Geodesic
On dynamic theories of free algebras
Sbornik. Mathematics, Tome 66 (1990) no. 2, pp. 313-327

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A certain canonical form is introduced for formulas of the logic $L_{\infty\omega}(n)$, and it is proved that every formula of this logic is equivalent on free algebras to some canonical formula. This permits one to establish that no finite number of programs is sufficient to express all inquiries expressible in dynamic logic and involving the free algebra under consideration. Also obtained, as a corollary, are results to the effect that an infinite memory increases the expressive possibilities of dynamic logic. Bibliography: 7 titles. 
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     author = {I. Kh. Musikaev and M. A. Taitslin},
     title = {On dynamic theories of free algebras},
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I. Kh. Musikaev; M. A. Taitslin. On dynamic theories of free algebras. Sbornik. Mathematics, Tome 66 (1990) no. 2, pp. 313-327. http://geodesic.mathdoc.fr/item/SM_1990_66_2_a0/