=Paper=
{{Paper
|id=Vol-2668/invited1
|storemode=property
|title=The Lattice Computing (LC) Paradigm
|pdfUrl=https://ceur-ws.org/Vol-2668/invited1.pdf
|volume=Vol-2668
|authors=Vassilis G. Kaburlasos
|dblpUrl=https://dblp.org/rec/conf/cla/Kaburlasos20
}}
==The Lattice Computing (LC) Paradigm==
https://ceur-ws.org/Vol-2668/invited1.pdf
The Lattice Computing (LC) Paradigm?
Vassilis G. Kaburlasos[0000−0002−1639−0627]
International Hellenic Univertity (IHU),
Human-Machines Interaction Laboratory (HUMAIN-Lab),
Department of Computer Science, 65404 Kavala, Greece
vgkabs@teiemt.gr
http://humain-lab.teiemt.gr
Abstract. The notion of Cyber-Physical Systems (CPSs) has been in-
troduced to account for technical devices with both sensing and reason-
ing abilities including a varying degree of autonomous behaviour. There
is a need for supporting CPSs by mathematical models that involve
both sensory data and cognitive data towards improving CPSs effective-
ness during their interaction with humans. However, a widely acceptable
mathematical modelling framework is currently missing. In the afore-
mentioned context, the Lattice Computing (LC) paradigm is proposed
for mathematical modelling in CPS applications based on lattice theory
by unifying rigorously numerical data and non-numerical data; the lat-
ter data include (lattice ordered) logic values, sets, symbols, graphs and
other. More specifically, the “cyber” components of a CPS involve non-
numerical data, whereas the “physical” components of a CPS involve
numerical data. A promising advantage of LC is its capacity to compute
with semantics represented by a lattice (partial) order relation.
Keywords: Cyber-Physical Systems · Mathematical modeling.
1 Introduction
A cyber-physical system (CPS) has been defined as a device with both sens-
ing and reasoning capacities [22]. Strategic initiatives regarding CPSs include
“Industrie 4.0” in Germany, the “Industrial Internet of Things (IIoT)” in the
United States, and “Society 5.0” in Japan [38]. CPSs typically focus on multi-
disciplinary applications in healthcare, agriculture, food supply, manufacturing,
energy, critical infrastructures, transportation, logistics, security, education [2].
Our interest is in models for driving CPSs, where by “model” we mean a
mathematical description of a world aspect [7]. A model describes a law, useful
to the extent it generalizes accurately. The development of a model, namely
modeling, is close to an art since a model needs to be both “detailed enough”, to
?
This project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No
777720.
Copyright c 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). Published in Francisco
J. Valverde-Albacete, Martin Trnecka (Eds.): Proceedings of the 15th International
Conference on Concept Lattices and Their Applications, CLA 2020, pp. 1–7, 2020.
2 Vassilis G. Kaburlasos
accurately describe phenomena of interest, and “simple enough”, to be amenable
to rigorous analysis. Models are applied in a world representation domain.
Classic modeling often regards physical phenomena. In particular, principles
/laws of physics, biology and other may be described by parametric algebraic
expressions that quantify a functional relation between real variables of interest.
Classic modeling ultimately rests upon the conventional measurement process,
which is carried out by comparing successively an unknown quantity to a known
prototype. For instance, an unknown length is measured by comparing it suc-
cessively to a known prototype (e.g. a meter) as well as to subdivisions of it.
The quotient and remainder of a measurement jointly define a real number –
that is how the set R of real numbers emerges. In conclusion, the physical-world
is represented by real numbers stemming from measurements, e.g. from inertial
sensors, gyroscopes, chronometers, thermometers, microphones, cameras, and
other. Therefore, classic models are developed in the Euclidean space RN , for
an integer number N .
However, when humans are involved, then, in addition to (multimodal) sen-
sory data during their interaction with one another, humans also employ cogni-
tive data such as: spoken language, relations, rules, moral principles, concepts
and symbols. Therefore, for a seamless interaction with humans, CPSs are ex-
pected to be able to cope with cognitive data. In other words, in addition to
numerical data stemming from (objective) “physical-world” measurements, a
CPS should be able to also deal with non-numerical data stemming from a (sub-
jective) “world-of-mind”. There follows a need to consider a “blended world”
model including both the physical-world and a (at least one) world-of-mind.
In response to the aforementioned considerations follows our Proposal P0
including two parts: first, sensory data are numerical, whereas cognitive data
may be non-numerical and, second, numerical data and non-numerical data are
unified in the context of mathematical lattice theory.
We remark that the emphasis below is mainly on the author’s own publi-
cations. For detailed comparisons with specific works from the literature the
interested reader might consult the references cited below.
2 The Lattice Computing (LC) Paradigm
Perhaps the most popular approach for dealing with non-numerical data is by
ad-hoc transforming them to numerical ones, thus risking the introduction of
irreversible and possibly harmful data distortions because original data semantics
may be lost. An alternative approach has been proposed based on the fact that
popular types of data are partially (lattice) ordered; in conclusion, the data-
unifying Lattice Computing (LC) information processing paradigm has been
proposed, in principle [7, 10].
Lattice Computing (LC) has been defined as “an evolving collection of tools
and methodologies that process lattice ordered data including logic values, num-
bers, sets, symbols, graphs, etc.,” [18, 39]. We point out that LC is not merely
an algorithm but rather it is an information processing paradigm. LC models
The Lattice Computing (LC) Paradigm 3
are expected to be useful in CPS applications including human-robot interaction
because LC models can (1) fuse formally numerical data (regarding physical sys-
tem components) and non-numerical data (regarding cyber system components),
(2) compute with semantics, represented by hierarchical partial-order relations,
(3) rigorously deal with ambiguity represented by partially-ordered information
granules, (4) naturally engage logic and reasoning, (5) process data fast, and (6)
deal with both “missing” and “don’t care” data values in a complete lattice [7].
The origins of LC are traced to an application of the fuzzy Adaptive Res-
onance Theory neural network, or fuzzy-ARTMAP for short, in health care
databases towards medical diagnosis [5]; in particular, the fuzzy-ARTMAP op-
erates by conditionally augmenting hyperboxes in the unit hypercube. It was
realized that the set of hyperboxes is lattice-ordered; hence, improvements were
sought using lattice theory. A naive theory of perception was proposed in the
unit hypercube by introducing novel tools such as an “inclusion measure” func-
tion for computing a fuzzy degree of inclusion of a hyperbox into another one;
moreover, the notion “fuzzy lattice” was introduced in RN . Nevertheless, the
work in [5] was primarily oriented toward medical diagnosis rather than toward
theoretical substantiation.
Subsequent work has extended the applicability domain from RN to a Carte-
sian product lattice L = L1 × ... × LN involving disparate, complete lattices. A
series of fuzzy lattice neurocomputing (FLN) models was launched and effective
applications were demonstrated in pattern recognition [19, 20]. Next, while re-
taining the basic tools of a FLN model, interest shifted to machine learning [23,
36]. A breakthrough analysis of fuzzy numbers using lattice-ordered “general-
ized intervals” further turned interest to fuzzy inference systems [11–13]. Later
work has introduced a granular extension of Kohonen’s Self-Organizing Map to
linguistic data [16]. Moreover, fuzzy lattice reasoning (FLR) was introduced [24]
and further employed in a number of applications [13, 15, 17, 18, 20, 26, 29].
Currently, there is a global interest in lattice theory applications in differ-
ent domains including (Fuzzy) Logic and Reasoning, Mathematical Morphology,
Formal Concept Analysis, Computational Intelligence, as outlined next.
Lattice theory has been instrumental in logic [1]. Furthermore, in the intro-
duction of fuzzy set theory, it was pointed out that “fuzzy sets (over a universe
of discourse) constitute a distributive lattice with 0 and 1” [43]. Moreover, it was
shown how a L(lattice)-fuzzy set generalizes the notion of a fuzzy set [4, 42].
Lattices are popular in mathematical morphology (MM) especially regarding
image processing applications [40].
Formal concept analysis (FCA), that is a lattice theory-based field of applied
mathematics [3], is based on complete lattice analysis. In the context of FCA
several schemes have been proposed for knowledge acquisition, classification, and
information retrieval in databases.
Computational Intelligence includes neural computing, which is typically car-
ried out in the Euclidean space RN . However, there is no evidence that biological
neurons operate in RN . Rather, there is evidence that biological neurons carry
out lattice- meet (min) and join (max) operations. Hence, lattice algebra was
4 Vassilis G. Kaburlasos
employed for modeling biological neurons [37]. Different authors have pursued
neural computing in the framework of fuzzy lattices [20, 34, 35], where a fuzzy lat-
tice stems from a conventional one by fuzzifying the crisp partial order relation.
The latter techniques were extended to fuzzy inference system (FIS) analysis
and design [12, 13].
Compared to the employment of mathematical lattice theory in either logic
/reasoning or MM or FCA, additional features in LC include (1) complete and/or
non-complete lattices, (2) lattices of either finite- or infinite- cardinality, and (3)
rigorous mathematical instruments including metric distances as well as fuzzy
order functions, based on positive valuation functions for tuning performance.
Moreover, LC techniques emphasize data unification based on the fact that pop-
ular mathematical lattices include: the Cartesian product RN , hyperboxes in
RN , Boolean algebras, measure spaces including probability spaces, distribution
functions, decision trees, and other.
Synergies/cross-fertilization in LC has been pursued [8–10, 14, 21, 25, 30].
3 Intervals’ Numbers (INs)
Currently, the far most popular LC models involve Intervals’ Numbers (INs).
Recall that an IN is a mathematical object that can represent either a fuzzy
interval or a distribution of samples [27, 28, 33]. An advantage of an IN is its
capacity to represent data statistics of all-orders using only few numbers; more
specifically, L numbers are used to define L intervals, where, typically L=32;
hence, a significant data reduction may result in a capacity to process big data
fast. No feature extraction is necessary since the all-order statistics, represented
by an IN, are implicitly employed as features. Lately, IN-based k nearest neighbor
classifiers have been introduced [31, 41].
Applications of INs have been reported regarding neural networks, fuzzy
inference systems as well as machine learning [6, 13, 15, 17, 18, 26, 32].
4 Conclusion
The LC paradigm has been proposed for modeling in CPS applications based
on a rigorous unification of disparate types of data. New instruments have been
introduced in a mathematical lattice such as metric distances as well as fuzzy
order functions, based on positive valuation functions. Potential future appli-
cations regard effective representations of abstract notions such as “(human)
intention” as well as associations of symbols with brain activity patterns toward
improving CPSs in practice.
The Lattice Computing (LC) Paradigm 5
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