=Paper= {{Paper |id=Vol-2987/paper7 |storemode=property |title=Multi-Frame Modal Symbolic Learning |pdfUrl=https://ceur-ws.org/Vol-2987/paper7.pdf |volume=Vol-2987 |authors=Giovanni Pagliarini,Guido Sciavicco,Ionel Eduard Stan |dblpUrl=https://dblp.org/rec/conf/gandalf/PagliariniSS21 }} ==Multi-Frame Modal Symbolic Learning== https://ceur-ws.org/Vol-2987/paper7.pdf

Multi-Frame Modal Symbolic Learning
Giovanni Pagliarini1,2 , Guido Sciavicco1 and Ionel Eduard Stan1,2
1
University of Ferrara, Italy
2
University of Parma, Italy

Abstract
Symbolic learning is the sub-field of machine learning that deals with symbolic algorithms and mod-
els, which have been known for decades and successfully applied to a variety of contexts. The main
limitation of symbolic models is the fact that they are essentially based on classical propositional logic,
which implies that data with an implicit dimensional component, such as temporal (e.g., time series)
or spatial data (e.g., images), cannot be properly dealt with within the standard symbolic framework.
Recently, modal symbolic learning models have been proposed as a natural extension of classical ones
to naturally deal with dimensional data, and successfully applied to temporal and spatial data. In this
paper, we discuss the possibility of further extending such learning models to deal with multi-frame
dimensional data, to be able to natively learn from instances represented by more than one dimensional
description.

Keywords
Dimensional Data, Modal Logics, Modal Symbolic Learning

1. Introduction
The most iconic and fundamental separation between sub-fields of machine learning is the
one between functional and symbolic learning. Functional learning is the process of learning a
function that represents the theory underlying a certain phenomenon. Symbolic learning, on
the other hand, is the process of learning a logical description that represents that phenomenon.
Whether one or the other approach should be preferred raised a long-standing debate among
experts, rooted in the fact that functional methods tend to be more versatile and statistically
accurate than symbolic ones, while symbolic methods are able to extract models that can be
interpreted, explained, and enhanced using human knowledge. From a logical standpoint,
classical symbolic learning schemata are all characterized by the use of propositional logic
(they are, in fact, sometimes called propositional methods), and can be classified along three
main directions: the structure of the models (from strongly structured ones, such as decision
trees [1, 2], to strongly unstructured ones, such as sets of independent rules [3]), the type of logic
(crisp versus fuzzy [4, 5]), and the type of learning method (from purely deterministic to purely
randomized). Dimensional data, such as temporal or spatial data, cannot be dealt with in a native
way using propositional methods. Examples of naturally dimensional data include temporal
OVERLAY 2021: 3rd Workshop on Artificial Intelligence and Formal Verification, Logic, Automata, and Synthesis,
September 22, 2021, Padova, Italy
" giovanni.pagliarini@unife.it (G. Pagliarini); guido.sciavicco@unife.it (G. Sciavicco); ioneleduard.stan@unife.it
(I. E. Stan)
0000-0002-8403-3250 (G. Pagliarini); 0000-0002-9221-879X (G. Sciavicco); 0000-0001-9260-102X (I. E. Stan)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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histories of patients, or objects described by spatial images; in all such cases, the dimensional
component is usually implicit. The general to-go strategy to treat dimensional data with
propositional models such as decision-trees is to flatten the dimensional component, effectively
hiding it. This consists in massaging the data set in such a way that new variables are created for
each dimensional variable 𝐴, which contain the values of 𝐴 at different times, spatial locations,
and so on; so for example, an instance that consists of a single-variate time series 𝐴 with 𝑁
(ordered) points ends up being represented as the (unordered) collection 𝐴(1), 𝐴(2), . . . , 𝐴(𝑁 ).
Such a representation is called lagged in the temporal case, and flattened in the spatial one, and
it allows one to use off-the-shelf propositional methods for the learning phase. Recently a new
line in symbolic learning has emerged, in which classical symbolic algorithms are enhanced so
that dimensional variables can be dealt by leveraging more expressive reasoning capabilities.
To this end, propositional logic is replaced with propositional modal logic [6] in the learning
schema, allowing one to natively express the relationships that emerge among the different
worlds that describe each instance (e.g., time points, time intervals, extended areas, etc.). Modal
logic can be declined into more practical languages, such as temporal or spatial logics, without
loosing its basic principles, and the definition of modal symbolic learning schema immediately
becomes a definition of a temporal, spatial, or spatial-temporal one. Modal decision trees,
for example, have been studied in the temporal case in [7], and applied to real data in [8].
Dimensional data, however, is more complex than what can be captured by a single description.
The temporal histories of patients, for instance, should be paired with patients’ static data;
the spatial descriptions of objects, as a different example, may require images from different
angles; more in general, the instances of real-world data sets are often very complex, and require
learning methods that can deal with such complexity.
In this paper we pave the way to a generalization of modal symbolic methods to the multi-
frame case, and present some practical cases in which such a complex schema can be useful.

2. Multi-Frame Modal Symbolic Learning
A multi-frame dimensional data set ℐ is a finite collection of instances 𝐼1 , . . . 𝐼𝑚 , each of which
is associated with (i.e., described by) 𝑟 Kripke models 𝑀1 , . . . , 𝑀𝑟 . In turn, in each model
𝑀𝑗 = (𝑊𝑗 , 𝑅1𝑗 , . . . , 𝑅𝑠𝑗 𝑗 , 𝑉𝑗 ), each world is characterized by the value of 𝑛𝑗 distinct attributes
𝐴𝑗1 , . . . , 𝐴𝑗𝑛𝑗 ; as in the classical propositional case, the attributes define the propositional alpha-
bet. We say that ℐ is labeled if it is partitioned in a finite number of classes 𝒞 = {𝐶1 , . . . , 𝐶𝑘 }.
In other words, a single instance is described by more-than-one dimensional information. We
assume that the number and the type of descriptions are consistent among instances; so, for
example, a multi-frame dimensional data set may contain instances described by three frames
each. To a multi-frame dimensional data set with 𝑟 frames we associate 𝑟 (in general, distinct)
modal languages. The 𝑗-th language is a unary modal logic with 𝑠𝑗 existential modalities
♢1 , . . . , ♢𝑠𝑗 , and their corresponding universal versions □1 , . . . , □𝑠𝑗 . Modalities are interpreted
by the relations 𝑅1𝑗 , . . . , 𝑅𝑠𝑗 𝑗 , so that:

𝑀𝑗 , 𝑤 ⊩ 𝑝 iff 𝑝 ∈ 𝑉𝑗 (𝑤);
𝑀𝑗 , 𝑤 ⊩ ¬𝜙 iff 𝑀𝑗 , 𝑤 ̸⊩ 𝜙;
𝑀𝑗 , 𝑤 ⊩ 𝜙 ∨ 𝜓 iff 𝑀𝑗 , 𝑤 ⊩ 𝜙 or 𝑀𝑗 , 𝑤 ⊩ 𝜓;
𝑀𝑗 , 𝑤 ⊩ ♢𝑗𝑡 𝜙 iff ∃𝑣 s.t. 𝑤𝑅𝑡𝑗 𝑣 and 𝑀𝑗 , 𝑣 ⊩ 𝜙,

where ♢𝑗𝑡 is the 𝑡-th modality of the 𝑗-th language. Multi-frame modal symbolic learning
consists of enhancing modal symbolic learning methods with the possibility of learning from
more-than-one dimensional description at the same time, and describing the learned knowledge
using the correct logic. Multi-frame dimensional data sets capture dimensional situations quite
naturally; but just as it happens in modal symbolic learning, in the multi-frame case too we
need to concretize the learning models and the associated languages to specific modal logics to
adapt them to the real-world cases. Dimensional data is generally represented in an implicit
form. This means that, for example, time series and images are usually linearized, and expressed
as sets of numbers; different, but equivalent, representations can be used to capture dimensional
situations (by rows and by columns are just two very popular examples). A key observation is
that the information in real data, in general, is not point-based (think, for example, to time
series or images: the values of single time points or single pixels are not really informative).
One around this problem is to employ modal logics in which worlds do not correspond to single
points, but to sets of points. Halpern and Shoham’s interval temporal logic (𝐻𝑆) [9] allows one
to express properties of intervals in the temporal case. By generalizing 𝐻𝑆 to any number of
dimensions, we obtain a family of logics that we can denote by 𝐻𝑆 𝑑 , where 𝑑 ∈ N; 𝐻𝑆 0 is just
propositional logic, 𝐻𝑆 1 is the original Halpern and Shoham logic of intervals, and 𝐻𝑆 2 is
the natural logical generalization of Rectangle Algebra [10]. In 𝐻𝑆 𝑑 , worlds are hyperrectagles
with edges parallel to the axis, and connected by the 𝑑-dimensional generalizations of Allen’s
interval relations. In this way, a set of propositional letters emerges naturally. Fixed a frame, a
world, and an attribute 𝐴, these are of the type:

𝑓 (𝐴) ◁▷∼𝛾 𝑎
where 𝑎 is a value of the domain of 𝐴, 𝑓 is a function, 𝛾 ∈ (0, 1] ⊂ R, and ◁▷ ∈ {<, ≤, =, ≥, >},
and ∼ ∈ {<, ≤, ≥, >}. Propositional letters of this type are interpreted over hyperrectangles;
for example, if 𝑑 = 1 (that is, data is one-dimensional, e.g., temporal), 𝑓 is the identity function,
◁▷ is <, ∼ is >, 𝛾 is 0.8, 𝐴 is the temperature, and 𝑎 = 37.5, then 𝐴 <>0.8 𝑎 represents
the proposition more than the 80% of the values of the temperature during the current interval
are below 37.5. By varying the function 𝑓 , one can produce more complex assertions on
hyperrectangles.
For a better understanding of how multi-frame modal symbolic learning can be useful in
practical case, consider, again, the medical example, as in Fig. 1 (top). In this situation, we have
a three-frame data set. The first frame is static, and it is associated to propositional logic (𝐻𝑆 0 );
the second frame is one-dimensional, and, specifically, temporal: we associate it with 𝐻𝑆 1 , so
that the information concerning the temporal attributes is learned by intervals; the third frame
is two-dimensional, that is, spatial: by associating it to 𝐻𝑆 2 , we can learn patterns of rectangles
and colors in them. Instances are labeled, making this data set suitable for knowledge extraction
by classification. One can observe how data are represented in concrete form (Fig. 1, bottom
left), and how we can extract knowledge in symbolic form from them (Fig. 1, bottom, right),
with modal decision trees or sets of modal rules in 𝐻𝑆 𝑑 .
Patient/Instance Static data Temporal data Image (R,G,B) Class

Age (A): 37, Weight (W): 70 Temp (T):
𝑃1 𝐶1
Gender (G): M
Press (P):

Age (A): 49, Weight (W): 81 Temp (T):
𝑃2 𝐶2
Gender (G): M
Press (P):

... ... ... ... ...

ion
en e
res ret
tat
rep conc

knowledge
extraction

∙

[𝐴]
𝑃1 37,70,M

𝑇≥
38;38.5;38;38;38.5;38.5
decision on a temporal interval

⟩𝑇
110;115;110;115;110;115

⟨𝐴

38
210;190;190;185;185;220;220
215;195;210;175;175;210;210 ∙ ∙

00
[𝐵 ,
≤2
200;180;180;185;190;225;225

𝐸]
...

⟩𝑅

𝑅>
𝑃2 49,81,M

,𝐸

200
38;38;38.5;38;37.5;38
ial rectangle
⟨𝐵
110;115;115;110;115;110 decision on a spat ∙ ∙
212;195;195;180;189;225;221
210;190;190;185;185;220;220 ⎧
212;195;195;180;189;225;221 ⎨ ⟨𝐵⟩ 𝑃 > 110 ∧ [𝐴, 𝑂] 𝐺 < 200 ⇒ . . .
... ⟨𝐴, 𝐴⟩ 𝐵 < 215 ∧ ⟨𝐿⟩ 𝑇 < 39 ⇒ . . .
...
⎩

Figure 1: Example of multi-frame dimensional data set.

3. Conclusions
In this paper we paved the way towards multi-frame modal symbolic learning. We defined
multi-frame dimensional data sets, the learning framework, and described how to concretize it
to deal with implicit dimensional data in a intuitive way.

Acknowledgments
We thank the INdAM GNCS 2020 project Strategic Reasoning and Automated Synthesis of Multi-
Agent Systems for partial support.
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