Extended Regular Expressions: Succinctness and Decidability

Abstract

Most modern implementations of regular expression engines allow the use of variables (also called back references). The resulting extended regular expressions (which, in the literature, are also called practical regular expressions, rewbr, or regex) are able to express non-regular languages.The present paper demonstrates that extended regular-expressions cannot be minimized effectively (neither with respect to length, nor number of variables), and that the tradeoff in size between extended and "classical" regular expressions is not bounded by any recursive function. In addition to this, we prove the undecidability of several decision problems (universality, equivalence, inclusion, regularity, and cofiniteness) for extended regular expressions. Furthermore, we show that all these results hold even if the extended regular expressions contain only a single variable.

A Polynomial Time Match Test for Large Classes of Extended Regular Expressions

“…We observe that there exists a JFA(4) M with L(M ) = L(α). This automaton M matches factors according to the complete matching order ((1, 4), (2,5), (6,7), (8,9), (3,10)). The trick is that after matching the factors related to the matching position (6, 7), i. e., the factors corresponding to the occurrences of x 4 , the counter responsible for factors corresponding to x 4 is reused to match the factors related to the matching position (8,9).…”

“…On the other hand, the original introduction of backreferences has been motivated by practical needs, which implies that such radical solutions cannot be used in various applied settings. Furthermore, as demonstrated by Freydenberger [7], regular expressions with backreferences allow the specification of regular languages in a much more compact manner than their counterparts without backreferences. More precisely, the size difference between extended regular expressions and equivalent "normal" regular expressions is not bounded by any recursive function.…”

Expressiveness and static analysis of extended conjunctive regular path queries

“…Aho [1], Freydenberger [9]). These expressions extend the regular expressions with variable binding and repetition operators; e. g., for every expression α, the extended expression (α)%x xx generates the language of all www with w ∈ L(α) (α generates some w ∈ L(α), %x assigns that w to x, and the subsequent uses of x repeat this w -hence, xx generates ww).…”

Document Spanners: From Expressive Power to Decision Problems

“…We show that every vstar-free xregex can be converted into an equivalent core spanner. Since all undecidability results by Freydenberger [12] also apply to vstar-free xregexes, these undecidability results carry over to core spanners. This also has various consequences for the minimization and the relative succinctness of classes of spanner representations.…”