The Theory of Simulation
Nina Podolskaya
Feb 25, 2009
Virtually all of the major unsolved
problems in science today concerns
complex, self-organizing systems,where
vast numbers of components interact
simultaneously, with each shift in one
agent influencing the other.
Steven Strogatz, cited in Nature, Vol. 421,
p. 782
When Simulation May Be Useful
When we are investigating the complicated
phenomena of life and
• No explicit model of a phenomena
or
• A model of a phenomena may not yet be
derived
or
• Analytic solutions is not appropriate due to
system complexity
What Is a Simulation
Simulation is a mechanism which interacts
many state transition models of individual
subsystems (i.e. system components) and
thus generates system dynamical
phenomena
What Is a Dynamical System
A dynamical system is a pair consisting of a
state space, with each point representing
a state of the system, and a fixed rule
governing the time evolution of these
points. The time can be continuous or
proceed in discrete steps.
Synthetic Investigation Properties
Compared to Analytical One
• No explanation, only representation
• Dynamical representation is implicit and
constructive: relations are not explicitly
encoded in subsystems but are the result
of interactions between the subsystems
Basic Parts of Simulation
• To be aware that a simulation is generating
dynamical phenomena at higher level than
elemental interactions are described
• To have methods to identify the elements of
underlying system that create the phenomena of
interest
• To formulate models of the important
subsystems
• To create the framework in which the simulation
of the subsystems in interaction is composed
and embody the system representation in this
framework so that phenomena of interest can be
generated and analyzed
Examples of Modeling
• Monomers and solvent molecules
• Vehicles, roadway segments, signals and
travel goals in a traffic system
Interactions
• Interactions can be viewed as discrete
events among the subsystems undergoing
local state changes and communicating
the state changes to its neighbors in some
space
Event Driven Simulation
• Event can be externally or internally generated
• Event can also be a time step
=>
• Time stepped simulation is a special case of an
event driven simulation
• Continuous space, continuous time, dynamical
systems equally well can be treated in the given
framework
• Discrete space, discrete time systems do not
have any other general formal framework within
which the dynamical properties can be
generated and analyzed
Simulation
Simulation
• We consider simulation as an iterated mapping
of a (usually large and complicated) system
• Simulation is an iterative system in which the
simulated system is represented and its
dynamics calculated
=>
• The simulation and the simulated system are
both dynamical systems and the interplay
between the dynamics of the coordination of the
simulation updates and the dynamics calculated
in the time series of system states are essential
issues
What Comprises a Simulation
• Existence of some ∑R, a real or natural system
in the world that we are interested in
• ∑(SiЄM), models of subsystems Si of this system
and rules that define interactions among the
subsystems
• ΣS, a simulation of ΣR involving Σ(SiЄM) and some
update functional U
• ΣC, a formal (and equivalent physical) computing
machine on which ΣS, is implemented
Objects
• The objects (elements or subsystems) in a
simulation are defined as
Si=Si (fi,Iii,xi,ti), i and j=1, …, n
where fi is the representation of the
dynamics of the i-th object and where Iii is
the i-th object’s interaction rules with other
objects j. Interaction and dynamics
operate on xi, the state of i-th object. ti is
the local object time coordinate
∑(SiЄM)
Si is an element in the system ∑(SiЄM), that is
Si is a model of the i-th element of the set
of modeled system elements in M, i=1, …,
n. Thus, the algorithmic part of Si is
equivalent to fi and Iij
U
• An object update functional U is the state
transition
Si(ti+Δi) ← Si (ti), i=1,…,n
or
Si(ti+Δi)=U(Si(ti)), i=1,…,n
where U, the update functional, defines,
organizes and executes the formal
interactive procedure that prescribes the
state transition
ΣS
• A simulation is the iteration of object
updates over the entire set of objects
{Si(ti+1)} ← {Si (ti)}, i=1,…,n
or
{Si(ti+1)}=U({Si(ti)}), i=1,…,n
ΣC
• The formal, or equivalent physical
implementation, of the mechanisms of the
iteration procedure that prescribe the
interactions and consequent object state
transitions ant their storage. ΣC is normally
a physical and conceptually digital
computer of some kind.
Emergence
Observational functions
• Si1Є ∑M, i=1,…,n
• U – update functional at level of
description L1
• O1 – observational function by which the
objects can be inspected
S2 ← U{Si1(fi,Iij,xi,ti), O1}, i,j=1,…,n
S2 - a second order structure occurring at
level L2. Thus new structure may be
subjected to a possible new kind of
observer O2.
Emergent property
• Property P is emergent iff P Є O2(S2), but
P not Є O1(Si1).
• Examples:
– Dynamical properties of a polymer in solution
(low level: monomer-monomer interactions,
monomer-solvent interactions)
– Properties of a congestion in a traffic system
(low level: vehicle-vehicle interactions,
vehicle-roadway and –signal interactions)
Simulation and Emergence
• Simulation is a representational
mechanism that is distinguished by its
capacity to generate relations that are not
explicitly encoded
• Simulation is a natural method to study
emergence
Examples
• S2 – polymer described through monomermonomer and monomer-solvent
interactions. P2 could be polymer
elasticity.
• S2 – traffic jam described through vehiclevehicle and vehicle-roadway interactions.
P2 could be lifetime of a traffic jam.
Examples in Science
• L1 – Statistical Mechanical, L2 –
Thermodynamical description of a matter
• L1 – Lattice Gas Automata for fluid particle
dynamics, L2 – Navier Stokes equations
for macroscopic fluid dynamics
Simulatability
Distributable Update
• Let φ(Sq1,…, Ss1) be a hierarchically distinct
representation of a subset of the interacting
objects (Sq1,…, Ss1). Thus, φ(Sq1,…, Ss1) defines
a sub-aggregation (an aggregation model) of
some of the objects.
• If
U({S11,…, Sn1})={U(Si1),U(Sj1),…,U(φ(Sq1,…, Ss1)),
U(Sl1),…, U(Sp1)}
for some order of the objects, then the update U is
distributable over the decomposition ΣM of the
system and each object and object aggregation
can be updated independently of each other
Simulatable System
• A system is simulatable iff there exists a
pair (M,U) such that U distributes over Si Є
M
=>
• If no distributable U exists for some
subaggregation of the objects M which
allows the update to distribute, then the
system is non-simulatable
• If no M exists so that a given U can
distribute then the system is also nonsimulatable
Example
• Model polymer defined on a 2D lattice:
polymer in some solvent (heat bath). Each
of the monomers update in parallel – nonsimulatable: k steps of update cycle
implies that polymers above a certain finite
length are non-simulatable.
• Two step parallel scheduling (red/green) –
polymers of any length can be simulated
Universal Simulator
Definition
• Universal simulator US is a machine that
is able to resolve all causal dependencies
among the objects Sj1.
A Universal Simulator Structure
qi – counter associated with the object Si
qf counts the failed update trails
IG – interaction graph
Determining Simulatability
• Update of S11: if it doesn’t depend on the
state of any other object – it is updated
and counter q1 incremented. If it has
dependencies – qf incremented and S21 is
attempted to be updated…
• US can determine whether a system is
simulatable or not at most ∑i=0n-1(n-1)(n-i)
operations
Dependency Jacobian
• US defines object update dependency Jacobian
Dq=(∂qi/ ∂qj)n×n≡(Δqi/ Δ qj)n×n
where the derivative ∂qi/ ∂qj expresses how many
updates of object j are necessary to update
object i once;
Δqi/ Δ qj=0 indicates that the update of the object i
is independent of object j;
Δqi/ Δ qj≥1 tells that the update of i needs the state
of object j at current time before it can be
updated;
Δqi/ Δ qj ≡ 1 for i=j
Simulatability Determined by US
• A system ∑S is simulatable if Dq is a
diagonal matrix (unit matrix).
• If Dq contains sub-matrices on the
diagonal and ∑S is found non-simulatable
it is indication that ∑S could become
simulatable by the construction of subaggregations including the objects
contained within each of the sub-matrices
Bibliography
• S.Rasmussen, C.L.Barret Elements of a
theory of simulation, 1995
• S.Rasmussen, J.R.Smith Lattice polymer
automata, 1994
• K.Nagel, S.Rasmussen Traffic at the edge
of chaos, 1994