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As concerns this final result, however, it is not completely clear to me its
real significance. Does the disjunctive Datalog program corresponding to the
considered restricted form of set-theoretic formulae constitute a decision
procedure for these formulae? Is termination of the Datalog program always
guaranteed? Does the Datalog reasoning engine provide a complete proof
procedure? In general, mapping set-theretical operations to some first-order
logic, e.g. Prolog, does not guarantee by itself one is able to get a complete
solving procedure for the set formulae. Is the mapping to disjunctive Datalog
different from this?

Conversely, let us consider
the two sets of constants $V={\consta, \constb}$ and $T={t}$
and let $\hinter={\nonpairin{\consta}{\constb}, \seteq{\consta}{t}}$,
$\hinter'=\hinter \cup {\nonpairin{t}{\constb}}$ be two
Herbrand interpretations.

\hinter non può essere considerata una sk-representation perchè contiene \seteq{\consta}{t} con t in T

Many of the definitions and results reviewed in Section 2 are
well-known and maybe not all necessary for the paper. This section should be
significantly shortened.

As concerns Section 3, most of its content comes from
paper [5]. The only change seems to be a "slight difference" in the definition
of the skeletal representation. This does not justify the need to repeat many
of the results already reported in [5]. Maybe one could try to describe more
concisely what are the major differences with the results in [5].

Since all the paper is about formulae satisfiability, the use of the term programs for Datalog formulae could be avoided. Maybe just mentioned in a footnote.

Negation is allowed just in head. Replace negation with integrity constraints in the reduction. Update the whole correspondence.

Review1- In section 4 please make more explicit the connection between Herbrand interpretation and skeletal representation. An example could help.

Review2 - It is not clear to me where the condition |T|=2|V| in Theorem 3 and later comes from.

The discussion about semantics of set-theoretic languages in the
Introduction seems not necessary for the paper and in any case not convincing.
What the authors claim (i.e., using the Von Neumann standard cumulative
hierarchy of sets for the interpretation of set terms) can be true "in general"
(as properly acknowledged), but it is not the only possible approach. Thus, the
conclusion "Because of this, results and techniques ... are not easily
reusable ..." seems rather hasty. If this discussion is important for the
paper, then it should be faced in a wider and deeper way. For the same reasons,
the second paragraph in the Abstract, "In general, the semantics ... realm."
could be completely removed.

Introduzione: eliminerei la frase "Despite of these... has been implemented yet". Trovarla nell’introduzione mi sembra un po’ fuorviante. Da questa affermazione sembrerebbe nell'articolo verrà descritta un’implementazione. Se proprio si deve parlare di implementazione, mi limiterei a farlo nelle conclusioni (come lavori futuri).

Consts D (\psi) = Consts D (\phi){a} is probably wrong, as a may occur as constant in \psi

"is satisfied by the Herbrand interpretation" should be "is satisfied by a Herbrand interpretation"

As for (C2), why not writing the regularity axiom?

Missing D in the last two equivalences

The brief descriptions about Disjunctive Datalog
at the end of Sect. 2 and of Sect. 3 should be maintained and possibly
extended. In particular, the syntactic restrictions on formula (7) highlighted
in Sect. 3.2 could be described more extensively, possibly providing some
examples of the considered formulae.

Theorem 2 which is referred in the text as Theorem 8.7.1 of [10] seems more Theorem 8.6.5 of [10].

Section 4 is the only truly original part of the paper. Some examples here and, in general, throughout the paper would help the reader to better appreciate the claimed results.

Review2: In section 3.1 some examples of skeletal representations and realizations should be added.

Also the reference to
"applications of techniques and results devised in the context of logic
programming" is too vague, and it should be further investigated or deleted.

Section 4 terminates with a very quick hint at another interesting result about
the correspondence between a restricted form of the considered set formulae and
disjunctive Datalog programs. Although it appears as one of the most
interesting achievements of the paper, it is confined to a few lines at the end
of the paper. Applicability of this result, even in informal terms, should be
further discussed and some examples should be shown to make the result more
appreciable and accessible.

The authors acknowledge this need in the Conclusions "Such a correspondence,
and its consequences, has to be further investigated.". In my opinion, this
issue should not be left for future work: it is appropriate and it is possible
to investigate it, at least partially, in this paper

Sezione 2.1: la frase "this result can be further refined... formulae." risulta sibillina, se si hanno in mente classi di formule diverse da quelle trattate nell’articolo, queste dovrebbero essere rese esplicite.

Is it possible to write a formula of the form “x=[x_1,x_2]”? From (3) this seems not allowed. Is it possible to write a formula expressing “x\in y” meaning that either “x\in \pi(y)” or “x=[x_1,x_2]\wedge [x_1,x_2]\in y”?
In other words are the axioms (C1), (C2), … expressible in the language?

Which is the difference between variables and constants? Since both Vars and Consts are just two denumerably infinite sets with no conditions on the interpretation of the constants and you study the satisfiability problem where both the interpretation for constants and variables are free, I am missing the point of making a distinction.
I checked in [5] and there constants are not mentioned.
Since, you impose here syntactic restrictions on quantifiers, are there formulae which were allowed in [5] and cannot be expressed here?

In the Abstract authors state that "a decision
procedure for ... $\forall_0^\pi$ has been provided", while in the Introduction
they state that "no decision procedure for it [the $\forall_0^\pi$ subclass]
has been implemented yet''. Are they both correct?

Lemma 3 is a consequence of Lemma 2 in [5]. Then Theorem 3 follows. These are important results in this paper. In my opinion Lemma 2 of [5] has to be included to make the paper self-contained. Moreover, the proofs of Lemma 2 of [5], Lemma 3 and Theorem 3 should be added.

Review1 - Proof of Lemma 4, as well as the (long) proofs of Lemma 5 and Lemma 6, should be removed
from the main text and put in an Appendix at the end of the paper or made
available online.

Review2- I am quite convinced by the proof of Lemma 5. However, I do not see where (S4) follows from. I understand that \chi_2 encodes properties of the equality and consequences of the extensionality, but what does ensure that \chi_2 is enough?

The definition of Terms_L, and as a consequence that of Forms_L, sounds strange.
All the set Consts is included in Terms_L, not just Consts_L. This implies that formulae
can involve constants which are not in Consts_L. Then the definition of interpretation I for L give meaning only to elements of Consts_L. Hence, a pair (I,A) cannot evaluate formulae involving constants which are in Consts\Consts_L.
At this point one could solve these problems replacing “Consts” with “Consts_L” in the definition of “Terms_L”.
However, with such change Theorem 1 would not work (as it is written). In fact, having only terms build from Consts_L for the language L having just one constant “a”, no predicates and no functional symbols we would obtain D^H={a}. This makes the formula
\exists x(x\neq a) not satisfiable. As a matter of fact, in [10] the Theorem is formulated over a language L^{PAR} extended with “parameters” (a sufficient number of constants).

"for each Skolemized" dovrebbe essere "for some Skolemized" ?

Review2: Please add in section 3 some examples of pairing functions. \pi_0 and \pi_{n+1} come later in section 3.1. Are there other possible?