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

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

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

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, ...

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

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

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

(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

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.

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

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)

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

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

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

      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

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

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