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A Living Russian Grammar by Natalia Bitekhtina, Larisa Grushevskaya, Yulia Sheina

By Natalia Bitekhtina, Larisa Grushevskaya, Yulia Sheina

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To denote variables and upper-case bold letters to denote sets of variables X, Y, Z. , ‘Moses’, ‘Isaac’, etc. Terms are either constants or variables and are denoted t, t1 , t2 , etc. An atom a is of the form p(t1 , . . , tn ) where p is a predicate symbol of arity n and each ti is a term. We use pred(a) to denote the predicate of a and we use ar(a) as a shorthand notation for ar(pred(a)). For an atom a, we denote by ti (a) a term which appears at position i. Terms which are mapped to a constant are said to be bound; other terms are free.

In general, it might be possible that q2 and further nodes can be matched in subtree(d2 ). The function call TM(d2 , q2 , q4 ) checks that possibility. ) But q2 is not labeled with a so the return value of the two TM calls is q1 . After this initial phase, HM(d2 , q1 , q5 ) tries to improve qtree and qhedge iteratively. It calls HM(d1 , q2 , q4 ) and improves qhedge to be q2 , because q2 and d1 are both labeled with b. Further improvements fail as there is no c-labeled node in the subhedge of d2 .

Towards a contradiction, assume that there is an u such that D |=u Q, but u was not reported by TMatch-All. By an easy induction it can be shown that for every data node d0 in D there is a call TMatch-All for d0 ’s subtree and Q. In particular, there was a call TMatch-All(u, qfrom , qroot ). Since 30 M. G¨ otz, C. Koch, and W. lastChild, qfrom , qroot ) < qroot − 1, (because otherwise qroot and u would have been compared and u would have been written to the output). In general, we have that HMatch-All(d, q1 , q2 )=min (HMatch(d, q1 , q2 ), qroot −1).

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