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arXiv 2107.04025cs.FL
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非确定性和无歧义Petri网在无限词上的表达力

On the expressive power of non-deterministic and unambiguous Petri nets over infinite words

Olivier Finkel, Michał Skrzypczak

AI总结:

本文证明非确定性Petri网与图灵机的ω-语言拓扑复杂度相同,且相关判定问题高度不可判定;相反,无歧义Petri网可确定化为Muller计数器机,其ω-语言为Δ₀₃集合。

摘要:

我们证明了(非确定性)Petri网的$\omega$-语言与(非确定性)图灵机的$\omega$-语言具有相同的拓扑复杂度:非确定性Petri网的$\omega$-语言类的Borel和Wadge层级等于非确定性图灵机的$\omega$-语言类的Borel和Wadge层级。我们还表明,确定一个Petri网$\omega$-语言的拓扑复杂度是高度不可判定的。此外,从上述结果的证明中我们推断出,Petri网的$\omega$-语言的等价性和包含性问题都是$\Pi_2^1$-完全的,因此也是高度不可判定的。另外,我们表明,当考虑无歧义Petri网时,情况恰恰相反,无歧义Petri网具有在每个输入上至多存在一个接受运行的语义性质。我们提供了一种将它们确定化为带有计数器复制的确定性Muller计数器机的过程。因此,我们得出无歧义Petri网可识别的$\omega$-语言是$\Delta^0_3$集合。

英文摘要:

We prove that $ω$-languages of (non-deterministic) Petri nets and $ω$-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $ω$-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $ω$-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net $ω$-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for $ω$-languages of Petri nets are $Π_2^1$-complete, hence also highly undecidable. Additionally, we show that the situation is quite the opposite when considering unambiguous Petri nets, which have the semantic property that at most one accepting run exists on every input. We provide a procedure of determinising them into deterministic Muller counter machines with counter copying. As a consequence, we entail that the $ω$-languages recognisable by unambiguous Petri nets are $Δ^0_3$ sets.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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