LaMI MGS: a Programming Language for the Transformations of Topological Collections

98 Pages·2001·4.42 MB·English

LaMI
Laboratoire de MØthodes Informatiques
MGS: a Programming Language for the
Transformations of Topological Collections
Jean-Louis Giavitto & Olivier Michel
email(s) : giavitto ou michel @lami.univ-evry.fr
o
Rapport de Recherche n 61-2001
Mai 2001
CNRS (cid:21) UniversitØ d’Evry Val d’Essonne
523, Place des Terrasses
F(cid:21)91000 Evry France

MGS: a Programming Language for the
Transformations of Topological Collections
Jean-Louis Giavitto & Olivier Michel
LaMI u.m.r. 8042 du CNRS
UniversitØ d’Evry Val d’Essone
91025 Evry Cedex, France.
[giavitto,michel]@lami.univ-evry.fr
◦
LaMI technical report N 61-2001, May 2001
Abstract
We present the (cid:28)rst results in the development of a new declarative programming
language called MGS. This language is devoted to the simulation of biological processes,
especially those whose state space must be computed jointly with the running state of
the system (for instance, in morphogenesis).
MGS proposes a uni(cid:28)ed view on several computational mechanisms. Some of them
are initially inspired by biological or chemical processes (Gamma and the CHAM, Lin-
denmayer systems, Paun systems and cellular automata). They share the property of
specifying local transformations in space and time.
The basic computation step in MGS replaces in a collection A of elements, some sub-
collection B, by another collection C. The collection C is function solely of B and its
adjacentelementsinA. ThepastingofC intoA−B dependsontheshapeoftheinvolved
collections. This step is called a transformation.
Theorganisationoftheelementsintothecollectionsisviewedfromatopologicalview-
point. A formal framework to specify this notion of topological collections is proposed.
By changing the topological structure of the collection, the underlying computational
model is changed.
The speci(cid:28)cation of the collection to be substituted can be done in many ways. We
propose here a pattern language based on the neighborhood relationship induced by the
topology of the collection. Several features to control the transformation applications are
then presented.
Keywords
Topological collection, transformation, declarative programming language, simulation
of biological processes, dynamical systems, dynamical structure, Gamma, CHAM, P sys-
tem, L system, cellular automata, rewriting, rule based programming, combinatorial al-
gebraic topology, chain complex, chain group.
The authors of this research report can be contacted at:
La.M.I., CNRS UMR 8042
UniversitØ d’(cid:201)vry Val d’Essonne
Tour (cid:201)vry 2 / 4eme etage
523 Place des terrasses de l’agora
91000 (cid:201)vry Cedex France
Tel: [phone]
Fax: [phone]
TheMGSinterpretersarefreelyavailable,bysendingademandto giavittoor [email protected].
The MGS home page is located at url http://www.lami.univ-evry.fr/~mgs .
Versions of this report:
Revision september 2001: Added example 4.10 and appendix B, some additional references and ac-
(cid:15)
knowledgements, corrections of typos.
Initial Version: may 2001.
(cid:15)
Copyrights2001Jean-LouisGiavitto,OlivierMichel;LaMI(cid:21)UniversitØd’EvryVald’EssonneandCNRS.
Table of Contents
1 Motivations 1
1.1 Dynamical Systems and their State Structures. . . . . . . . . . . . . . . . . . . . . . . 1
1.2 DS with a Dynamical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Chemical Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Game of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Embryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Protein transport and Golgi Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Cell Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Summary of the Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 MGS Basic Ideas 11
2.1 The Concept of Transformation in a Collection . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Collections as Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 The MGS Project: Modeling Biosystems with a Dynamical Structure with Topological
Collections and their Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Organization of the Rest of this Report . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 An MGS Quick Tour 17
3.1 Functions, Sentences and Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Monoidal Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 The topologies of Monoidal Collections . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 User-De(cid:28)ned Monoidal Subtypes . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.4 Structural Recursion on Monoidal Collections . . . . . . . . . . . . . . . . . . . 20
3.3 Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Pattern, Rule and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.1 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Managing the Applications of a Transformation . . . . . . . . . . . . . . . . . . . . . . 25
4 Examples of MGS Programs 27
4.1 Maximal Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Map and Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Sorting a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Eratosthene’s Sieve on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Eratosthene’s Sieve on a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.7 Maximum Segment Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.8 Tokenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.9 Token moving on a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.10 Morphogenesis Triggered by a Turing Di(cid:27)usion-Reaction Process . . . . . . . . . . . . 35
5 Topological Collections and their Transformations 39
5.1 Organization of this section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Cellular Spaces and Combinatorial Structure of Complexes . . . . . . . . . . . . . . . 40
5.3 Star, Link and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Chain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Chain Group with Coe(cid:30)cient in an Arbitrary Abelian Group . . . . . . . . . . . . . . 45
5.6 Example of the C(K,Z/2,∂) Chain Complex . . . . . . . . . . . . . . . . . . . . . . . 47
5.7 The Structure of the Chain Group with Coe(cid:30)cient . . . . . . . . . . . . . . . . . . . . 49
5.8 Duality: Cochain, Coboundary and Cochain Complex . . . . . . . . . . . . . . . . . . 51
5.9 Arbitrary Labeling the Cells of a Complex . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.10 Topological Collections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.11 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.12 The Example of a 2D Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Comparaison with Other Approaches 65
6.1 The topology of Sets and Multisets: the programming language Γ and the CHAM . . 65
6.2 Nesting of Multisets: P systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 The Topology of Sequences: L systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4 The Topology of Arrays: Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5 Production Systems, Rewriting systems and All That . . . . . . . . . . . . . . . . . . 68
6.6 A Comparison with the Multi-Agent Modeling Paradigm. . . . . . . . . . . . . . . . . 69
7 Conclusion 73
A An MGS Grammar 75
B Full Code of the Turing+Morphogenesis Example 79
C Review of Some Notions Related to the Group Structure 83
List of Figures
1 Evolution of a predator-prey system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 DS with a simple dynamical structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Some rules for a lattice gas automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 During the neurula stage, the neural plate is folded to shape a tube. . . . . . . . . . . 6
5 Plant growth in presence of obstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6 Protein tra(cid:30)cking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 A basic transformation of a collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8 Transformation and iteration of a transformation. . . . . . . . . . . . . . . . . . . . . . 12
9 The subtyping hierarchy of collection kinds. . . . . . . . . . . . . . . . . . . . . . . . . 19
10 The Eratos program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
11 Tokenisation of a sequence of letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
12 The Turing di(cid:27)usion-reaction process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13 The Turing di(cid:27)usion-reaction process coupled with a morphogenesis. . . . . . . . . . . 37
14 Examples of complexes build from polygons. . . . . . . . . . . . . . . . . . . . . . . . 41
15 An abstract complex.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
16 An abstract complex cannot handle orientation. . . . . . . . . . . . . . . . . . . . . . . 43
17 Examples of star and link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
18 Connection and shape of a set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
19 Examples of a non-homological and an homological complex. . . . . . . . . . . . . . . 45
20 Application of the boundary operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
21 Oriented complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
22 The dual ∂ and δ operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
23 The labeling of the cell of an abstract complex. . . . . . . . . . . . . . . . . . . . . . . 55
24 Depiction of the boundary and coboundary operation on chains. . . . . . . . . . . . . 56
25 Parts of a complex involved in a substitution. . . . . . . . . . . . . . . . . . . . . . . . 60
26 Substitutions in a line graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
27 Modelling of 2D grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
28 A simpli(cid:28)cation of the neurulation for simulation purposes. . . . . . . . . . . . . . . . 69
List of De(cid:28)nitions
1. Bounded Poset (P,<) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2. Abstract Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3. Subcomplex, Star and Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4. Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5. Locally (cid:28)nite complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6. Chain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7. Chain Group with Coe(cid:30)cient in an abelian group G . . . . . . . . . . . . . . . . . . . 46
8. Compatible Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9. The free Chain Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
10. Cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11. Dual Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
12. Coboundary Operator δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
13. Cochain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
14. Topological Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
15. Split, Patch and Subcollection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
16. Shape-preserving, Pointwise and Local Operations . . . . . . . . . . . . . . . . . . . . 58
17. Renaming Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
18. Split, Patch and Non-Local Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . 59
19. Simple Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1 Motivations
We want to developp a framework dedicated to the simulation of dynamical systems with
a dynamical structure. The application area we have in mind is the simulation of some
biological processes, especially those whose state space must be computed jointly with the
running state of the system. This technical report is organized as follow:
Section 1 gives our motivations. After a very brief presentation of the notions related to
the dynamical systems, we introduce the notion of dynamical structure through some
examples.
Section 2 sketchesanuni(cid:28)edframeworktodescribedynamicalsystemwithadynamicstruc-
ture. The notions of collection, subcollection and transformation are described.
Section 3 contains a brief description of the MGS programming language. This language
implements a subset of the previous ideas.
Section 4 illustrates the MGS language through paradigmatic examples.
Section 5 presents the (cid:28)rst development of a formal description of the MGS constructions,
using mathematical notion developped in the (cid:28)eld of algebraic topology. Our main goal
in this section is to introduce some of the topological notions upon which a theory of
transformations can be build.
Section 6 makesacomparisonwithotherapproaches: theGammalanguageandtheCHAM,
Lindenmayer systems, Paun systems and cellular automata. A comparaison with the
approach of multi-agent systems, often advocated in the simulation of complex dynam-
ical systems, is also developped.
Section 7 concludes this report by giving some directions opened by this work.
1.1 Dynamical Systems and their State Structures.
A dynamical system (or DS in short) corresponds to a phenomenon that evolves in time.
The phenomenon is located on a system characterized by (cid:16)observables(cid:17). The observables
are called the variables of the system, and are linked by some relations. The value of the
variables evolves with the time. The set of the values of the variables that describes the
system constitutes its state. The state of a system is its observation at a given instant. The
state has often a spatial extent (the speed of a (cid:29)uid in every point of a pipe for example).
The temporal sequence of state changes is called the trajectory of the system.
Intuitively, a DS is a formal way to describe how a point (the state of the system) moves
in the phase space (the space of all possible states of the system). It gives a rule telling us
where the point should go next from its current location (the evolution function). These
notions are illustrated in Fig.1.
We are interested in the simulation of such systems. This requires the speci(cid:28)cation of the
system state and the evolution function. This speci(cid:28)cation can be very di(cid:30)cult to achieve
1
Trajectories x(t) and y(t)
Constraints in the evolution of x and y
20
x(t) dx
=Ax−Bxy
dt
15 dy
=−Cy+Dxy
dt
10
Solving the constraints gives the trajectory
of x(t) and y(t) starting from some initial
5
y(t) state. The evolution of a variable is periodic.
50 100 150 200 250 300
Evolution in the phase space (x,y)
12
Thethreecurvescorrespondtothecyclicevo-
10
lution of the system starting from three dif-
ferent initial conditions. A point in this plot
8
corresponds to a state (x,y). A curve corre-
6 sponds to the evolution (x,y)(t). The peri-
odicity of the trajectories of x and y gives a
4
closedcurve. Thereisafourthcurvereduced
to a (cid:28)xed point. The image by the evolu-
2
tionfunctionofthispointisitself. Thispoint
is characterized by dx/dt = dy/dt = 0 (no
5 10 15 20
change).
Figure 1: Evolution of a predator-prey system (a DS with a static structure).
The system is characterized by two variables: x corresponds to the number of predators and y to the
numberofpreysinsomeecologicalsystem. Thenumberofpreyschangesbecauseofthegrowthofthe
populationandbecausethepreysareeatenbythepredators. Thenumberofbirthsisproportionalto
thenumberofpreysandthedecreaseisproportionaltothenumberofprey-predatorencounters,which
is itself proportional to the product xy. The number of predators decreases because the competition
between predators and the increase is proportional to the chance of prey-predator encounters. The
resulting di(cid:27)erential equations specify the evolution function. They can be integrated to plot the
trajectory of x and y (top picture) and the state evolution (bottom picture). The structure of the
system is static in the sense that the state of the system is always described as an element of R2.
2