We discuss substitutions as a further type of operations, added to matrix insertion-deletion systems. For such systems, we additionally discuss the effect of appearance checking. This way, we obtain new characterizations of the families of context-sensitive and the family of recursively enumerable languages. To reach computational completeness, not much context is needed for systems with appearance checking.
It is well known that for a given bottom-up tree automaton it can be decided whether or not there exists deterministic top-down tree automaton that recognized the same tree language. Recently it was claimed that such a decision can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their procedure and corresponding property is wrong. Here we correct this mistake and present a correct property which allows to determine in polynomial time whether or not a given tree language can be recognized by a deterministic top-down tree automaton. Furthermore, our new property is stated for arbitrary deterministic bottom-up tree automata, and not for minimal such automata (as before).
In this paper, we discuss the addition of substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems, i.e., in addition to insertions and deletions we allow single letter replacements to occur. We investigate the effect of the addition of substitution rules on the context dependency of such systems, thereby also obtaining new characterizations of and even normal forms for context-sensitive (CS) and recursively enumerable (RE) languages and their phrase-structure grammars. More specifically, we prove that for each RE language, there is a system generating this language that only inserts and deletes strings of length two without considering the context of the insertion or deletion site, but which may change symbols (by a substitution operation) by checking a single symbol to the left of the substitution site. When we allow checking left and right single-letter context in substitutions, even context-free insertions and deletions of single letters suffice to reach computational completeness. When allowing context-free insertions only, checking left and right single-letter context in substitutions gives a new characterization of CS. This clearly shows the power of this new type of rules.
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