Polycyclic and Bicyclic Valence Automata

Structurally and Arithmetically Controlled Grammars

“…Besides, in that paper, they also proved that context free valence grammars over commutative or finite monoids have an equivalent power as valence grammars over commutative groups or finite group respectively. Later, after a few years which is in 2008, Render and Kambites [41] continued the study done by [45], where they investigate the languages class recognized by polycyclic and bicyclic valence automata (or identically regular valence grammar) with rational target as well as together with the closure properties and rational subset membership decidability problem of those grammar where as their main results they showed that such automata have accepted exactly the languages of contextfree for the case polycyclic monoids of rank two or more and the languages class which including the languages of partially blind one counter for the case bicyclic monoids (polycyclic monoid of rank 1) [41].…”

“…Despite their diversity, all of the introduced regulated grammars can be classified into several types depending on their common characteristics like (1) control by prescribed sequences such as matrix grammars [12][13][14][15][16][17], regularly controlled grammars [18], vector grammars [19], different variants of Petri net controlled grammars [20][21][22][23][24][25][26] and Parikh vector controlled grammars [27], (2) control by context conditions such as conditional grammars and ordered grammars [28], random context grammars [29], tree controlled grammars [30][31][32][33][34][35][36][37], semi-conditional grammars [38] and string-regulated graph grammars [39], (3) control by computed sequences such as programmed grammars [40] and valence grammars [41][42][43][44][45][46][47], (4) control by memory such as indexed grammars [48], (5) control by partial parallelism such as scattered context grammars [49], Russian parallel grammars [50], Indian parallel grammars [51] and global indexed grammars [52] and many other regulated grammars ...…”