Summary
Using methods from linear algebra and crossing-sequence arguments it is shown that logarithmic space is necessary for the recognition of all context-free nonregular subsets of {a1}* ... {an}*, where {a1,...,an} is some alphabet. It then follows that log n is a lower bound on the space complexity for the recognition of any bounded or deterministic non-regular context-free language.
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Abbreviations
- ℕ:
-
the set of natural numbers
- ℕ0=:
-
\(\mathbb{N} \cup \left\{ 0 \right\}\)
- ℤ:
-
the set of integers
- \(\mathbb{Q}\) :
-
the set of rational numbers
- \(\mathbb{Q}_ + \) :
-
the set of nonnegative rational numbers
- [r 1, r 2]:
-
the set \(\{ x \in \mathbb{Q}|r_1 \leqq {\text{x}} \leqq r_2 \}\)
- [r 1,r2):
-
the set \(\{ x \in \mathbb{Q}|r_1 \leqq {\text{x}} \leqq r_2 \}\)
- δ + A :
-
the set of all vectors {δ + α¦αεA} for a given vector δ and a set of vectors A
- M :
-
the complement of the set M
- w R :
-
the reversal of the word wε σ*, e.g. (x1...xi)R=x n ...x 1
- O: f :
-
O(g) for two functions f,g: ℕ0→ℕ0 means lim sup \(\mathop {lim}\limits_{n \to \infty } sup\frac{{f(n)}}{{g(n)}} < \infty ; o: f = o(g) \Leftrightarrow \mathop {lim}\limits_{n \to \infty } \frac{{f(n)}}{{g(n)}} = 0\)
References
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Alt, H. Lower bounds on space complexity for contextfree recognition. Acta Informatica 12, 33–61 (1979). https://doi.org/10.1007/BF00264016
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DOI: https://doi.org/10.1007/BF00264016