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Decidability Questions for Insertion Systems and Related Models

Malcher Andreas

Fundamenta Informaticae

2021

Vol. 180, nr 1-2

53--76

Insertion systems or insertion grammars are a generative formalism in which words can only be generated by starting with some axioms and by iteratively inserting strings subject to certain contexts of a fixed maximal length. It is known that languages generated by such systems are always context sensitive and that the corresponding language classes are incomparable with the regular languages. On the other hand, it is possible to generate non-semilinear languages with systems having contexts of length two. Here, we study decidability questions for insertion systems. On the one hand, it can be seen that emptiness and universality are decidable. Moreover, the fixed membership problem is solvable in deterministic polynomial time. On the other hand, the usually studied decidability questions such as, for example, finiteness, inclusion, equivalence, regularity, inclusion in a regular language, and inclusion of a regular language turn out to be undecidable. Interestingly, the latter undecidability results can be carried over to other models which are basically able to handle the mechanism of inserting strings depending on contexts. In particular, new undecidability results are obtained for pure grammars, restarting automata, clearing restarting automata, and forgetting automata.

Clearing Restarting Automata

Černo P., Mráz F.

Fundamenta Informaticae

2010

Vol. 104, nr 1/2

17-54

Restarting automata were introduced as a model for analysis by reduction, which is a linguistically motivated method for checking correctness of a sentence. We propose a new restricted version of restarting automata called clearing restarting automata with a very simple definition but simultaneously with interesting properties with respect to their possible applications. The new model can be learned very efficiently from positive examples and its stronger version can be used to learn effectively a large class of languages. We relate the class of languages recognized by clearing restarting automata to the Chomsky hierarchy.