Parameterized Regular Expressions and Their Languages

• Between each two consecutive [even] or [odd] strings there is exactly one [read] and one[head] in the slots devoted to [action] in form (1).• No other cell can change its context, except for those marked with [read] or [head], and• Thecontent thatchangesinthecellsmarkedby[read]and[head]respects thetransitionfunction δ of M.Moreover, we also add a set of expressions E 5 , accepting words such that:• The initial configuration of M is encoded as the first step of the computation representedby w.• The last computation ends in a final state of M.It is a tedious, but straightforward task to define the sets E 4 and E 5 of expressions.Furthermore, the fact that ⋂ e∈E L ✷(e) is empty if and only if M accepts on input ā followsimmediately from the remarks given along the construction. This finishes the proof. ✷The generic reduction used in the proof of Expspace-hardness of Nonemptiness ✷ alsoprovides lower bounds on the minimal sizes of words in languages L ✷ (e) (note that thelanguage L ✸ (e) always contains a word of linear size in |e|).Corollary 1. There exists a polynomial p : N → N and a sequence of parameterized regularexpressions {e n } n∈N such that each e n is of size at most p(n), and every word in the languageL ✷ (e n ) has size at least 2 2n .Before explaining the proof, we note that the single-exponential bound is easy to see(it was hinted at in the first paragraph of the introduction, and which was in fact used inconnection with querying incomplete graph data in [7]). For each n, consider an expressione n = (0|1) ∗ x 1 ...x n (0|1) ∗ . If a word w is in L ✷ (e n ), then w must contain every word in{0,1} n as a subword, which implies that its length must be at least 2 n +(n−1).Proof of Corollary 1: Clearly, for each n ∈ N, it is possible to construct a deterministicTuring machine M n over alphabet Σ = {0,1} that on input 1 n works for exactly 2 2n steps,using 2 n cells. Furthermore, it is possible to specify this machine using polynomial size withrespect to n.Next, using the construction in the reduction of Theorem 1, construct a set of parameterizedregular expressions E (Mn,1 n ) such that the single word w n ∈ ⋂ e∈E (Mn,1 n )L ✷ (e) representsa run (or, more precisely, a sequence of configurations) of M n on input 1 n . Note that eachset E (Mn,1 n ) is of size polynomial with respect to n. Moreover, according to the reductionin the proof of Theorem 1, ⋂ e∈E (Mn,1 n )L ✷ (e) contains a single word, of length greater than2 2n , representing the single run of M n on input 1 n .For each n, we define e n as the expression such that L ✷ (e n ) is empty if and only if⋂e∈E (Mn,1 n )L ✷ (e) is empty, constructed as in the proof of Lemma 2, so that e n is of size12