1. Autonomous units and their semantics – the parallel case
WADT 2006
Autonomous units and their semantics – the parallel case
Hans-Jörg Kreowski & Sabine Kuske
Theoretical Computer Science, Department of Mathematics and Computer Science
research within the CRC 637 (12 research groups in computer science, economics, information technology, mathematics, and production engineering)
based on graph transformation
new programming and modelling paradigms (ubiquitous & mobile computing,
multi-agent systems, communication networks, etc)
logistic processes, in particular

2. Vision
apply rule-based graph transformation to model logistic processes (or others) without assuming central control
formal with rigorous semantics
adequate and general
generalizing existing approaches
admitting comparisons
supporting verification
visual
WADT 2006: the parallel case
ICGT 2006: the sequential case 4t HH HB 3t H DO 1t

3. Ideas syntax semantics autonomous units logistic processes
model
autonomous units ...
exist and coexist in a common environment
may cooperate or compete
have goals that they try to reach
have capabilities to work on the environment
run potentially nondeterministic
are self-controlling to cut down the nondeterminism

4. Concepts environment: graph (e.g. transport net) goals:
graph class expressions
(specifying what should become true)
capabilities:
internal: rules for local changes
external: import to get help
(not considered further in this talk)
control:
conditions to choose next action
finite state machines,
priorities,
„as long as possible“,
fitness functions, ...

5. community of autonomous units
Syntactic schema
community of autonomous units
overall goal (if there is any),
terminal environments
set of autonomous units
Com = (Aut, Init, Goal)
both specified by
graph class expressions
initial environments
autonomous unit
aut = (goal, rules, control)
individual

6. Example (graph algorithm)
Shortest Paths
Aut:
Init:
directed graphs with distance function x
Goal:
implies x  dist(path)
path

7. Example (place/transition systems)
P/T System
Aut:
aut(t)
rule: t
t  T
Init:
m0 (initial marking)

8. (free commutative semi-group over R with +) *
Semantic requisites
graph transformation approach
environments: G set of graphs
set R of rules with rule application relation:
  G G for r  R+
(free commutative semi-group over R with +) * r
* parallelism is plugged in here by allowing
the application of multisets of rules
rather than single rules

9. Semantic requisites
set X of graph class expressions with SEM(g)  G for g  X
e.g. terminal labels, reduced forms, graph properties,
graph grammars, etc.
set C of control conditions specifying
SEMP, CHANGE(c)  Gseq
for all P  R+ and CHANGE  G G
e.g. finite-state automata with rules as inputs,
valuation functions (to choose some best step), etc.

10. changes of environment
Semantic requisites
changes of environment
CHANGE  G G
Moreover, there is a set MR  R of (meta-)rules such that
CHANGE =  and
 =  for some 0  MR and all r  R MR
r+0 r

11. Parallel semantics of autonomous units
parallel process of aut = (goal, rules, control)
with dynamic environment given by CHANGE:
G0, G1, G2, …  SEM(control) with (for all i)
Gi  Gi+1 for some r  rules+ and r´  MR
or (Gi, Gi+1)  CHANGE
r+r´
PARCHANGE(aut) set of all parallel processes of aut
reachability of goal: Gj  SEM(goal)

12. Parallel semantics of communities
parallel process of Com = (Aut, Init, Goal):
s = G0, G1, G2, … with G0  SEM(Init)
and, for all i, there is some multi-set r of rules of units of Com with Gi  Gi+1 and, for each aut = (goal,rules,control)Aut,
s  SEMrules,CHANGE(control) with CHANGE = PAR(Com-aut) r
PAR (Com) set of all parallel processes of Com
reachability of overall goal: Gj  SEM(Goal)
Observation: PAR(Com) = PARCHANGE(aut)/Init for aut  Aut if CHANGE = PAR(Com-aut) and the control condition is compatible

13. with arbitrary parallelism of minimum and sum
Examples
PAR(Shortest Paths) = PARPAR(minimum)(sum)/Init = PARPAR(sum)(minimum)/Init
with arbitrary parallelism of minimum and sum
PAR(Shortest Paths) contains only finite sequences
G0,…,Gn such that in Gn if and only if the shortest path from A to B in G0 has distance x (correctness)
PAR(P/T System) contains all sequences firing
activated multisets of transitions starting in m0 A B x

14. Multi-agent systems
Communities of autonomous units are models of
the axiomatic approach to multi-agent systems (Wooldridge et al.)
all agents act in parallel on the global environment,
but one agent performs only one action in each step
the perceive-do procedure of agents can be used as control condition

15.       Example (NP-complete problems)
travelling salesman problem (TSP)
rules: 
start
(r1) (r2) (r3) (r4) (r5) (r6)
start 
start
run x x
run  ok
run x x ok
run  ok
run x x y
run 
run x
run
start  ok ok x x
control:
initially no node labels & r1 and r2 once & iterate (r3; r4! || r5) & r6 eventually & terminally only ok nodes & branch and bound

16. Example (NP-complete problems)
TSP solves the travelling salesman problem
nondeterministically in polynomial time (NP)
probability of a successful run is exponentially bad
build a community of k copies of TSP
and run them in parallel (still NP)
probability of a successful run converges against 1
with increasing k
There are other examples of swarm intelligence

17. Cellular automata G,A,init CAU(G,A,init) celaut-2-CAU
with a regular graph
G = (V,E,s,t,l) of type k
(ordered neighbours),
a finite-state automaton
A = (Q,Qk,d), and
an initial marking init
G with a marking represents
an environment, initial
environment by initial marking,
each node v induces an
autonomous unit au(v)
with rules reflecting d
Correctness
The community yields environments
by maximum parallel steps that correspond
to the markings derived by the cellular automaton