Hybrid modes in cooperating distributed grammar systems: combining the t-mode with the modes ⩽k and =k

Abstract

We continue our research on hybrid derivation modes of cooperating distributed (CD) grammar systems started in (Fernau et al., Proc. 3rd Internat. Conf. Developments in Language Theory, Aristotle University of Thessaloniki, 1997, pp. 261–272; Theoret. Comput. Sci. 259 (2001) 405). Here, we focus on (hybrid) systems with context-free components working in a hybrid t- and Δk-mode, Δ∈{⩽,=}. We prove that CD grammar systems working in the mode (t∧=1), which means that each component cannot perform more than one derivation step in a row, characterize the context-free programmed languages with finite index. Moreover, we show that additional external hybridization of this mode with, e.g., the =1-mode (or =2-mode, respectively) leads to a characterization of the context-free ordered languages (or the context-free programmed languages with appearance checking, respectively). In particular, hybrid CD grammar systems with classical and internally hybrid modes have universal computational power, a result which is unknown for hybrid CD grammar systems whose components work only in classical modes, since such systems are computationally equivalent to recurrent programmed grammars with appearance checking, as shown in (Theoret. Computer. Sci. 259 (2001) 405). Furthermore, we obtain a characterization of ordered languages within the framework of CD grammar systems.