From: AAAI Technical Report FS-92-01. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.
DISTRIBUTED
Dipartimento
AI
MODELS FOR PERCOLATION
S.Bandini,
G.Cattaneo,
G.Tarantello
di Scienze dell’Informazione
- Universit£
email: bandini@hermes.mc.dsi.unimi.it
Abstract
The challenge of modelling complex physical situations by a qualitative approach represents a
good platform for testing alternative or integrative
methods to quantitative approaches. Weselected
a well-knowntheoretical and physical problem as a
field for studying classical solutions and their limits, proposing and testing a qualitative distributed
approach based on molecular ontology. The problem is percolation and the physical situation we
selected is coffee percolation phenomenon.
Introduction
Percolation is a complex phenomenon which has been
studied from several theoretical, applicative and computational standpoints. Several formal tools have been
tested or developed as possible solution to the problem of modeling and simulating it. Molecular ontology theory [1],[7], developed to obtain qualitative and
intuitive computational models for simulating the behaviour of substances (fluids, electrons, etc.), presents
some advantages in simulation tasks where a pure numerical approach presents intrinsic difficulties in terms
of computational size or collecting boundary conditions. Then, we are studying percolation for testing
our theory and comparing it with classical approaches.
Percolation:
the picture
Percolation is the simplest model of disordered system
whosetheoretical problems are not exactly solved [17].
It is an example of phase transition phenomenonand
many real problems, like materials conductivity and
magnetic properties, involve similar processes. According to a classical definition, given a square bidimensionai grid, percolation problem is that of determining
the lowest concentration of filled places pc such that an
unique infinite path of filled places can exist. Such a
path is called percolation cluster and pc is called percolation threshold. Formulating the problem in this way
*Supported by CNR(Prog. Fin. Sistemi Informatici
Calcolo Paxallelo, sott. 7,OB3)and by Iliycaff$ S.p.A.
di Milano
we obtain site percolation; another possibility is bond
percolation where the links, rather than the places, are
filled. Besides these classic percolation models, several
different models have been defined in order to represent particular situations. For instance, directed percolation, where a direction is assigned to the lattice
links, is a good model for a fire spreading in a forest
under a strong wind in one direction, or for a random
network of resistors that can be crossed in only one
way. In this case percolation threshold separates insulation from conduction [6]. Migration and sedimentation phenomena, like electrolysis,
can be modeled by
diffusion limited aggregation mechanisms [16] where a
migrating particle is represented by a random walker
jumping on the lattice from one site to another one (as
for the Brownian movement), and adhering to the aggregate whenis adjacent to it. Finally, invasion percolation [13], finds its best application in the problem of
washing out the petroleum rocks. Here, water spreads
throughout the rock and petroleum comes out from the
opposite side. The porous rock is modeled by a lattice
where the sites are pores connected by thin channels
(the links) and a random number, assigned to each
link, represents channel resistance to water flowing: at
each instant, water flows along the lowest resistance
path, removing petroleum.
Rest:
a cup of coffee
A more circumscribed, but no less complex percolation problem is "espresso" coffee percolation. A coffee
producing company offered to us the very interesting
instance of studying this process in terms of a scientific problem: several difficulties in espresso cooking
simulation by classic numerical models suggested, as
alternative, Artificial Intelligence qualitative methods.
The beverage (the final espresso we drink) is obtained
by a suitable percolation device where high temperature and hydrostatic pressure water flows throughout
a given quantity of ground and rosted coffee particles
(the "cake"). Each coffee particle is a piece of porous
material containing air and small oil drops, so, our phenomenonis similar in some aspects to the petroleum
case. Nevertheless, it is more complex owing to pro-
From: AAAI Technical Report FS-92-01. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.
ceases like oil emulsion, glucide dissolution, the formation of air bubbles and particles compacting.
Percolation:
the tools
Percolation as a static phenomenonis studied by numerical analysis in order to determine the exact value
of percolation threshold: exact results have been obtained only in some particular cases like monodimensional systems or the Bethe lattice [17]. However, if
we focus our attention on simulating real processes, as
espresso percolation, we must adopt a dynamic point
of view. One possibility in this direction comes from
fractal theory [4],[14]. In fact, percolation clusters
are often autosimilar on variable scales as are deterministic fractals (for example,Sierpinskji triangle [4]),
thus, somebodystudies percolation using iterated function sets and considering fractal dimension [6]. On
the other hand, a suitable non completely quantitative approach for modeling percolation in porous media [15],[9],[12], is obtained by cellular automata for
fluid dynamics: the HPP[10] and FHP[8] Lattice Gas
models respecting Navier-Stokes equation. Weare expanding these models to represent water flow throughout porous coffee particles with its chemical interactions. However, even if computational simulation of
these models by means of CAM[18] showed the possibility of representing some aspects of a coffee cake,
it was not possible to represent, in a qualitative way,
important features of the knowledge about this phenomenon. Thus, we captured the suggestion of employing a local and distributed approach but integrating it
by means of methods of knowledge- based simulation
[5].
Percolation:
lular automata for fluid dynamics suggested of adopting a local and distributed model and experimenting
molecular ontology tool.
an alternative
Molecular ontology model [1],[7], developed in Qualitative Physics context [11], is defined in terms of a
set of individual entities ("molecules"), lying on the
sites of an adjacency graph, and changing their position as they react to some external qualitative fields
[2]. They can communicate by messages exchange and
link together to form aggregates [3]. Their actions are
prescribed by a suitable set of local behaviour rules
wich can embody, in a qualitative way, the knowledge
of domain experts about espresso percolation. So, if a
distributed set of molecules is used in order to model
coffee particles and active water clusters, cake geometrical modification effect, determined essentially by high
pressure presence and consisting of the local movements of coffee particles, can be obtained by a suitable
communication between the molecules allowing them
to knowif there are free neighbour sites, (containing
air, for instance), to be occupied.
Conclusions
Weare investigating an AI qualitative approach in simulation of percolation. The advantages offered by cel-
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