On Practical Regular Expressions

We report on simulation, hierarchy, and decidability results for Practical Regular Expressions (PRE), which may include back references in addition to the standard operations union, concatenation, and star. The following results are obtained: PRE can be simulated by the classical model of nondeterministic finite automata with sensing one-way heads. The number of heads depends on the number of different variables in the expressions. A space bound O(n log m) for matching a text of length m with a PRE with n variables based on the previous simulation. This improves the bound O(nm) from (C\^ampeanu and Santean 2009). PRE cannot be simulated by deterministic finite automata with at most three sensing one-way heads or deterministic finite automata with any number of non-sensing one-way heads. PRE with a bounded number of occurrences of variables in any match can be simulated by nondeterministic finite automata with one-way heads. There is a tight hierarchy of PRE with a growing number of non-nested variables over a fixed alphabet. A previously known hierarchy was based on nested variables and growing alphabets (Larsen 1998). Matching of PRE without star over a single-letter alphabet is NP-complete. This strengthens the corresponding result for expressions over larger alphabets and with star (Aho 1990). Inequivalence of PRE without closure operators is Sigma^P_2-complete. The decidability of universality of PRE over a single letter alphabet is linked to the existence of Fermat Primes. Greibach's Theorem applies to languages characterized by PRE.

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