LIPIcs.CPM.2024.13.pdf
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One of the classical algorithmic problems in formal languages is the context-free recognition problem: for a given context-free grammar and a length-n string, check if the string belongs to the language described by the grammar. Already in 1975, Valiant showed that this can be solved in {O}̃(n^ω) time, where ω is the matrix multiplication exponent. More recently, Abboud, Backurs, and Vassilevska Williams [FOCS 2015] showed that any improvement on this complexity would imply a breakthrough algorithm for the k-Clique problem. We study the natural online version of this problem, where the input string w[1..n] is given left-to-right, and after having seen every prefix w[1..t] we should output if it belongs to the language. The goal is to maintain the total running time to process the whole input. Even though this version has been extensively studied in the past, the best known upper bound was {O}(n³/log²n). We connect the complexity of online context-free recognition to that of Online Matrix-Vector Multiplication, which allows us to improve the upper bound to n³/2^{Ω(√{log{n}})}.
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