Dissecting Power of a Finite Intersection of Context Free Languages
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Let ^k,α denote a tetration function defined as follows: ^1,α=2^α and ^k+1,α=2^^k,α, where k,α are positive integers. Let Δ_n denote an alphabet with n letters. If L⊆Δ_n^* is an infinite language such that for each u∈ L there is v∈ L with | u|<| v|≤^k,α| u| then we call L a language with the growth bounded by (k,α)-tetration. Given two infinite languages L_1,L_2∈Δ_n^*, we say that L_1 dissects L_2 if | L_1∩ L_2|=∞ and |(Δ_n^*∖ L_1)∩ L_2|=∞. Given a context free language L, let κ(L) denote the size of the smallest context free grammar G that generates L. We define the size of a grammar to be the total number of symbols on the right sides of all production rules. Given positive integers n,k with k≥ 2, we show that there are context free languages L_1,L_2,…, L_3k-3⊆Δ^*_n with κ(L_i)≤ 40 k such that if α is a positive integer and L⊆Δ_n^* is an infinite language with the growth bounded by (k,α)-tetration then there is a regular language M such that M∩(⋂_i=1^3k-3L_i) dissects L and the minimal deterministic finite automaton accepting M has at most k+α+3 states.
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