1. Characterizations of Bounded Semilinear Languages by One-way and Two-way Deterministic Machines Oscar H. Ibarra 1 and Shinnosuke Seki 2 1. Department of Computer Science, University of California, Santa Barbara, USA 2. Department of Systems Biosciences for Drug Discovery, Kyoto University, Japan

2. Conceptual diagram of a counter machine 2011/8/18Automata and Formal Languages (AFL 2011)2 finite state control a b input tape p c1c1 c2c2 ckck can be tested for zero counters …

3. Conceptual diagram of a counter machine 2011/8/18Automata and Formal Languages (AFL 2011)3 finite state control a b input tape p c1c1 c2c2 ckck can be tested for zero counters q …

4. Variants of counter machine 2011/8/18Automata and Formal Languages (AFL 2011)4 finite state control a b input tape p c1c1 c2c2 ckck counters … Pushdown counter machines 2-way counter machines stack end markers

5. Formal definition of pushdown counter machines 2011/8/18Automata and Formal Languages (AFL 2011)5

6. Reversal-bounded counter machines 2011/8/18Automata and Formal Languages (AFL 2011)6 Definition [Baker and Book 1974, Ibarra 1978]Definition (Equivalent * ) A finite automaton augmented with 2 counters is Turing-complete. [Minsky 1961] * This equivalence needs a help of finite state control to remember how many reversals each counter has made.

7. 2011/8/18Automata and Formal Languages (AFL 2011)7 Lemma

8. Classes of counter machines 2011/8/18Automata and Formal Languages (AFL 2011)8

9. Classes of counter machines (cont.) 2011/8/18Automata and Formal Languages (AFL 2011)9

10. Semilinear sets 2011/8/18Automata and Formal Languages (AFL 2011)10

11. Bounded semilinear languages 2011/8/18Automata and Formal Languages (AFL 2011)11 Definition

12. Finite-turn machines 2011/8/18Automata and Formal Languages (AFL 2011)12 Finite-turnFinite-crossing Finite-turn Finite-crossing

13. Main theorem 2011/8/18Automata and Formal Languages (AFL 2011)13 Main Theorem

14. Main theorem 2011/8/18Automata and Formal Languages (AFL 2011)14 Main Theorem 1 25 34

15. 2011/8/18Automata and Formal Languages (AFL 2011)15 Language Accepted by result 1 [Ibarra 78] Use result 1 and non- deterministic guess. Linear Diophantine- equations turn out to be solvable by these machines

16. Letter-bounded to bounded 2011/8/18Automata and Formal Languages (AFL 2011)16

17. 2011/8/18Automata and Formal Languages (AFL 2011)17 Language Accepted by result 1 [Ibarra 78] Use result 1 and non- deterministic guess. Linear Diophantine- equations turn out to be solvable by these machines How can a deterministic machine non-deterministic decompose an input? Our answer is “ this decomposition does not require non- determinism. ”

18. Deterministic decomposition 2011/8/18Automata and Formal Languages (AFL 2011)18 Theorem [Fine and Wilf 1965]

19. Main theorem 2011/8/18Automata and Formal Languages (AFL 2011)19 Main Theorem

20. 2011/8/18Automata and Formal Languages (AFL 2011)20 Lemma stack

21. 2011/8/18Automata and Formal Languages (AFL 2011)21 Lemma

22. 2011/8/18Automata and Formal Languages (AFL 2011)22 Theorem

23. Main theorem 2011/8/18Automata and Formal Languages (AFL 2011)23 Main Theorem (Proved!!) Open

24. Acknowledgements This research was supported by National Science Foundation Grant Natural Sciences and Engineering Research Council of Canada Discovery Grant and Canada Research Chair Award in Biocomputing to Lila Kari Funding Program for Next Generation World-Leading Researchers (NEXT Program) to Yasushi Okuno 2011/8/1824Automata and Formal Languages (AFL 2011)

25. References 2011/8/18Automata and Formal Languages (AFL 2011)25 [Baker and Book 1974] B. S. Baker and R. V. Book. Reversal-bounded multipushdown machines. Journal of Computer and System Sciences 8: 315-332, 1974. [Fine and Wilf 1965] N. J. Fine and H. S. Wilf. Uniqueness theorem for periodic functions. Proc. Of the American Mathematical Society 16: 109-114, 1965. [Ibarra 1978] O. H. Ibarra. Reversal-bounded multicounter machines and their decision problems. Journal of the ACM 25: 116-133, 1978.

26. References 2011/8/18Automata and Formal Languages (AFL 2011)26 [Minsky 1961] M. L. Minsky. Recursive unsolvability of Post’s problem of “tag” and other topics in theory of turing machines. Annals of Mathematics 74: 437-455, 1961.