Review Comment:
The paper has been improved and can be accepted. Some minor (mostly linguistic) remarks are listed below.
p 2, Example 1: Q = \exists y play (x,y) is rather confusing -- on the one hand, the query has an answer variable (x), which is not listed as an argument of Q; on the other hand, the authors prefer to omit the quantifiers altogether in the rest of the paper. One can suggest to use Q(x) (and respectively, Q'(x)) in Example 1.
p 2, Example 1: font usage for movie, play, actor should be unified
p 3, above Paper contributions: what is the precise reference in [RK13] (i.e., theorem number)?
p 4, after the 3 items: "Moreover, if we delete ... then we obtain"
p 5, Example 5: the use of quantifiers in queries is rather confusing again --- Q has some, but Q_1 does not
p 5, Example 5: "despite Q being not"
p 5, par 5: "When additionally the head of a rule R"
p 5, par 6: "fewer queries"
p 6, par 2: "denote its sets of variables, constants and terms, respectively"
p 6, Def 1: noted -> denoted
p 6, after Def 1: quantifiers are omitted not only in rule, but also in CQs
p 7, Example 3: "and consider the following preorder"
p 8, Example 4: it seems that one rule R_1 is enough to active the effect; perhaps, it is also worth pointing out that R_1 is not fus
p 8, proof of Th 1: the cardinalities of covers, strictly speaking, are not smaller -- they do not exceed cardinalities of the rewriting sets
p 8, footnote 3: "f.i." -> "e.g.," or "for instance,"
p 9: the elements of rew(Q,R) are not just some queries, they are BCQs (also, the terminology should be consistent throughout the section)
p 9, Def 7: just a suggestion to use rew_{\leq k} for W_k --- otherwise, there are too many symbols with similar meanings
p 10, Def 8: sentences should not begin with a formula -- one could use, for example, "A rewriting operator rew is said to be prunable if …"
p 11, proof of Property 5: there is no need to repeat the 4 invariants in the proof
p 12, above Property 6: "owns a finite cover" -> "has a finite cover"
p 12, proof of Property 6: "proved by straightforward induction"
p 13, proof of Property 9: "we are sure that" -> "we obtain"
p 13, proof of Theorem 10: "We conclude with" -> "The claim then follows from"
p 14, Def 11: "satisfying the following three conditions"
p 19, Example 8, last line: remove "so"
p 19, Example 9: it is not clear why there is an "e.g." after "yields"
p 21, Def 15: Given such *a* compatible
p 26, par 2: "if a rule R has an atomic head then every atom in Q..."
p 27, last line of Sec 7.2: "hereafter" -> "below"
pp 29 and 31: Tables 2 and 4 use "Base" for an ontology -- why? set of rules might be better (but it is certainly not a knowledge base)
p 30, line -5: "being less" (no of)
p 31, Table 4: runtime in ms?
p 31, Table 4: 0ms looks a bit strange -- the timing for rapid presumably included JVM startup time, parsing, etc.; and it is hard to believe that that could be done instantaneously for PURE... by the way, is PURE an abbreviation? Is it available to download?
p 32: [CGL12] is a journal version of [CGL09]
pp 32-33: abbreviate names in [CLR03], [OP11], [VSS14] (and check publication titles)
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