1. A Small Universal Splicing P System A. Alhazov, Y. Rogozhin, S. Verlan HiroshimaParis Chi ş in ă u

2. What is small? We consider the number of rules as a descriptional complexity measure.

3. What is universal? The universality is a property of a class of “computability models” C that means that there is a fixed element U that is able to “simulate” any element E of C, providing that an “appropriate” encoding of E and its input is given. Universality is not the same as computational completeness.

4. What is universal (2)? From Robinson, 1991: A Turing machine U is called universal if it can simulate each Turing machine T. We are to give U a code for the structure of T and a code for the initial tape of T. The machine U must halt if T would have halted, and the final tape of U must include a code for the output of T. In constructing a small universal machine, such as Minsky’s, we must allow rather elaborate methods of encoding the input and decoding the output. How can we be sure that we are not cheating, by doing the real work in the encoding and decoding? This can be avoided if we insist that the encoding and deconding processes always belong to certain classes. Then the bulk of a difficult calculation will remain with the Turing machine.

5. What is universal (3)? So the concrete definition of the universality depends a lot on the definition of the encoding and decoding functions. Generally, we assume that these functions should be computable by the underlying model. In most of the cases elementary functions are used.

6. Small universal Turing machines

7. Small universal register machines [Korec, 1996]

8. Small universal P systems Strings: ◦ Universal splicing P systems (8 rules) [RogozhinVerlan2005] Multisets: ◦ Universal antiport P systems (30,23 rules) [Csuhajetal2007], [AlhazovVerlan2008]

9. What is splicing? (T. Head) u v u' v' x y x'y' (xuvy,x'u'v'y') |- (xuv'y',x'u'vy)

10. Splicing P system a P system the objects are strings the operation is splicing with targets (can be associated to the edges of the communication graph)

11. Simulating device Tag systems ◦ Rules: a i → P i ◦ Transition: a i bw ⇒ wP i ◦ Halt: when a 1 is the first letter.

12. Example T=(2,{c,a,b},P), a → bb b → abc c → STOP bcabbbbbbbbbcccaaaa

13. Coding data Unary encoding: c(a i )=α i β String: w => Xββc(w) βY

14. Main idea Rotate-and-simulate method: ◦ Start with Xββ c(a i bw) β Y. ◦ Replace β Y by words of form c(P j ) βα j Y, 1 ⩽ j ⩽ n (a guess). ◦ In a cycle eliminate at the same time one α from the beginning of the word and one α from the end of the word. ◦ If reached β from both ends, then the guess was good. ◦ Erase the remaining symbol.

15. The idea (2) The end: ◦ X ββ αβ c(w) β Y ⇒* X ’ αβ c(w) Y’

16. X ββ α 3 β α 2 β α 3 β β YX ββ α 3 β α 2 β α 3 β α 3 β α 3 β β α 2 YX α3β α2β α3β α3β α3β β α2 YX α3β α2β α3β α3β α3β β α2 YX α3β α2β α3β α3β α3β β α YX α3β α2β α3β α3β α3β β α YX α2β α2β α3β α3β α3β β α YX α2β α2β α3β α3β α3β β α YX α2β α2β α3β α3β α3β β YX α2β α2β α3β α3β α3β β YX αβ α 2 β α 3 β α 3 β α 3 β β YX ββ α 3 β α 2 β α 3 β α 2 β α 3 β α β β α 3 YX α3β α2β α3β α2β α3β αβ β α3 YX α3β α2β α3β α2β α3β αβ β α3 YX α 3 β α 2 β α 3 β α 2 β α 3 β α β β α 2 YX α 2 β α 2 β α 3 β α 2 β α 3 β α β β α 2 YX α2β α2β α3β α2β α3β αβ β α YX α2β α2β α3β α2β α3β αβ β α YX αβ α 2 β α 3 β α 2 β α 3 β α β β α YX αβ α2β α3β α2β α3β αβ β YX αβ α2β α3β α2β α3β αβ β YX β α2β α3β α2β α3β αβ β YX β α2β α3β α2β α3β αβ β YX β αβ α3β α2β α3β αβ β YX β αβ α3β α2β α3β αβ β YX β β α3β α2β α3β αβ β YX β β α3β α2β α3β αβ β Y Previous example (a → bb, b → abc, c → STOP, bab)

17. The construction 1 2 3 X ββα XZ 3.1: εβYβY Z’ ε 1.1: X βαε XβXβ Z 1.3: X ββαβαβ X’Z 3.2: XαXαε XZ 2.1: εαYαY Z Y 1.2:

18. Some remarks Number of rules6 Size of rules(3,2,2,1) The size of the alphabet8 Number of axiomsfinite What is the trick? Usage of sets instead of multisets !

19. Conclusion The result gives a surprisingly small universal computationally complete device, competing with TM. It could be difficult to improve as the rotate-and-simulate procedure needs at least 4 rules.