Characterising REGEX languages by regular languages equipped with factor-referencing

Deterministic regular expressions with back-references

“…The thus defined class of deterministic regex is denoted by DRX, and L(DRX) refers to the corresponding language class. The underlying automaton model for this approach is a slight modification of the memory automata (MFA) proposed by Schmid [46] as a characterisation for the class of regex-languages. More precisely, we introduce memory automata with trap-state (TMFA), for which the deterministic variant, the DTMFA, is better suited for complementation than the deterministic MFA.…”

“…We define the semantics of regex using the ref-words (short for reference words) by Schmid [46]. A ref-word is a word over (Σ ∪ Ξ ∪ Γ), where Γ : ={[ x , ] x , | x ∈ Ξ}.…”

“…In this way, a ref-word r compresses a word over Σ, the so-called dereference D(r) of r, which can be obtained by replacing every variable occurrence x by the corresponding factor v (note that v might again contain variable occurrences, which need to be replaced as well), and removing all symbols [ x , ] x ∈ Γ afterwards. See [46] for a more detailed definition, or the following Example 1 for an illustration. Finally, the language of a regex α is defined by L(α) : ={D(r) | r ∈ R(α)}.…”

“…Aho [2]).Regex were first examined from a theoretical point of view by Aho [2], but without fully defining the semantics. There were various proposals for semantics, of which we mention the first by Câmpeanu, Salomaa, Yu [10], and the recent one by Schmid [46], which is the basis for this paper. Apart from defining the semantics, there was work on the expressive power [10,11,25], the static analysis [11,23,24], and the tractability of the membership problem (investigated in terms of a strongly restricted subclass of regex) [21,22].…”

The Teaching Complexity of Erasing Pattern Languages with Bounded Variable Frequency

“…Shinohara [35] later introduced the class of erasing pattern languages, proving that the class of all such languages generated by regular patterns (patterns in which every variable occurs at most once) is polynomial-time learnable in the limit. Patterns and allied notions -such as that of an extended regular expression [1,9,33,14], which has more expressive power than a pattern -have also been studied in other fields, including word combinatorics and pattern matching. For example, the membership problem for pattern languages is closely related to the problem of matching 'patterns' with variables (based on various definitions of 'pattern') in the pattern matching community [6,2,12,10,11].…”