=Paper=
{{Paper
|id=Vol-2969/paper32-CAOS
|storemode=property
|title=A Multipreference Semantics from Common Sense Reasoning to Neural Network Models: An Overview
|pdfUrl=https://ceur-ws.org/Vol-2969/paper32-CAOS.pdf
|volume=Vol-2969
|authors=Laura Giordano,Valentina Gliozzi,Daniele Theseider Dupré
|dblpUrl=https://dblp.org/rec/conf/jowo/0001GD21
}}
==A Multipreference Semantics from Common Sense Reasoning to Neural Network Models: An Overview==
<pdf width="1500px">https://ceur-ws.org/Vol-2969/paper32-CAOS.pdf</pdf>
<pre>
A Multipreference Semantics from Common Sense
Reasoning to Neural Network Models: an Overview
Laura Giordano1 , Valentina Gliozzi2 and Daniele Theseider Dupré1
1
    DISIT - Università del Piemonte Orientale, Viale Michel 11, I-15121, Alessandria, Italy
2
    Dipartimento di Informatica, Università degli Studi di Torino, Corso Svizzera 185, I-10149,Torino, Italy


                                         Abstract
                                         In this short paper we report about a “concept-wise" multipreference semantics for weighted condition-
                                         als and its use to provide a logical interpretation to some neural network models, Self-Organising Maps
                                         (SOMs) and Multilayer Perceptrons (MLPs). For MLPs, a deep network can be regarded as a conditional
                                         knowledge base, in which the synaptic connections correspond to weighted conditionals.

                                         Keywords
                                         Common Sense Reasoning, Preferential semantics, Typicality in Description Logics, Neural Network
                                         models




1. Introduction
Preferential approaches to common sense reasoning [1, 2, 3, 4, 5, 6, 7] have their roots in
conditional logics [8, 9], and have been recently extended to Description Logics (DLs), to deal
with inheritance with exceptions in ontologies, by allowing non-strict form of inclusions, called
defeasible or typicality inclusions.
   Different preferential semantics [10, 11] and closure constructions [12, 13, 14, 15, 16, 17, 18]
have been proposed for such defeasible DLs and, in this paper, we report about a concept-
wise multipreference semantics [19], which has been recently introduced as a semantics of
ranked knowledge bases in a lightweight DL to account for preferences with respect to different
concepts, and has been proposed as a semantics for some neural network models.
   We have considered both an unsupervised model, Self-organising maps (SOMs)[20], which
is considered as a psychologically and biologically plausible neural network model, and a
supervised one, Multilayer Perceptrons (MLPs) [21]. Learning algorithms in the two cases are
quite different but our aim is to capture, through a semantic interpretation, the behavior of the
network resulting after training and not to deal with the learning process. We will see that this
can be accomplished in both cases in a similar way, based on the multi-preferential semantics.
   In both cases, considering the domain of all input stimuli presented to the network during
training (or in the generalization phase), one can build a semantic interpretation describing the
input-output behavior of the network as a multi-preference interpretation, where preferences are

CAOS 2021: 5th Workshop on Cognition And OntologieS, held at JOWO 2021: Episode VII The Bolzano Summer of
Knowledge, September 11-18, 2021, Bolzano, Italy
" laura.giordano@uniupo.it (L. Giordano); valentina.gliozzi@unito.it (V. Gliozzi); dtd@uniupo.it (D. Theseider
Dupré)
                                       © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
    CEUR
    Workshop
    Proceedings
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                  ISSN 1613-0073
                                       CEUR Workshop Proceedings (CEUR-WS.org)
associated to concepts. For SOMs, the learned categories 𝐶1 , . . . , 𝐶𝑛 are regarded as concepts so
that a preference relation (over the domain of input stimuli) is associated to each category [22, 23].
For MLPs, each neuron in the deep network (including hidden neurons) can be associated with
a concept and with a preference relation on the domain [24].
   The idea is that, given two input stimuli 𝑥 and 𝑦, and two categories/concepts, e.g., Horse
and Zebra, the neural model can assign to 𝑥 a degree of membership in the category Horse
which is higher than the degree of membership of 𝑦, so that 𝑥 can be regarded as a being more
typical than 𝑦 as a horse (x <Horse y), while vice-versa 𝑥 can be regarded as a being less typical
than 𝑦 as a zebra (y <Zebra x ). A preferential interpretation can be built over the domain of
input stimuli and can be used for checking properties such as: are the instances of category 𝐶1
also instances of category 𝐶2 ? Are typical instances of category 𝐶1 also instances of category
𝐶2 ? This verification can be done by model-checking given a multipreference interpretation
describing the input-output behavior of the network [23].
   For MLPs, the relationship between our logic of commonsense reasoning and deep neural
networks is even stronger, as a deep neural network can itself be regarded as a conditional
knowledge base, i.e., as a set weighted conditionals. This has been achieved by developing a
concept-wise fuzzy multipreference semantics for a DLs with weighted defeasible inclusions.
   The strong relationship between neural networks and conditional logics of commonsense
reasoning raises several issues from the standpoint of knowledge representation, from the
standpoint of neuro-symbolic integration, as well as from the standpoint of explainable AI
[25, 26, 27]. We will hint at some of these issues in the extended abstract after shortly describing
the approach.


2. The Concept-Wise Multipreference Semantics
The concept-wise multipreference semantics (cw𝑚 -semantics) has been introduced as a seman-
tics for ranked ℰℒ knowledge bases [19], and later extended to weighted knowledge bases [24].
In both cases the knowledge base contains (besides standard inclusions, called strict) defeasible
or typicality inclusions of the form T(𝐶) ⊑ 𝐷 (meaning “the typical 𝐶s are 𝐷s" or “normally
𝐶s are 𝐷s") with a rank (resp. a weight). They correspond to KLM conditionals 𝐶 |∼ 𝐷 [4].
Ranks (weights) of defeasible inclusions represent their strength (plausibility/implausibility).
The preferential semantics of ranked and weighted knowledge bases are defined in terms of
concept-wise multipreference interpretations, based on different constructions.
   Concept-wise multipreference interpretations (cw𝑚 -interpretations) are defined by adding to
standard DL interpretations, which are pairs ⟨∆, ·𝐼 ⟩, where ∆ is a domain, and ·𝐼 an interpre-
tation function, the preference relations <𝐶1 , . . . , <𝐶𝑛 associated with a set of distinguished
concepts 𝐶1 , . . . , 𝐶𝑛 , representing the relative typicality of domain individuals with respect to
these concepts. Each preference relation <𝐶𝑖 is a modular and well-founded strict partial order
on ∆. Preferences with respect to different concepts do not need to agree; as we have seen, a
domain element 𝑥 may be more typical than 𝑦 as a horse, but less typical as a zebra. A global
preference relation < can be defined starting from the <𝐶𝑖 ’s, and concept T(𝐶) is interpreted
as the set of all <-minimal 𝐶 elements. A simple notion of global preference < exploits Pareto
combination of the preference relations <𝐶𝑖 , but a more sophisticated notion of preference
combination has been considered in [19], by exploiting a modified Pareto condition which takes
into account the specificity relation among concepts (e.g., that concept Penguin is more specific
than concept Bird ). It has been proven [19] that global preference in a cw𝑚 -interpretation
determines a KLM-style preferential interpretation, and cw𝑚 -entailment satisfies the KLM
postulates of a preferential consequence relation [4].


3. A Preferential Interpretation of Self-Organising Maps
Once the SOM has learned to categorize, one can look at the result of the categorization as
a concept-wise multipreference interpretation over a domain of input stimuli, in which a
preference relation is associated to each concept (each learned category), and the combination of
the preferences into a global one (following the approach described above) defines a KLM-style
preferential model of the SOM. More precisely, once the SOM has learned to categorize, to assess
category generalization, Gliozzi and Plunkett [28] define the map’s disposition to consider a
new stimulus 𝑦 as a member of a known category 𝐶 as a function of the distance of 𝑦 from the
map’s representation of 𝐶. The relative distance 𝑟𝑑(𝑥, 𝐶𝑖 ) of a stimulus 𝑥 from a category 𝐶𝑖
can be used to build a binary preference relation <𝐶𝑖 among the stimuli in ∆ with respect to
category 𝐶𝑖 [22, 29], by letting 𝑥 <𝐶𝑖 𝑦 if and only if 𝑟𝑑(𝑥, 𝐶𝑖 ) > 𝑟𝑑(𝑦, 𝐶𝑖 ) (𝑥 is more typical
than 𝑦 with respect to category 𝐶𝑖 if its relative distance from category 𝐶𝑖 is lower than the
relative distance of 𝑦).
   This preferential model can be exploited to learn or validate conditional knowledge from
empirical data, by verifying conditional formulas over the preferential interpretation constructed
from the SOM. Both a two-valued and a fuzzy semantics have been considered [23]. In both
cases, model checking can be used for the verification of inclusions (either defeasible inclusions
or fuzzy inclusion axioms) over the respective models of the SOM (for instance, do the most
typical penguins belong to the category Bird with at least a degree of membership 0.8?). Starting
from the fuzzy interpretation of the SOM, a probabilistic account can also be given based on
Zadeh’s probability of fuzzy events [30].


4. A Preferential Interpretation of Multilayer Perceptrons
For MLPs, a deep network is considered after the training phase, when the synaptic weights
have been learned. The input-output behaviour of the network can be captured in a similar
way as for SOMs by constructing a preferential interpretation over the domain ∆ of the input
stimuli considered during training (or generalization) [24]. Each neuron 𝑘 of interest can be
associated to a concept 𝐶𝑘 and, for each distinguished concept 𝐶𝑗 , a preference relation <𝐶𝑗
is defined over the domain ∆ based on the activity values, 𝑦𝑗 (𝑣), of neuron 𝑗 for each input
𝑣 ∈ ∆. In a similar way, a fuzzy interpretation of the network can be constructed over the
domain ∆, as well as a fuzzy-multipreference semantics.
   All the three semantics allow the input-output behavior of the network to be captured by
interpretations built over a set of input stimuli through simple constructions, which exploits
the activity level of neurons for the stimuli. In particular, for the fuzzy-multipreference inter-
pretations, the idea [24] is to extend a fuzzy DL interpretation with a set of induced preferences.
In a fuzzy DL interpretation 𝐼, the interpretation of a concept 𝐶ℎ is a mapping 𝐶𝑖𝐼 : ∆ → [0, 1],
associating to each 𝑥 ∈ ∆ the degree of membership of 𝑥 in 𝐶ℎ . The activation value of unit ℎ
for a stimulus 𝑥 in the network (assumed to be in the interval [0, 1]) is taken as the degree of
membership of 𝑥 in concept 𝐶ℎ . The fuzzy interpretation also induces an ordering <𝐶ℎ on the
domain ∆, for each 𝐶ℎ , to be regarded as the preference relation associated to concept 𝐶ℎ . This
allows a notion of typicality to be defined in a fuzzy interpretation. Let us call ℳ𝑓,Δ
                                                                                      𝒩 the fuzzy
multipreference interpretation built from the network 𝒩 over a domain ∆ of input stimuli.
   As for SOMs, logical properties of the neural network (both typicality properties and fuzzy
axioms) can then be verified by model checking over such an interpretation. Evaluating proper-
ties involving hidden units might be of interest, although their meaning is usually unknown.
In the well known Hinton’s family example [31], one may want to verify whether, normally,
given an old Person 1 and relationship Husband, Person 2 would also be old, i.e., whether
T(𝑂𝑙𝑑1 ⊓ 𝐻𝑢𝑠𝑏𝑎𝑛𝑑) ⊑ 𝑂𝑙𝑑2 is satisfied. Here, concept 𝑂𝑙𝑑1 (resp., 𝑂𝑙𝑑2 ) is associated to a
(known, in this case) hidden unit for Person 1 (and Person 2), while Husband is associated to an
input unit. If the properties of interest involve some specific units, only the concepts associated
to those units may be considered in the language to build the interpretation.
   All the three kinds of interpretations considered above for MLPs describe the input-output
behavior of the network. However, the fuzzy multipreference interpretation ℳ𝑓,Δ      𝒩 described
above can be also proven to be a model of the neural network 𝒩 in a logical sense, by mapping
the multilayer network into a weighted conditional knowledge base.

4.1. Weighted 𝒜ℒ𝒞 Knowledge Bases
In this section, we shortly recall the definition of weighted conditional knowledge bases through
an example, and give some hints about the two-valued and fuzzy multipreference semantics,
referring to [24] for a detailed description for ℰℒ.
   A weighted 𝒜ℒ𝒞 knowledge base 𝐾 over a set 𝒞 = {𝐶1 , . . . , 𝐶𝑘 } of distinguished 𝒜ℒ𝒞
concepts is a tuple ⟨𝒯 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜⟩, where the TBOX 𝒯 is a set of 𝒜ℒ𝒞 inclusion axiom,
the ABox 𝒜 is a set of 𝒜ℒ𝒞 assertions and, for each distinguished concept 𝐶𝑖 ∈ 𝒞, 𝒯𝐶𝑖 is a set
of weighted typicality inclusions of the form T(𝐶𝑖 ) ⊑ 𝐷, with a positive or negative weight (a
real number). In the fuzzy case, 𝒯 and 𝒜 contain fuzzy axioms.
   Consider the weighted knowledge base 𝐾 = ⟨𝒯 , 𝒯𝐵𝑖𝑟𝑑 , 𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 , 𝒜⟩, over the set of distin-
guished concepts 𝒞 = {Bird , Penguin}, with empty ABox and with 𝒯 containing the inclusions
Penguin ⊑ Bird and Black ⊓ Grey ⊑ ⊥.
   The weighted TBox 𝒯𝐵𝑖𝑟𝑑 contains the following weighted defeasible inclusions:
   (𝑑1 ) T(Bird ) ⊑ Fly, +20
   (𝑑2 ) T(Bird ) ⊑ ∃has_Wings.⊤, +50
   (𝑑3 ) T(Bird ) ⊑ ∃has_Feathers.⊤, +50;
𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 contains the defeasible inclusions:
   (𝑑4 ) T(Penguin) ⊑ Fly, - 70
   (𝑑5 ) T(Penguin) ⊑ Black , +50;
   (𝑑6 ) T(Penguin) ⊑ Grey, +10;
   The meaning is that a bird normally has wings, has feathers and flies, but having wings and
feathers (both with weight 50) for a bird is more plausible than flying (weight 20), although
flying is regarded as being plausible. For a penguin, flying is not plausible (inclusion 𝑑4 has a
negative weight -70), while being black or being grey are plausible properties of prototypical
penguins, in fact, 𝑑5 and 𝑑6 have positive weights, resp. 50 and 10, so that being black is more
plausible than being grey.
   A two-valued semantics for weighted 𝒜ℒ𝒞 knowledge bases can be defined by developing
a semantic closure construction in the same spirit as Lehmann’s lexicographic closure [32],
but more similar to Kern-Isberner’s semantics of c-representations [7, 33], in which the world
ranks are generated as a sum of impacts of falsified conditionals. Here, the (positive or negative)
weights of the satisfied defaults are summed, but in a concept-wise manner, so to determine the
plausibility of a domain elements with respect to certain concepts. In this way, the modular
structure of the knowledge base can be considered. More precisely, for a domain element 𝑥 in
∆, and a distinguished concept 𝐶𝑖 , the weight 𝑊𝑖 (𝑥) of 𝑥 wrt 𝐶𝑖 is defined as the sum of the
weights 𝑤ℎ𝑖 of the typicality inclusions T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ in 𝒯𝐶𝑖 verified by 𝑥 (and is −∞ when 𝑥
is not an instance of 𝐶𝑖 ). From the weights 𝑊𝑖 (𝑥) the preference relation ≤𝐶𝑖 can be defined by
letting: for 𝑥, 𝑦 ∈ ∆, 𝑥 ≤𝐶𝑖 𝑦 iff 𝑊𝑖 (𝑥) ≥ 𝑊𝑖 (𝑦). The higher the weight of 𝑥 wrt 𝐶𝑖 the higher
its typicality relative to 𝐶𝑖 . This closure construction defines preferences <𝐶𝑖 (strict modular
partial orders) and allows for the definition of concept-wise multipreference interpretations as in
Section 2.
   In the fuzzy case, the fuzzy logic combination functions are used for complex concepts to
compute the 𝑊𝑖 (𝑥)’s and to determine the associated preference relations. To guarantee that
the preferences determined from the knowledge base are coherent with the fuzzy interpretation
of concepts, a notions of coherent (fuzzy) multipreference interpretation (cf𝑚 -interpretation) is
also introduced [24].

4.2. MLPs as Conditional Knowledge Bases
Let us describe how the multilayer network 𝒩 can be mapped to a weighted conditional
knowledge base 𝐾 𝒩 , i.e., to a set of weighted typicality inclusions. The idea is to consider,
for each unit 𝑘, all the units 𝑗1 , . . . , 𝑗𝑚 , whose output signals are the input signals of unit 𝑘,
with synaptic weights 𝑤𝑘,𝑗1 , . . . , 𝑤𝑘,𝑗𝑚 . Let 𝐶𝑘 be the concept name associated to unit 𝑘 and
𝐶𝑗1 , . . . , 𝐶𝑗𝑚 be the concept names associated to units 𝑗1 , . . . , 𝑗𝑚 . One can define, for unit 𝑘, a
set 𝒯𝐶𝑘 of 𝑚 typicality inclusions, with their associated weights, as follows: T(𝐶𝑘 ) ⊑ 𝐶𝑗1 with
𝑤𝑘,𝑗1 , . . . , T(𝐶𝑘 ) ⊑ 𝐶𝑗𝑚 with 𝑤𝑘,𝑗𝑚 . The network 𝒩 can than be mapped to a conditional
knowledge base 𝐾 𝒩 containing, for each neuron 𝑘, a set of typicality inclusions 𝒯𝐶𝑘 as defined
above.
   Let us consider the fuzzy multipreference interpretation ℳ𝑓,Δ        𝒩 built from 𝒩 over a domain
∆ of input stimuli, as described above. Let us further assume that, in the construction, all units
are considered and a concept 𝐶𝑘 is introduced in the language for each unit 𝑘. It has been
proven [24] that the interpretation ℳ𝑓,Δ      𝒩 is a cf -model of the knowledge base 𝐾 , under
                                                        𝑚                                       𝒩

some condition on the activation functions in 𝒩 . In particular, the properties that are entailed
from 𝐾 𝒩 are properties satisfied by ℳ𝑓,Δ     𝒩 , for any choice of the input stimuli in the domain ∆.
5. Discussion and Conclusions
In [22, 23, 24] we have studied the relationships between a preferential logic of common sense
reasoning and two different neural network models, Self-Organising Maps and Multilayer
Perceptrons, showing that a multi-preferential semantics can be used to provide a logical
model of the neural network behavior after training. Such a model can be used to learn or to
validate conditional knowledge from the empirical data used for training and generalization,
by model checking of logical properties. A two-valued KLM-style preferential interpretation
with multiple preferences and a fuzzy semantics have been considered, based on the idea of
associating preference relations to categories (in the case of SOMs) or to neurons (for Multilayer
Perceptrons). Due to the diversity of the two models we would expect that a similar approach
might be extended to other neural network models and learning approaches. The plausibility
of concept-wise multipreference semantics is supported by the fact that self-organising maps
are considered as psychologically and biologically plausible neural network models. This
multipreference semantics has been shown to satisfy the KLM properties in the two-valued case
[19], and most of the KLM properties in the fuzzy case, depending on their reformulation and
on the fuzzy combination functions considered [34].
   Much work has been devoted, in recent years, to the combination of neural networks and
symbolic reasoning [35, 36, 37], leading to the definition of new computational models [38,
39, 40, 41] and to extensions of logic programming languages with neural predicates [42, 43].
Among the earliest systems combining logical reasoning and neural learning are the Knowledge-
Based Artificial Neural Network (KBANN) [44] and the Connectionist Inductive Learning and
Logic Programming (CILP) [45] systems and Penalty Logic [46], a non-monotonic reasoning
formalism used to establish a correspondence with symmetric connectionist networks. The
relationships between normal logic programs under the stable model semantics [47] and neural
networks have been investigated by Garcez and Gabbay [45, 35] and by Hitzler et al. [48].
   The correspondence between neural network models and fuzzy systems has been first in-
vestigated by Kosko in his seminal work [49]. We have adopted the usual way of viewing
concepts in fuzzy DLs [50, 51, 52], and we have used fuzzy concepts within a multipreference
semantics, based on a semantic closure construction in the line of Lehmann’s semantics for
lexicographic closure [32] and strictly related to Kern-Isberner’s c-representations [7, 33]. Fur-
thermore, we have adopted a preferential semantics with multiple preferences, in order to make
it concept-wise: each distinguished concept 𝐶𝑖 has its own set 𝒯𝐶𝑖 of (weighted) typicality
inclusions, and an associated preference relation <𝐶𝑖 . This allows a preference relation to be
associated to each category (e.g., in the preferential interpretation of SOMs) or to neurons (in
a deep network). A combination of fuzzy logic with the preferential semantics of conditional
knowledge bases has been first studied by Casini and Straccia [53], who have developed a
rational closure construction for propositional Gödel logic.
   For Multilayer Perceptrons, it has been proven [24] that a deep network can itself be regarded
as a weighted conditional knowledge base (under some conditions on the activation function).
This opens to the possibility of adopting a conditional logics as a basis for neuro-symbolic
integration. While a neural network, once trained, is able and fast in classifying the new stimuli
(that is, it is able to do instance checking), all other reasoning services such as satisfiability,
entailment and model-checking are missing. These capabilities would be needed for dealing with
tasks combining empirical and symbolic knowledge, such as, for instance: proving whether the
network satisfies some (strict or conditional) properties; learning the weights of a conditional
knowledge base from empirical data, and combine the defeasible inclusions extracted from a
neural network with other defeasible or strict inclusions for inference.
   To make these tasks possible, the development of proof methods for such logics is a prelimi-
nary step. In the two-valued case multipreference entailment is decidable for weighted ℰℒ⊥
knowledge bases, and proof methods for reasoning with weighted conditional knowledge bases
in ℰℒ⊥ can, for instance, exploit Answer Set Programming (ASP) encodings of the concept-wise
multipreference semantics [54], using asprin [55] to achieve defeasible reasoning, an approach
already considered for ranked ℰℒ+   ⊥ knowledge bases [19]. In the fuzzy case, an open problem
is whether the notion of fuzzy-multipreference entailment is decidable (even for the small
fragment of ℰℒ without roles), and under which choice of fuzzy logic combination functions.
Undecidability results for fuzzy description logics with general inclusion axioms [56, 57, 58]
motivate the investigation of decidable approximations of fuzzy-multipreference entailment.
   An interesting issue is whether the mapping of deep neural networks to weighted conditional
knowledge bases can be extended to more complex neural network models, such as Graph neural
networks [38], or whether different logical formalisms and semantics would be needed. Another
issue is whether the fuzzy-preferential interpretation of neural networks can be related with
the probabilistic interpretation of neural networks based on statistical AI. This is an interesting
issue, as the fuzzy DL interpretations we have considered in [24], where concepts are regarded
as fuzzy sets, also suggests a probabilistic account based on Zadeh’s probability of fuzzy events
[30]. We refer to [23] for some results concerning a probabilistic interpretation of SOMs and to
[59] for a preliminary account for MLPs.


Acknowledgments
We thank the anonymous referees for their helpful comments and suggestions. This research
has been partially supported by INDAM-GNCS Projects 2019 and 2020.


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