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This is It is sometimes convenient to talk about the positive closure of R, denoted R+ . ∞ It is deﬁned as 1 Ri, or RR∗. Using these three operations, we can deﬁne the set of regular expressions (res) over an alphabet Σ. 1. if a ∈ Σ, then a is also a re denoting the set containing only a. 2. if r and s are res, denoting sets L(r) and L(s) respectively, then (r) is a re denoting L(r) r | s is a re denoting the union of L(r) and L(s) rs is a re denoting the concatenation of L(r) and L(s) r∗ is a re denoting the Kleene closure of L(r).

To construct the initial state, s0 ∈ S, it puts q0N into s0 and then augments s0 with every state in QN that can be reached from q0N by following one or more -transitions. The algorithm abstracts this notion of following -transitions into a function, called -closure For a state, qi , -closure(qi ) is the set containing qi and any other states reachable from qi by taking only -moves. Thus, the ﬁrst step is to construct s0 as -closure(q0N ). 36 CHAPTER 2. LEXICAL ANALYSIS Once S has been initialized with s0 , the algorithm repeatedly iterates over the elements of S, extending the partially constructed dfa (represented by S) by following transitions out of each si ∈ S.

Sf - α Hγ* - s Hγ* 6 β δHj ⇒ 6 βδHj t0 f In this particular case, we can eliminate the -move by combining the two states, sf and t0 , into a single state. To accomplish this, we need to make sf the source of each edge leaving t0 , and sf the sink of any edge entering t0 . This produces the simpliﬁed nfa shown on the right. Notice that coalescing can create a state with multiple transitions on the same symbol: si , , @α@ R sj - α sm sk ⇒ ,, @α@R α sij sk sm If sk and sm are distinct states, then both sij , sk , α and sij , sm , α should remain.