Geodesic
On solutions of systems consisting both of word equationa and of word length inequalities
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VI, Tome 40 (1974), pp. 24-29

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It is proved that systems named in the title are undecidable. Moreover some undecidable set $M$ can be represented in the form $a\in M\Leftrightarrow\exists x_1\dots x_n P$ where $P$ is a system of the above mentioned form. However some recursive set cannot be represented in this form. 
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     author = {N. K. Kossovski},
     title = {On solutions of systems consisting both of word equationa and of word length inequalities},
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     volume = {40},
     year = {1974},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_40_a4/}
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N. K. Kossovski. On solutions of systems consisting both of word equationa and of word length inequalities. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VI, Tome 40 (1974), pp. 24-29. http://geodesic.mathdoc.fr/item/ZNSL_1974_40_a4/