Towards the Complexity of Petri Nets and One Counter Machines via Coordinated Table Selective Substitution Systems

Abstract

We investigate computational resources used by Turing machines (TMs) and alternating Turing machines (ATMs) to accept languages generated by coordinated table selective substitution systems with two components. We prove that the class of languages generated by real-time (<tex>$RL;\ 0S$</tex>)-systems, an alternative device to generate <tex>$\lambda$</tex>-free labeled marked Petri nets languages, can be accepted by nondeterministic TMs in <tex>$\mathcal{O}(\log n)$</tex> space and <tex>$\mathcal{O}(n\log n)$</tex> time. Consequently, this proper subclass of Petri nets languages (known also as <tex>$\mathcal{L}$</tex>-languages) is included in <tex>$NSPACE (\log n)$</tex>. The class of languages generated by (<tex>$RL; RB$</tex>)-systems for which the nonterminal alphabet of the <tex>$RL$</tex>-grammar is composed of only one symbol and the nonterminal alphabet of the <tex>$RB$</tex>-grammar is composed of two symbols, can be accepted by ATMs in <tex>$\mathcal{O}(\log n)$</tex> time and space. Consequently, this proper subclass of one counter languages generated by one counter machines with only one control state is included in <tex>$U_{E^{\ast}}$</tex>-uniform <tex>$\mathcal{NC}^{1}$</tex>, hence in <tex>$SPACE (\log n)$</tex>.