带置换规则的Lambek演算文法：识别能力及其与带状态的分支向量加法系统的联系
Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States
AI总结：
本文证明带置换规则的Lambek演算文法（LP-grammars）等价于线性受限的带状态分支向量加法系统（lBVASSAM），并通过构造生成非半线性向量集的例子证明LP-grammars的生成能力严格强于上下文无关语言的置换闭包，同时给出其范式并证明语言类对交集封闭。
摘要：
在（Van Benthem, 1991）中证明了上下文无关语言的所有置换闭包都可以由带置换规则的Lambek演算文法（LP-grammars）生成；然而，据我们所知，其逆命题是否成立尚未得到确定。本文中，我们证明LP-grammars等价于线性受限的带状态且带附加存储的分支向量加法系统（简称lBVASSAM），后者是带状态的分支向量加法系统的一种修改形式。随后给出一个这样的lBVASSAM的例子，它生成一个非半线性的向量集合；这表明LP-grammars生成的类比上下文无关语言的置换闭包更多。此外，LP-grammars与lBVASSAM的等价性使我们能够给出LP-grammars的一种范式，并由此证明LP-grammars等价于不含乘积的LP-grammars。最后，我们证明由LP-grammars生成的语言类在交集运算下封闭。
英文摘要：
In (Van Benthem, 1991) it is proved that all permutation closures of context-free languages can be generated by grammars over the Lambek calculus with the permutation rule (LP-grammars); however, to our best knowledge, it is not established whether the converse holds or not. In this paper, we show that LP-grammars are equivalent to linearly-restricted branching vector addition systems with states and with additional memory (shortly, lBVASSAM), which are modified branching vector addition systems with states. Then an example of such an lBVASSAM is presented, which generates a non-semilinear set of vectors; this yields that LP-grammars generate more than permutation closures of context-free languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to present a normal form for LP-grammars and, as a consequence, prove that LP-grammars are equivalent to LP-grammars without product. Finally, we prove that the class of languages generated by LP-grammars is closed under intersection.
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