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Context-free complexity of finite languages. (English) Zbl 0536.68065

Necessary conditions are given for an infinite language to have a sublinear complexity. To achieve the main result, characterizations for the number of productions of finite languages are derived. The main result: Let L be an infinite language over \(\Sigma\) with sublinear complexity. Then for all k there is an n(k) and there are words \(x_ i\), \(y_ i\), \(z_ i\), \(i=1,...,k,\) such that for all \(n>n(k)\) one of the following two cases holds: \[ (1)\quad(x_ i,z_ i)\neq(x_ j,z_ j),\quad y_ i\neq y_ j\quad for\quad i\neq j\quad and\quad \cup^{k}_{i=1}x_ i\{y_ 1,...,y_ k\}z_ i\subseteq L_ n, \]
\[ (2)\quad x_ i\neq y_ i\quad for\quad i=1,...,k\quad and\quad \{x_ 1,y_ 1\}...\{x_ k,y_ k\}\subseteq L_ n. \]

MSC:

68Q45 Formal languages and automata
68Q25 Analysis of algorithms and problem complexity
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References:

[1] Bucher, W., A note on a problem in the theory of grammatical complexity, Theoret. comput. sci., 14, 337-344, (1981) · Zbl 0469.68082
[2] Bucher, W.; Culik, K.; Maurer, H.; Wotschke, D., Concise description of finite languages, Theoret. comput. sci., 14, 227-246, (1981) · Zbl 0469.68081
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