Systems with discrete geometry

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Systems with discrete geometry Nicoletta
Sabadini, RFC Walters Final meeting ART Udine
7-8 January 2008
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1. Introduction

• Systems in computer science may be described in
  two ways
• as a composition of processes eg by process
  algebra,
• as a geometry with an associated state space.
• The main result of this lecture is a relation
  between these points of view.
• The geometry in (ii) is usually finite (or at
  most, finitely recursively defined).
• The geometry may represent distribution or
  control.

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Flowchart
Petri net
Digital circuit
Analog circuit
Dining philosophers
(all except the last diagram taken from wikipedia
pages)?
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Notice that though all these examples have finite
geometry, in each case there is a state space
understood, and in the flow chart, analog
circuit, and (perhaps) the Petri net, the state
space is infinite. The state space is continuous
in the analog circuit. Our result will allow us
to describe all these examples and their
behaviours in a uniform way. In the flow chart
the geometry might be better called control. In
other examples the geometry is spatial
distribution. In the dining philospher example
we see two levels of geometry, distribution and
control.
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2. What geometry?
In each case the geometry is (not a topological
space but) a monoidal graph, i.e. it consists it
consists of a set A of arcs (the components), a
set V of vertices (the wires) the source and
target of each component is a word in the
vertices. Monoidal graphs form a presheaf
category MonGraph. If there are input and output
wires, the geometry is a cospan of monoidal
graphs, between objects which are discrete graphs
(no arcs). Denote the category of such cospans
as Cspn(MonGraph).
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monoidal graph
cospan of monoidal graphs
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Let Rm,n be the cospan of graphs with one
component, and with m inputs and n
outputs. Proposition 1 Cspn(MonGraph) is the
free wscc category generated by the Rm,n
(m,n0,1,2,) A wscc category is one with the
structure of a symmetric monoidal category, and
for each object X the structure of a commutative
monoid and comonoid satisfying the Frobenius and
separable axioms.
(partial results in RSW04, GH97)?
separable
Frobenius
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This result says that monoidal graphs may be
represented as wscc expressions already a
relation between geometry and algebra.
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3. Categories of state spaces
Consider a category, preferably a topos, E whose
objects are apt for being state spaces of
systems. Suppose also there is an object I (or
objects) which parametrize behaviours i.e.
Hom(I,X) is the set of behaviours of object
X. Example 1. EGraphs. The objects which
parametrize motion are the graphs In0-gt1-gt-gtn.
A behaviour in X is a path in the graph X.
Behaviours, by virtue of a famillial cocategory
structure on the In, form a category.
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Example 2. Let C be a topos of smooth spaces, and
smooth maps. Then take E to be C/T, T the tangent
space functor. The behaviour parametrizer is
R-gtR2, t-gt(t,1), R the real numbers. If Y Rn
then TYYxY. Then a behaviour of (p,v)X-gtTYYxY
is a function fR-gtX such that (pf)?vf In
the special case that XY and (p,v)(1,v) then
(p,v) is just a vector field on X and f satisfies
f?vf We will see that the more general case
allows us to consider mixed equations and
differential equations, and hence components
which a computer scientist might call
non-deterministic. Katis, Sabadini, Walters, On
the algebra of systems with feedback boundary,
Rendiconti del Circolo Matematico di Palermo
Serie II, Suppl. 63 (2000)
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We may picture such state spaces and behaviours
as follows
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Example 3. EMonGraph. The parametrizers of
motions are the linear multigraphs those in
which an object occurs at most once as a domain
and once as a codomain, and which have no loops.
There is a famillial cocategory structure on
these graphs which induces a category structure
on the behaviours. The behaviours form the free
symmetric monoidal category on the monoidal
graph. This is a coherence theorem string
diagram calculations. c/f Monanari, Meseguer
Petri nets
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4. Systems with geometry and state space
I have described two separate things geometry and
state space dont get confused by the last
example where monoidal graphs occur in two
places. Now to combine them. What is a geometry
with a state space? The answer is that there are
two cases a covariant one and a contravariant
one. A covariant state space associated with the
geometry G in the category E is a monoidal graph
morphism from G to cospan(E).
Contravariant is a morphism to span(E). The
covariant case regards control, the contravariant
regards distribution.
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Contravariant case each component has a state
space which is reflected on the boundary so
each component yields a span of state spaces.
Examples circuits, dining philosophers.
Covariant case The input and output boundaries
are initial and final states, hence a
component yields a cospan. Examples
flowcharts, Petri nets.
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The total state space of a system is then given
by a colimit in the covariant case, and a limit
in the contravariant case. The behaviours are
given by homming in from the parametrizing
object(s). Example EGraphs
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Example Esmooth spaces
E
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IIn analogue circuits what seems to be a diagonal
is in fact a component which satisfies the
Kirchoff laws for voltages and currents. The
alowed expressions are compact closed ones, not
the full wscc expressions.
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5. Discrete spaces with state vs algebras of
processes
What now is the promised connection between the
two points of view? The following 2
results Proposition 2. Cspn(MonGraph/G) is the
free wscc category on the monoidal graph G. This
is a generalization of the earlier result. Now if
C is a wscc category, let C denote the
underlying monoidal graph of C. The identity
monoidal graph morphism induces a wscc functor
Cspn(MonGraph/C)-gtC. Proposition 3. The
induced functor Cspn(MonGraph/span(E))-gtspan(E)
? is the total state space functor described
above (limit case). Similarly for cospan (colimit
case).
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The meaning of the theorem is that total space,
and hence the behaviour of a system described as
a monoidal graph morphism phi G-gtspan(E) can be
calculated in two ways (i) by calculating a
limit of a diagram (ii) by expressing G in terms
of the wscc operations in Cspn(MonGraph/span(E)
) and evaluating the same expression instead in
span(E). (i) is the geometric point of view
(ii) is the compositional point of view. A
special case of this was presented at CT2007.
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6. Examples
1. The Boolean circuit?
2. The analogue circuit?
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7. Remarks
1. What is missing is an understanding of the
relation between limits and colimits.
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8. The hundred errors of CCS (or Taking
Milner seriously)
Comment commenced 19 December 2007 I have been
thinking a bit more about CCS. I agree that
processes are fundamental. A mathematical theory
of such is of great importance, perhaps equal to
the importance of a theory of functions. One of
the main issues is the parallel composition of
processes. This means that the attempts to
develop such a theory should be subject to
the fiercest debate. Any mistake at the beginning
would lead to endless confusion. In this spirit
I would like to argue that there are many
misdirections taken in the theory proposed by
Robin Milner. Perhaps not 100 mistakes but many.
In this page I would like to record my opinion.
So far I have described only 10 errors but the
page will gradually be updated.
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I am very happy to receive corrections and
comments on my list in particular with respect
to my misconceptions about CCS, but also with
regarding divergence of opinion about the nature
of processes. I can immagine that this list may
be regarded as solely negative "it is easy to
criticize". Instead it should be understood as
taking the programme of Milner and others
seriously an attempt to help in understanding
the fundamental notion of "process", and
parallelism a program initiated and researched
mainly by the concurrency community.
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Mistake 1. The parallel communicating
composition of processes should not involve
broadcast as the basic communication method. One
consequence of broadcast is that parallel
composition (even of two atomic processes) needs
to be complicated in order to be associative.
Mistake 2. A theory of processes should begin
with a clear model. Syntax should come second.
This applies even at the stage of developing a
theory of processes. Mistake 3. Interleaving is
a mistake.
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Mistake 5. In CCS there is no distinction between
channel and signal on a channel. What does the
handshake - channels or signals? Mistake 6. The
geometry of processes should be explicit and
distinct from states of processes. Mistake 7.
In CCS there is no distinction between
fx1AxA-gtAxA and fA-gtA. As a result there is no
distinction between fxgAxA-gtAxA and gfA-gtA such
that gffg. Hence any meaning of causality is
lost.
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Mistake 8. P Q should not be equal to Q P
at most they should be isomorphic. Mistake 9.
There is no distinction between non-communicating
parallel and communicating parallel. Mistake 10.
Composition should hide the interface of
communication.
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Mistake 11. Lack of Zero communication. There is
a single tau, but it does not play the role of
indicating that two processes have zero
communication. Mistake 12. n processes should be
able to synchronize in a single action. This is
related to the following Mistake 13. There
should be processes for each channel which are
the identity with respect to parallel - that
transmit at the same time as they receive.
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Mistake 14. The simplest examples of processes
are functions. So the category of sets and
functions should be an algebra of processes. It
has sequential and parallel composition which are
related through a distributive law. Mistake 15.
The relation between sequential and parallel
should be a distributive law/exactness
condition. Mistake 16. The operation of hiding a
channel has the effect of preventing action, not
just hiding. This seems to me to be unphysical,
very difficult to implement. Seems a purely
mathematical operation.
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Mistake 17. The only notion of cycle is through
recursion. Real processes cycle through returning
to the same state. Recursion is a different
phenomenon, also important. An algebra of
processes should have both. Mistake 18. A
process which consists of an infinite number of
processes in parallel may be defined. Mistake
19. The meaning of a CCS term is given by a
behaviour, not a process. A process should have a
behaviour, but not just be a behaviour. Mistake
20. Programs should be expressions in the algebra
of processes. Instead, in process algebras
programs come first (that is, the free algebra
comes first), and processes are defined to be
elements of a quotient algebra.
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End of lecture