1. Automaton-like P Colonies: New Variants
Lucie Ciencialová1 and Erzsébet Csuhaj-Varjú2
1Institute of Informatics, Silesian University at Opava,
Opava, Czech Republic
2Faculty of Informatics, Eötvös Loránd University,
Budapest, Hungary
The work of E. CS-V. was supported by Grant No of the National Research,
Development, and Innovation Office, Hungary.

2. Automaton-like P colonies (APCol systems)
P colonies where the environment is given by a string. The members of the P colony (the agents) may insert, delete, or change symbols of the string. If the string is reduced to the empty word, then it is accepted by the APCol system. The set of strings (over an alphabet) accepted by the APCol system is its accepted language.

3. Automaton-like P colonies (APCol systems)
Agents + Environment
Environment:
multiset of objects over a finite alphabet
one object, e, in infinite number of copies
other objects in a finite number of copies

4. Automaton-like P colonies (APC0L systems)
Agents:
finite collection of finite multisets
(usually, pairs); the size remain unchanged
each agent is represented by its state (actual
multiset) and by a finite set of programs
(a,c) - contents (state) of the agent
(a->b; c<->d) – program (rules)
a is changed to b in the agent and c is exchanged with d in the string

5. Automaton-like P colonies (APCol systems)

6. Automaton-like P colonies (APCol systems)

7. Automaton-like P colonies (APC0L systems)
Result:
≈RE can be obtained with two agents.
(2-counter machine can be simulated)

8. Automaton-like P colonies (APCol systems) – a new variant
New definition of the accepted language: A string is accepted if all agents of the APCol system (the P colony) read through all of its letters. Verifying working mode

9. Automaton-like P colonies (APCol systems) – other new variant
Interacting agents:
Special rules for exchanging objects
Agent i Agent j
(a,b) (c,d)
b and c are allowed to be exchanged
(a,c) (b,d)

10. Observation: similarity to multihead finite automata

11. Observation: similarity to multihead finite automata
Each agent corresponds to a head of the multihead automaton.
The states (contents) of the agents correspond to the states of
the multihead automaton.
The agents „read” the symbols by changing it for some indexed
version of the letter; the index identifies the agents that had
already visited the symbol.
The string is accepted if every agent visited all symbols
of the string.

12. Similarity to multihead finite automata
Any language of an APCol system with interacting agents and
working in verifying mode can be accepted by a
jumping multihead finite automaton
and conversely.
Jumping multihead finite automaton:
each head scans all symbols but „jumps” are allowed.
Finite multihead automata characterize NSPACE(log n).

13. Similarity to multihead finite automata – Questions
How to simulate standard finite multihead
automata by APCol systems?
How to simulate two-way finite multihead
How to simulate pushdown multihead
What about size complexity?

14. Further problems, new variants
What about self-verifying APCol systems?
How to define multitape automata in terms of APCol systems?

15. Let us change object type
Instead of simple object
Let us use stack A A 1 A

16. Agent with capacity c
c sets of stacks with depth at most 2, only one stores 1
If the capacity is 2 and agent contains object Be 1 e A B C … 1 e A B C …

17. Rules, programs Rules - rewriting and communication Three phases
Detection of condition of applicability
Selection one of the applicable programs
Attempt to perform actions – agents perform actions in random order, not all agents are successful

18. Condition for use of programs
Program: < A → B; C ↔ D >
Because of stack agent can read only symbol on the top
IF (A is inside) and (C is inside) and (D is outside) then …
IF (X[A] v Y[A]) & (X[C] v Y[C]) & D then ... 1 e A B C … D 1 X 1 e A B C … Y

19. Actuator functions Program: < A → B; C ↔ D >
Divide program into rules and perform one by one
Erase and add symbols to the stacks
IF X[A] then E(X[A]) & A(X[B]) …
IF Y[C] then E(Y[C]) & A(Y[D]) & E(D) & A(C) C 1 D 1 1 e A B C … X 1 e A B C … Y

20. Notes
It is possible to construct one such complex rule for whole P colony
This system seems to be still computationally complete
It is not maximally parallel

21. Agent creation, dissolution
String processing P colony
After applying program agent can make copy of itself or dissolve its membrane
Can we solve 3-SAT?

22. Thank you for your attention