Abstract
The construction of an ε -free nondeterministic finite automaton (NFA) from a given NFA is a basic step in the development of compilers and computer systems. The standard conversion may produce an ε -free NFA with up to Ω ( n 2 ⋅ | Σ | ) transitions for an NFA with n states and alphabet Σ . To determine the largest asymptotic gap between the minimal number of transitions of NFAs and their equivalent ε -free NFAs has been a longstanding open problem. We show that there exist regular languages L n that can be recognized by NFAs with O ( n log 2 n ) transitions, but ε -free NFAs need Ω ( n 2 ) transitions to accept L n . Hence the standard conversion cannot be improved drastically. However, L n requires an alphabet of size n , but we also construct regular languages K n over { 0 , 1 } with NFAs of size O ( n log 2 n ) , whereas ε -free NFAs require size n ⋅ 2 c ⋅ log 2 n for every c < 1 / 2 .
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