Parameterized Regular Expressions and Their Languages

espect to the original expression e ′ (the number of possible valuations for e ′ is |Σ| |W| , whereW is the set of variables in e ′ ), the above procedure runs in Expspace.For the lower bound we present a reduction from the complement of the acceptanceproblem of a Turing machine. Let L be a language that belongs to Expspace, and let M bea Turing machine that decides L in Expspace. Given an input ā we construct in polynomialtime with respect to M and ā a parameterized regular expression e M,ā over some alphabet∆ such that L ✸ (e M,ā ) consists of all the strings over ∆ if and only if M does not acceptinput ā.Just as in section 4.1, we assume that M = (Q,Γ,q 0 ,{q m },δ), where Q = {q 0 ,...,q m }is the set of states, Γ is the tape alphabet (that contains the distinguished blank symbol B),the initial state is q 0 , q m is the unique final state, and δ : (Q\{q m })×Γ → Q×Γ×{L,R} isthe transition function. Notice that we assume without loss of generality that no transitionis defined on the final state q m . Since M decides L in Expspace, there is a polynomial S()such that, for every input ā over Σ, M decides ā using space of order 2 S(|ā|) .Let ā = a 0 a 1···a k−1 ∈ Σ ∗ be an input for M (that is, each a i , 0 ≤ i ≤ k−1 is a symbolin Γ). For notational convenience we will assume from now on that S(|ā|) = n.We also find it convenient to introduce the following notation. For an alphabet Σ ={b 1 ,...,b p }ofsymbols, weabusenotationanddenotebyΣtheregularexpression(b 1 | ··· |b p ).Thus, for example, assume that Γ = {b 1 ,...,b p }∪{B}. Then, when we write (Γ∪(Γ×Q))we represent the language given by (b 1 | ··· | b p | B | (b 1 ,q 0 ) | ··· | (B,q m )). Furthermore,we reuse the notation introduced in Section 4.1, and write the shorthand [i] to denote thebinary representation of the number i as a string of n characters (i.e., [0] corresponds to theword 0 n , and [2] corresponds to the word 0 n−2 10).Our parameterized expression e M,ā uses the alphabet ∆ = {0,1,#,&,%}∪Γ∪(Γ×Q).The idea of the reduction is as follows. Using ∆, we represent a configuration of M by wordsin ∆ ∗ of the form:#·[0]·(Γ∪(Γ×Q))·&·[1]·(Γ∪(Γ×Q))·&·[2]·(Γ∪(Γ×Q))·&.[2 n −1]·(Γ∪(Γ×Q))·&·% (3)Intuitively, the strings [0],[1],...,[2 n − 1] indicate each one of the 2 n cells of M, and thesymbol following these strings represents either the content of the cell, or the content ofthe cell plus the state of M, if M is pointing into that particular cell of the tape in theconfiguration that is being encoded.Since each word of the form (3) represents a configuration of M, we can represent a runof M on input ā as a sequence of concatenations of words of the form (3), as long as each oneof these configurations is consistent with the computation of M. The idea of the reductionis to construct a parameterized regular expression e M,ā that represent all words w in ∆ ∗ thatare either not valid concatenations of subwords of the form (3), or, in case that they are,28