Review Comment:
The paper provides motivation for studying conjunctive query answering in a contextualized setting, and studies cases where this can be done by materialization. More specifically, after redefining the chase in this setting, the contributions are:
 the definition of safe, msafe and csafe quad systems;
 the study of the complexity of reasoning for such quad systems;
 how to recognize such quad systems;
 restriction of the problem to the case where bridge rules are rangerestricted
 a comparison with similar acyclicity notions in the framework of existential rules (and a proof tha safeness is more general in a sense than MFA)
Compared to the previous version of paper, a lot of shortcomings have been addressed and the quality has largely improved. In particular, the complexity study of the introduced classes has been corrected, and a thorough comparison with similar criteria known in the literature has been added. There are still quite some minor improvements to be done, and two technical points that need correction/better explanation:
 p.17, definition of augmented rules: adding c_c:(y_j, descendantOf,y_j) is very surprising. It is stated in the right column that it is done in order to "maintain also the reflexivity of `descendantOf' relation". However:
 it is stated p.8 that this relation is not reflexive
 it seems to me that if the relation is reflexive, then all safeness test trivially fails as soon as there is a blank node generated (since one can take b = b')
Could you elaborate on this?
 p.20, Th.5.3: the reduction provided in Dantsin et. al is not adaptable here, since it uses predicates of arbitrary arity. It actually seems to me that the hardness result does not hold.
As for the significance for the paper, it could be improved by clearly answering the following questions: why are the introduced safeness notions important with respect to the zoology of previously defined acyclicity notions? Does safeness covers interesting cases that are not covered by MFA, in particular when dealing with contexts? Is csafeness easy to recognize?
The style of the paper could be greatly improved by making shorter sentences and using less commas. At some points (some of them mentioned below), it becomes really tricky to keep track of the sentence.
"Recently, as a solution the aforementioned problem, SW community, adopts the use of a quad, an extension of a triple, as the primary carrier of knowledge": this sentence is quite hard to parse. Maybe a more light use of commas may help?
"A quad c:(s,p,o), thus adds the fourth component": ungrammatical use of the comma
p.2, col1, l9: "the latest Billion triple challenge datasets has been" > "have been"
"While reasoning with knowledge...": this sentence is also hard to parse
p.3, col 1, item 1: "Studying conjunctive query answering." > here a comma, instead of a point
p.3, col 1, item 4: \forall\exists rules: the terms "existential rules" or "tuplegenerating dependencies" seem to be more widely accepted than \forall\exists
p.4, col 2 : I have not found any explanation of the acronym "LIR" before (or during) its introduction
p.7 : "Although, we now know how" : ungrammatical use of a comma
p.8, col.1, undecidability results: the original reference from BeeriVardi should be cited (A Proof Procedure for Data Dependencies, J.ACM, 84). Moreover, the provided proof is very similar to the proof detailed in the research report associated with: Markus Krötzsch, Sebastian Rudolph, Pascal Hitzler: Conjunctive Queries for a Tractable Fragment of OWL 1.1. ISWC/ASWC 2007: 310323. I have not checked in detail the two pages and a half of proof.
p.8, col.2, l14: lemma 3.3 > Lemma 3.3 (references to specific figures, theorems, are capitalized)
p.8, Def 4.1: x[\mu^{ext(y)}]: this is actually the same thing as x[\mu], or am I mistaken? Moreover, please cut the sentence before the introduction of w. Moreover, announcing Example 4.3 just after the definition may be useful.
p.9, caption of Fig.1 : note > not
p.10, Ex. 4.6: this example seems to be unnecessarily long at this point. The same result could be obtain by removing r_3,r_4 and r_5. Explicitly writing the origin vectors of b_1 and b_2 may also ease the understanding.
p.14, section 4.1: this proof is very similar to the one of 2exptime hardness for weaklyacyclic rules (Towards more expressive ontology languages: The query answering problem, Cali, Gottlob, Pieris, Artificial Intelligence Volume 193, December 2012, Pages 87–128), Theorem 5.1
Recognition of the safety conditions: all the acyclicity notions so far have been provided with their complexity of recognition. Maybe adding it here as well would be beneficial
p.19, col.1, par.1, l2: dChase iterations is > are
Section 6: the comparison between acyclicity for existential rules and safeness for quad system is somewhat strange. Indeed, acyclicity criteria for existential rules are meant to be "dataindependent". In other terms, it ensures that whatever the quads are, a set of bridge rules ensure termination of the chase. This is not the case (nor the intention) for the safeness condition that are proposed here. Maybe expressing clearly this distinction may be useful to the reader.
p.21, col.1,l1: p_n(x,y) > p_n(x,z)
p.21, proof of Th 5.5: DP: the use of this abbreviation is not justified by an intense use of the expression "decision problem", and hinders the reading
p.21and elsewhere : s.t. > such that
p.21: of the from (2): this formula appears on page 4. It would ease the reading to repeat the formula here. Moreover, it would also be clearer to first define (in a separate definition) the transformation, and then to give its properties.
p.25: Bernardo et al. : Bernardo is the first name (it is correctly cited elsewhere)
p.26: "A deterministic algorithm.." : actually, the cited algorithm considers only a subclass of bts, called gbts.
p27: "recently, a more dynamic approach...": the approach is mainly used (at least in the proposed experiments) to detect a *dataindependent* termination, by starting from the socalled critical instance.
p.31, proofs of Section 4: the proofs [...] is
