Experiments and stability in group automata (Q2277864)

Let \({\mathcal A}=(Q,A,B,\delta,\lambda)\) be a Mealy automaton such that Q,B,A are \(\Omega\)-groups and \(\delta\) : \(Q\times A\to Q\), \(\lambda\) : \(Q\times A\to B\) are \(\Omega\)-homomorphisms. Define \(\delta^*: Q\times A^*\to Q\) as usual, and \(\lambda^*: Q\times A^*\to B^*\) such that \(\lambda^*(q,\Lambda)=\Lambda\), \(\lambda^*(q,a_ 1,...,a_ r)=\lambda (q,a_ 1)\lambda^*(\delta (q,a_ 1)\), \(a_ 2...a_ r)\). Two states \(q_ 1,q_ 2\in Q\) are equivalent, we write \(q_ 1\sim q_ 2\), if \(\lambda^*(q_ 1,\alpha)=\lambda^*(q_ 2,\alpha)\) for every \(\alpha \in A^*\). For \(U\subseteq Q\), a word \(\alpha \in A^*\) is called U-identifier experiment of first order, in short UIE1, (or U- identifier experiment of second order in short UIE2) for \({\mathcal A}\) if \(\lambda^*(q_ 1,\alpha)=\lambda^*(q_ 2,\alpha)\) implies \(q_ 1=q_ 2\) (or \(\delta^*(q_ 1,\alpha)=\delta^*(q_ 2,\alpha))\) for all \(q_ 1,q_ 2\in U\). We say that \({\mathcal A}\) is k-stable (or weakly k-stable) if \(\delta^*(q_ 1,\alpha)=\delta^*(q_ 2,\alpha)\) (or \(\delta^*(q_ 1,\alpha)\sim \delta^*(q_ 2,\alpha))\) for all \(q_ 1,q_ 2\in Q\) and \(\alpha \in A^*\) of length k. Set \(\delta_ 0(q)=\delta (q,0)\), \(\delta^ k_ 0(q)=\delta^*(q,00...0)\), \(\lambda_ 0(q)=\lambda (q,0)\) where 0 is the neutral element of the group A. Then there exists a UIE1 iff \(U\cap (\cap \{Ker \lambda_ 0\delta^ i_ 0\); \(i=0,...,k-1\})=\{0\}\) for some k, there exists a UIE2 iff \(U\cap (\cap \{Ker \lambda_ 0\delta^ i_ 0\); \(i=0,...k- 1\})\subseteq U\cap Ker \delta^ k_ 0\) for some k. The automaton \({\mathcal A}\) is k-stable iff Ker \(\delta\) \({}^ k_ 0=Q\), \({\mathcal A}\) is weakly k-stable iff Im \(\delta\) \({}^ k_ 0\subseteq Ker \lambda_ 0\). Consequences are derived if Q is a solvable group.