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Characterizations of Bounded Semilinear Languages by One-way and Two-way Deterministic Machines

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

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Characterizations of Bounded Semilinear Languages by One-way and Two-way Deterministic Machines

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  1. Characterizations of Bounded Semilinear Languages by One-way and Two-way Deterministic Machines Oscar H. Ibarra1 and Shinnosuke Seki2 Department of Computer Science, University of California, Santa Barbara, USA Department of Systems Biosciences for Drug Discovery, Kyoto University, Japan

  2. Conceptual diagram of a counter machine finite state control can be tested for zero p a b c2 … c1 ck input tape counters Automata and Formal Languages (AFL 2011)

  3. Conceptual diagram of a counter machine finite state control can be tested for zero p q a b c2 … c1 ck input tape counters Automata and Formal Languages (AFL 2011)

  4. Variants of counter machine • Pushdown counter machines • 2-way counter machines finite state control p a b c2 … c1 ck input tape counters stack end markers Automata and Formal Languages (AFL 2011)

  5. Formal definition of pushdown counter machines For a pushdown -counter machineis a 7-tuple , where a finite set of states input and stack alphabets the bottom stack symbol the initial state a set of accepting states and is a relation Automata and Formal Languages (AFL 2011)

  6. Reversal-bounded counter machines A finite automaton augmented with 2 counters is Turing-complete. [Minsky 1961] Definition [Baker and Book 1974, Ibarra 1978] For a positive integer , a counter machine is -reversal if for each of its counters, the number of alternations between non-decreasing mode and non-increasing mode – called reversal – is upper-bounded by in any computation. Definition (Equivalent*) For a positive integer , a counter machine is -reversal if for each of its counters, the number of alternations between non-decreasing mode and non-increasing mode – called reversal – is upper-bounded by in any accepting computation. * This equivalence needs a help of finite state control to remember how many reversals each counter has made. Automata and Formal Languages (AFL 2011)

  7. -reversal counter and 1-reversal counter One can simulate a counter that makes -reversals by counters that make reversal. This lemma enables us to focus on the -reversal counter machines. Lemma For any positive integer , an -reversal counter can be simulated by -reversal counters. Automata and Formal Languages (AFL 2011)

  8. Classes of counter machines For a machine class , by , we denote the set of all languages accepted by a machine in . Automata and Formal Languages (AFL 2011)

  9. Classes of counter machines (cont.) • is the class of deterministic counter machines. Note that . Automata and Formal Languages (AFL 2011)

  10. Semilinear sets A subset is a linear set if there exist -dimensional integer vectors such that A set is semilinear if it is a finite union of linear sets. Automata and Formal Languages (AFL 2011)

  11. Bounded semilinear languages A language is bounded if there exist and such that . If are letters, is especially said to be letter-bounded. Definition For a bounded language , let . If this set is semilinear, then we say that is bounded semilinear. Automata and Formal Languages (AFL 2011)

  12. Finite-turn machines Finite-turn Finite-crossing For an integer , a 2-way machine is -turn if the machine can accept an input with at most “turns” of the input head. A 2-way machine is finite-turn if it is a -turn for some . For an integer , a 2-way machine is -crossing if every accepted input admits a computation by the machine during which the input head crosses the boundary between any two adjacent symbols no more than times. A 2-way machine is finite-crossing if it is a -turn for some . Finite-crossing Finite-turn Automata and Formal Languages (AFL 2011)

  13. Main theorem Main Theorem The following 5 statements are equivalent for every bounded language : is bounded semilinear; can be accepted by a finite-crossing ; can be accepted by a finite-turn whose stack is reversal-bounded; can be accepted by a finite-turn ; can be accepted by a . Automata and Formal Languages (AFL 2011)

  14. Main theorem Main Theorem The following 5 statements are equivalent for every bounded language : is bounded semilinear; can be accepted by a finite-crossing ; can be accepted by a finite-turn whose stack is reversal-bounded; can be accepted by a finite-turn ; can be accepted by a . • and trivially hold since • , and • Let us prove 1 2 5 3 4 Automata and Formal Languages (AFL 2011)

  15. Proof direction for Automata and Formal Languages (AFL 2011)

  16. Letter-bounded to bounded • Lets.t. is semilinear. • Given a word , . • Let , and construct a for it. • for • reads ; • non-deterministically decomposes it as ; • runs on , and returns ’s decision as its decision for and . Automata and Formal Languages (AFL 2011)

  17. Proof direction for • How can a deterministic machine non-deterministic decompose an input? • Our answer is “this decomposition does not require non-determinism.” Automata and Formal Languages (AFL 2011)

  18. Deterministic decomposition • Let be a sufficiently-large constant. • This is efficiently computable from a given language s.t. is semilinear. • Let . • Based on this theorem, we can figure out that looking-ahead by some constant distance on an input tape settles the decomposition. Theorem [Fine and Wilf 1965] Let be primitive words. If and share their prefix of length , then . Automata and Formal Languages (AFL 2011)

  19. Main theorem Main Theorem The following 5 statements are equivalent for every bounded language : is bounded semilinear; can be accepted by a finite-crossing ; can be accepted by a finite-turn whose stack is reversal-bounded; can be accepted by a finite-turn ; can be accepted by a . Automata and Formal Languages (AFL 2011)

  20. Proof for • Proof idea • In a computation by a finite-crossing 2DPDA, contents of ’s stack consist of constant # of blocks of the form (pumped structure). • Each of these blocks correspond to a writing phase during which the stack is never popped. • # of “long” writing phases is bounded by a constant . • We build a , and let its counters to store , and let its finite state control to remember Lemma If a letter-bounded language is accepted by a finite-crossing , then it is accepted by a finite-crossing . stack Automata and Formal Languages (AFL 2011)

  21. Proof for The previous result is generalized for a broader class of finite-crossing . Lemma If a letter-bounded language is accepted by a finite-crossing , then it is accepted by a finite-crossing . Automata and Formal Languages (AFL 2011)

  22. Proof for Proof. Let be a finite-crossing for some s.t. . It is easy to construct a finite-crossing for . This language is letter-bounded, and hence, this machine can be converted into an equivalent finite-crossing . It is known that if a letter-bounded language is accepted by a finite-crossing , then the language is bounded semilinear. Thus, is semilinear. Theorem A bounded language that is accepted by a finite-crossing is effectively semilinear. Automata and Formal Languages (AFL 2011)

  23. Main theorem Main Theorem (Proved!!) The following 5 statements are equivalent for every bounded language : is bounded semilinear; can be accepted by a finite-crossing ; can be accepted by a finite-turn whose stack is reversal-bounded; can be accepted by a finite-turn ; can be accepted by a . Open Is every bounded language accepted by a finite-crossing bounded semilinear. Automata and Formal Languages (AFL 2011)

  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 Automata and Formal Languages (AFL 2011)

  25. References [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. Automata and Formal Languages (AFL 2011)

  26. References [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. Automata and Formal Languages (AFL 2011)

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