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On a representation of measurable automaton transformations by stochastic automata. (English) Zbl 0365.94076

Let \((X,\mathcal A)\) be a measurable, \((Y,\mathcal B)\) be a Borel space, and denote by \((X^*,\mathcal A^*)\) the direct sum of \(((X^n,\mathcal A^n))_{n\geq 0}\). If \(K\) is a transition probability from \((X^*,\mathcal A^*)\) to \((Y^*,\mathcal B^*)\), \(K\) is said to be a stochastic transformation iff for every \(v\in X^*\), \(K(v)(Y^{| v|}) = 1\), and if for every \(x\in X\), \(B\in\mathcal B^*\), \(K(vx)(B\times Y) = K(v)(B)\) holds \((| v|\) denoting the length of \(v)\). By means of a disintegration theorem due to D. Rhenius [Markoffsche Entscheidungsprozesse mit unvollständiger Information und Anwendungen in der Lerntheorie. Thesis, Hamburg (1971), see also Ann. Stat. 2, 1327–1334 (1974; Zbl 0294.49007)] it is shown that \(K\) is a stochastic transformation iff it is the behavior of an initial stochastic automaton. If \(Y\) is \(\sigma\)-compact and Polish, an automaton transformation \(R\) (i.e. the behavior of an initial complete nondeterministic automaton, cf. A. Schmitt [Computing 4, 56–74 (1969; Zbl 0213.02203)]) is said to be measurable iff \(R\) is closed-valued, and if \(X^*\ni v\rightarrow R(v)\subset Y^*\) is (weakly) measurable [C. J.Himmelberg, Fundam. Math, 87, 53–72 (1975; Zbl 0296.28003)].
The main result of this paper is that \(R\) is measurable iff there exists a stochastic transformation \(K\) such that \(\forall v\in X^*\) : \(R(v)=\text{supp}(K(v))\), \(\text{supp}\) denoting support. This is shown by means of measurable selections [D. H. Wagner, SIAM J. Control Optim. 15, 859–903 (1977; Zbl 0407.28006)] for the weakly measurable relation that assigns to every \(v\) the set of all probabilities which have their support in \(R(v)\). This result implies that a closed-valued relation \(F\) from \(X\) to \(Y\) is measurable iff there exists a transition probability from \((X,\mathcal A)\) to \((Y,\mathcal B)\) such that \(F(x) = \text{supp}(K(x))\) holds for every \(x\in X\).

MSC:

68Q45 Formal languages and automata
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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[1] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201
[2] Bourbaki, N.: ”General topology,” part 2. (1966) · Zbl 0145.19302
[3] Brauer, W.: Zu den grundlagen einer theorie topologischer sequentieller systeme und automaten. Report 31 (1970) · Zbl 0216.56503
[4] Claus, V.: Stochastische automaten. (1971) · Zbl 0226.94046
[5] Doberkat, E. -E: Optimal strategies in measurable learning systems on metric spaces. J. appl. Probability 14, 795-805 (1977) · Zbl 0385.68069
[6] Dunford, N.; Schwartz, J. T.: ”Linear operators,” part I. (1957) · Zbl 0128.34803
[7] Himmelberg, C. J.: Measurable relations. Fund. math. 87, 53-72 (1975) · Zbl 0296.28003
[8] Himmelberg, C. J.; Van Vleck, F. S.: Multifunctions with values in a space of probability measures. J. math. Anal. appl. 50, 108-112 (1975) · Zbl 0299.54009
[9] Hinderer, K.: Foundations of non-stationary dynamic programming with discrete time parameter. Lecture notes in operations research and mathematical systems no. 33 (1970) · Zbl 0202.18401
[10] Menzel, W.: An extension of the theory of learning systems. Acta informatica 2, 357-381 (1973) · Zbl 0261.68043
[11] Parthasarathy, K. R.: Probability measures on metric spaces. (1967) · Zbl 0153.19101
[12] Rhenius, D.: Markoffsche entscheidungsmodelle mit unvollständiger information und anwendungen in der lerntheorie. Doctoral dissertation (1971)
[13] Rieder, U.: Bayesian dynamic programming. Advances in appl. Probability 7, 330-348 (1975) · Zbl 0316.90081
[14] Rockafellar, R. T.: Integral functionals, normal integrands, and measurable selections. Lecture notes in mathematics no. 543, 157-207 (1976) · Zbl 0374.49001
[15] Schmitt, A.: Zur theorie der nichtdeterministischen und unvollständigen automaten. Computing 4, 56-74 (1969) · Zbl 0213.02203
[16] Starke, P. H.: Theorie stochastischer automaten. Elektron. informationsverarbeit. Kybernetik 1, 71-98 (1965) · Zbl 0149.01003
[17] Wagner, D. H.: Survey of measurable selection theorems. SIAM J. Control optimization 15, 859-903 (1977) · Zbl 0407.28006
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