrary monoids M, ${\mathcal{C}} M$ is the closure of ${\mathcal{R}}M$ as a $\mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${\mathcal{R}} M$ . We provide an algebraic representation of ${\mathcal{C}} M$ in a suitable product of ${\mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $\Delta_2$ of two pairs of bracket symbols. Namely, ${\mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${\mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $X\subseteq M$ by regular expressions over $X\cup\Delta_2$ interpreted in the product of ${\mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .

In the course of providing an (infinitary) axiomatization of the equational theory of the class of context-free languages, Grathwohl, Kozen and Henglein (2013) have introduced the class of mu-continuous Chomsky algebras. These are idempotent semirings where least solutions for systems of polynomial inequations (i.e. context-free grammars) can be computed iteratively and where multiplication is continuous with respect to the least fixed point operator mu. We prove that the matrix ring of a mu-continuous Chomsky algebra also is a mu-continuous Chomsky algebra.

A compression algorithm takes a finite structure of a class K as input and produces a finite structure of a different class K' as output. Given a property P on the class K defined in a logic /spl Lscr/, we study the definability of property P on the class K'. We consider two compression schemas on unary ordered structures (words), compression by runlength encoding and the classical Lempel-Ziv. First-order properties of strings are first-order on runlength compressed strings, but this fails for images, i.e. 2-dimensional strings. We present simple first-order properties of strings which are not first-order definable on strings compressed with the Lempel-Ziv compression schema. We show that all properties of strings that are first-order definable on strings are definable on Lempel-Ziv compressed strings in FO(TC), the extension of first-order logic with the transitive closure operator. We define a subclass /spl Fscr/ of the first-order properties of strings such that if L is defined by a property in /spl Fscr/, it is also first-order definable on the Lempel-Ziv compressed strings. Monadic second-order properties of strings are dyadic second order definable on Lempel-Ziv compressed strings.

We improve on J. Kilbury's proposal to interchange “predictor” and “scanner” in Earley's parser. This modification of Earley's parser can trivially be combined with those suggested by S. Graham, M. Harrison, and W. Ruzzo, leading to smaller parse tables and almost the power of lookahead 1. Along these lines we can also obtain Earley-parsers having partial lookahead r ≥ 1, without storing right contexts. Parse trees with shared structure can be stored in the parse tables directly, rather than constructing the trees from “dotted rules."