How to understand this formula in Direct Construction of Minimal Acyclic Subsequential Transducers?

Question

In this paper "Direct Construction of Minimal Acyclic Subsequential Transducers" https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=690EBF51DD6D52AB9160723AAE3A9541?doi=10.1.1.24.3698&rep=rep1&type=pdf

1. in Definition 3, what does !(u(r,a)) mean? I can't understand this formula as a whole.

2 In lemma 3, what does ¬!u(tk,a) mean? It come out without explaination.

Answer 1

In the same author's earlier paper, Direct Building of Minimal Automaton for a Given List , it is specified (under Background and Notations ) that

we use !μ(r,σ) to denote that μ(r,σ) is defined

I suppose it is the same in this paper.

I can only guess what ¬!μ(r,σ) might mean; I assume that ¬ is being used as a negation operator, which is what would be usual. That is, it means that μ(r,σ) is not defined.

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