It was recently proved that any straight-line program (SLP) generating a given string can be transformed in linear time into an equivalent balanced SLP of the same asymptotic size. We generalize this proof to a general class of grammars we call generalized SLPs (GSLPs), which allow rules of the form A→x\documentclass[12pt]{minimal}
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\begin{document}$$A \rightarrow x$$\end{document} where x is any Turing-complete representation (of size |x|) of a sequence of symbols (potentially much longer than |x|). We then specialize GSLPs to so-called Iterated SLPs (ISLPs), which allow rules of the form A→Πi=k1k2B1ic1⋯Btict\documentclass[12pt]{minimal}
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\begin{document}$$A \rightarrow \Pi _{i=k_1}^{k_2} B_1^{i^{c_1}}\cdots B_t^{i^{c_t}}$$\end{document} of size O(t)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(t)$$\end{document}. We prove that ISLPs break, for some text families, the measure δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document} based on substring complexity, a lower bound for most measures and compressors exploiting repetitiveness. Further, ISLPs can extract any substring of length λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, from the represented text T[1..n]\documentclass[12pt]{minimal}
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\begin{document}$$T[1\mathinner {.\,.}n]$$\end{document}, in time O(λ+log2nloglogn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(\lambda + \log ^2 n\log \log n)$$\end{document}. This is the first compressed representation for repetitive texts breaking δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document} while, at the same time, supporting direct access to arbitrary text symbols in polylogarithmic time. We also show how to compute some substring queries, like range minima and next/previous smaller value, in time O(log2nloglogn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(\log ^2 n \log \log n)$$\end{document}. Finally, we further specialize the grammars to run-length SLPs (RLSLPs), which restrict the rules allowed by ISLPs to the form A→Bt\documentclass[12pt]{minimal}
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\begin{document}$$A \rightarrow B^t$$\end{document}. Apart from inheriting all the previous results with the term log2nloglogn\documentclass[12pt]{minimal}
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\begin{document}$$\log ^2 n \log \log n$$\end{document} reduced to the near-optimal logn\documentclass[12pt]{minimal}
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\begin{document}$$\log n$$\end{document}, we show that RLSLPs can exploit balancedness to efficiently compute a wide class of substring queries we call “composable”—i.e., f(X·Y)\documentclass[12pt]{minimal}
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\begin{document}$$f(X \cdot Y)$$\end{document} can be obtained from f(X) and f(Y). As an example, we show how to compute Karp-Rabin fingerprints of texts substrings in O(logn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(\log n)$$\end{document} time. While the results on RLSLPs were already known, ours are much simpler and require little precomputation time and extra data associated with the grammar.
