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Non-Self-Embedding Grammars and Descriptional Complexity

Pighizzini Giovanni, Prigioniero Luca

Fundamenta Informaticae

2021

Vol. 180, nr 1-2

103--122

Non-self-embedding grammars are a subclass of context-free grammars which only generate regular languages. The size costs of the conversion of non-self-embedding grammars into equivalent finite automata are studied, by proving optimal bounds for the number of states of nondeterministic and deterministic automata equivalent to given non-self-embedding grammars. In particular, each non-self-embedding grammar of size s can be converted into an equivalent nondeterministic automaton which has an exponential size in s and into an equivalent deterministic automaton which has a double exponential size in s. These costs are shown to be optimal. Moreover, they do not change if the larger class of quasi-non-self-embedding grammars, which still generate only regular languages, is considered. In the case of letter bounded languages, the cost of the conversion of non-self-embedding grammars and quasi-non-self-embedding grammars into deterministic automata reduces to an exponential of a polynomial in s.

Decision Problems for Probabilistic Finite Automata on Bounded Languages

Bell P. C., Halava V., Hirvensalo M.

Fundamenta Informaticae

2013

Vol. 123, nr 1

1--14

We show that several problems concerning probabilistic finite automata of a fixed dimension and a fixed number of letters for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert’s tenth problem. We then consider the set of so called “F-Problems” (emptiness, infiniteness, containment, disjointness, universe and equivalence) and show that they are also undecidable for bounded (non-)strict cut-point languages on probabilistic finite automata. For a finite set of matrices { M1 , M2 , . . . , Mk }⊆ Q txt , we then consider the decidability of computing the maximal spectral radius of any matrix in the set X = { Mj11 Mj22··· Mjkk|j1, j2, . . . , jk≥0 } , which we call a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining if the maximal spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining whether it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata).