In matrix grammars, context-free rules have to be applied in a certain order. In simple semi-conditional (SSC) grammars, the derivations are controlled either by a permitting string or by a forbidden string associated to each rule. In SSC grammars, the maximum length i of permitting strings and the maximum length j of forbidden strings, the numbers of conditional rules and of nonterminals serve as measures of descriptional complexity and the pair $(i,j)$ is called the degree of such SSC grammars. Matrix grammars with appearance checking with three nonterminals are computationally complete; however, the matrix length is unbounded. Matrix SSC grammars (MSSC) put matrix grammar control on SSC rules. In this paper, we show that MSSC grammars with degrees $(2,1)$, $(2,0)$ and $(3,0)$ are computationally complete, restricting several other descriptional complexity measures. With our constructions, we even bound the matrix length for MSSC grammars.

在矩阵文法中，上下文无关规则必须以特定顺序应用。在简单的半条件 (SSC) 语法中，派生由与每个规则关联的允许字符串或禁止字符串控制。在 SSC 文法中，允许字符串的最大长度 i 和禁止字符串的最大长度j 、条件规则的数量和非终结符的数量作为描述复杂性的度量，并且对$(\mathrm{一世},j)$称为此类 SSC 文法的程度。具有三个非终结符的外观检查的矩阵文法在计算上是完整的；但是，矩阵长度是无限的。矩阵 SSC 语法 (MSSC) 将矩阵语法控制置于 SSC 规则上。在本文中，我们展示了带度数的 MSSC 文法$(2,1)$,$(2,0)$和$(3,0)$在计算上是完整的，限制了其他几个描述复杂性度量。通过我们的构造，我们甚至限制了 MSSC 语法的矩阵长度。