State grammars with stores

Abstract State grammars are context-free grammars where the productions have states associated with them, and a production can only be applied to a nonterminal if the current state matches the state in the production. Once states are added to grammars, it is natural to add various stores, similar to machine models. With such extensions, productions can only be applied if both the state and the value read from each store matches between the current sentential form and the production. Here, generative capacity results are presented for different derivation modes, with and without additional stores. In particular, with the standard derivation relation, it is shown that adding reversal-bounded counters does not increase the capacity, and states are enough. Also, state grammars with reversal-bounded counters that operate using leftmost derivations are shown to coincide with languages accepted by one-way machines with a pushdown and reversal-bounded counters, and these are surprisingly shown to be strictly weaker than state grammars with the standard derivation relation (and no counters). The complexity of the emptiness problem involving state grammars with reversal-bounded counters is also studied.

[1]  Matthias Jantzen,et al.  Petri net algorithms in the theory of matrix grammars , 2005, Acta Informatica.

[2]  Ralf Stiebe.  Slender matrix languages , 1999, Developments in Language Theory.

[3]  S. Ginsburg,et al.  Finite-Turn Pushdown Automata , 1966 .

[4]  Oscar H. Ibarra,et al.  Grammatical characterizations of NPDAs and VPDAs with counters , 2018, Theor. Comput. Sci..

[5]  Oscar H. Ibarra,et al.  The effect of end-markers on counter machines and commutativity , 2016, Theor. Comput. Sci..

[6]  Eitan M. Gurari,et al.  The Complexity of Decision Problems for Finite-Turn Multicounter Machines , 1981, J. Comput. Syst. Sci..

[7]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[8]  Etsuro Moriya,et al.  Some Remarks on State Grammars and Matrix Grammars , 1973, Inf. Control..

[9]  Gheorghe Paun,et al.  Regulated Rewriting in Formal Language Theory , 1989 .

[10]  Arto Salomaa,et al.  Matrix Grammars with a Leftmost Restriction , 1972, Inf. Control..

[11]  Grzegorz Rozenberg,et al.  On the Effect of the Finite Index Restriction on Several Families of Grammars , 1978, Inf. Control..

[12]  Oscar H. Ibarra,et al.  Reversal-Bounded Multicounter Machines and Their Decision Problems , 1978, JACM.

[13]  Takumi Kasai,et al.  An Hierarchy Between Context-Free and Context-Sensitive Languages , 1970, J. Comput. Syst. Sci..

[14]  Friedrich Otto,et al.  On state-alternating context-free grammars , 2005, Theor. Comput. Sci..

[15]  Sheila A. Greibach.  Remarks on Blind and Partially Blind One-Way Multicounter Machines , 1978, Theor. Comput. Sci..

[16]  Michael A. Harrison,et al.  Introduction to formal language theory , 1978 .