Review Comment:
The paper consists of two halves: in the first half, logical commitments
for knowledge graph embeddings are examined. There is a certain interaction
between the geometry of the embedding and the logic underlying the embedding.
A related survey is [7]; however, it reviews only approaches based on
threshold functions, which geometrically means that relations are
expressed via threshold balls in Euclidean space. By constrast, the
present paper takes a much broder view.
This examination culminates in Table 1, where eight different
approaches are listed with their geometries and logical
commitments. This is clearly an interesting survey, based on different
levels of logical commitments (Fig. 1), like different operators or
calculi.
However, the exact relation between geometry and logic remains quite
vague here. Specific examples are discussed only briefly. The different
underlying algebraic structures are not discussed in detail (with the
exception of ortholattices), and in Table 1, only some rows list
the corresponding algebraic structures.
Morevoer, the authors speak about logics for TBox, ABox, querying and
explainability (involving three different logics L, L_1, L_2), and
have some general considerations, but it is unclear how these general
considerations influence Table 1. For exmaple, in Table 1, only logic
L is specified  what about L_1 and L_2? Also, the general
considerations themselves remain too vague in various respects. For
example, the authors consider the knowledge based of all L_1sentences
true in I_E, where L_1 is the main logic in use. What would this
amount to in the examples? Is this knowledge base really relevant for
knowledge graph embeddings?
The authors introduce a definition for the central notion of the special
issue, explainability, but the definition does not make much sense in my
eyes.
Basically, at various places, the introduced notions and notations need
more explanation and examples in order to be digestablle. Moreover,
I doubt that the authors themselves have thought things through.
See also the detailed comments.
The second half of the paper examines orthologics, i.e. logics for
ortholattices and their relation to geometric models based on
cones. It is unclear to me why this specific embedding approach (which
clearly deserves its place in Table 1) deserves so much attention
here. The authors write "This will give us an opportunity to motivate
orthologics as framework to compare different approaches." (p.10
l. 26f) However, I cannot see that this claim has been
fulfilled. Moreover, even if one ignores this lack of motivation, one
must say that the paper does not develop this approach very well
w.r.t. its overall context. No knowledge graph embedding based on
cones and orthlogic is presented, let alone evaluated. Instead, the
authors conduct a purely theoretical investigation of orthologics and
conebased ortholattices, culminating in Table 2, with no link to
knowledge graph embeddings, nor explainability.
Hence, I think that this paper canot be accepted. I suggested that the
authors split the paper into two papers. These both need substantial
revision before they can be resubmitted.
Detailed comments:
p.2 l.1 must play > should play (otherwise, "at least" sounds a bit
peculiar here)
various places: logic properties > logical properties
p.2 l.14 a KGE
p.3 l.27ff.
It reamains unclear why ortholattices receive so much attention here.
p.4, Def. 1
You defined what it means that agent A_1 is explainable to agent A_2,
but in the definiens, A_1 does not occur, except from logic L_1
being supported by A_1. This does not make sense. Probably, A_1
needs to be related to the knowledge base KB somehow?
Moreover, the assumption that there is a common superlogic L of
logic L_1 (used by agent A_1) and logic L_2 (used by agent A_2) is
very strong.
Finally, the most intersting case of explainability is not covered
by Def. 1: namely the case when A_1 uses a subsymbolic system like
a neural network, and A_2 explains A_1's answers using some logic.
p.4 l. 45
"The above definition presumes a notion of logic that comes with a
syntax and semantics with the usual notions of
structures/interpretations and of models as wells as derived notions
such as entailment which can be used to formally define query
answering."
No, Def. 1 only presumes a notion of query answering. This could be
purely prooftheoretic or alorithmic, with no semantics. (Of course,
a semantics is highly desirable, but not presumed by Def. 1.)
p.4 l.48
"which gives rise to some logic L'_1" why? how? example?
p.5 l.27
logical properties
p.5 l.50, p.7 l. 31
use "how ... look" or "what ... look like", but not "how ... look like"
p.7 l.12
this is not really a union, more a tupling
p.7 l.22 The kingqueentype of word puzzles is more a type of analogy
reasoning, and not the usual kind of querying based on logics. Vector
arithmetic in word embeddings supports these types of analogies. It is
an open question how to capture this logically. But
1. this type of open question does not make sense at this place
2. the authors' reference to [30], a paper on mathematical induction (and not
induction in the sense of learning), does not make any sense here.
p.8 l.5 "sets concept symbols"  what is this?
p.8 l.25 "I ∈ gMod(EA) is true w.r.t. the pair (T , A)"
This does not make any sense to me. What is the relation between EA and
(T,A)? Probably you need something like gMod(EA,(T,A)) ?
This definitely needs more explanation and examples.
p.8. l.29 consistency in FOL is not semidecidable, only cosemidecidable
(that is, inconsistency is semidecidable).
p.8 l.32 "query answering amounts to model checking"
only for ground queries. The interesting case, queries with variables,
amounts to finding an answer substitution, which is more complex
than model checking.
p.8 l. 39f. The authors speak of embedding structures E and
"associated logical structures I_E" without defining what this
means. Probably, the construction of I_E depends very much on the
embedding? Then at least one or two more detailed examples would be
needed here.
p.8 l. 44f.
"Of course everything hinges upon finding
appropriate logics L, L_1 , L_2 and such that L_1 is sound and
complete for the class of structures of an embedding.",
It is not clear what soundness and completeness would mean here.
p.9, Table 1
The "Operators" column could be more precise.
E.g. "subBoolean algebra"s  are these e.g. infsemilattices?
Is negation as failure algebraic at all?
The row with "threshold balls" is a bit imprecise. Note that there
are approaches like TransR [1] that use a projection before applying
a threshold ball. Would this be useful for other approaches, too?
[1] Yankai Lin, Zhiyuan Liu, Maosong Sun1, Yang Liu, Xuan Zhu:
Learning Entity and Relation Embeddings for Knowledge Graph Completion
p.9 l.45 superfluous ")"
p.10 l.39 How can orthologics be classic, if they do not obey the
law of excluded middle? E.g. Kleene's threevalued logic should be a model.
p.11 l.2 I do not udnerstand the fourth axiom. How can A be equivalent
to ~~A&~A ???
p.11 l.22 nonorthomodular: this has not been defined
p.11 l.35 lowercase x in the formula
p.11 l.37 What is the "resulting logic"?
p.11 l.38 orthomodel: this has not been defined
p.12 Prop. 1
Where do you prove that this is an orthoframe?
In the proof, you tacitly use a characterisation of the closure of A
that should be made explicit.
p.13 l.46 "hence must be a Boolean algebra"  how does excluded middle follow?
p.14 l.48: Here you try some link to explainability. However, the link
remains completely unclear to me.
Moreover, no relation of orthologics to knowledge graph embeddings is
provided nor discussed.
