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State Complexity: Recent Results and Future Directions

Sheng YuDepartment of Computer Science

University of Western OntarioLondon, Ontario, Canada

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What is state complexity?

State complexity is a descriptional complexity.

The state complexity of a regular language L is the number of states of the minimal DFA that accepts L.

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What is state complexity? (continue)

The state complexity of a class of regular languages is the worst among the state complexities of all the languages in the class.

The state complexity of a collection of classes of regular languages is a function of the state complexities of the classes.

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Example:

Let L be the language accepted by the following 10-state DFA.

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Example: (continue)

It can be shown that the state complexity of LR is 210 (1024).

It can be proved that the state complexity of regular languages that are the reversals of 10-state DFA languages is 210.

It can also be shown that the state complexity of the class of languages that are the reversals of n-state DFA languages, n>1, is 2n.

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The study of state complexity related problems has a long history

From 1950s to early 1990s

From early 1990s to now

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From 1950s to early 1990s

In 1959, Rabin and Scott proved that the number of states in a DFA that is transformed from an n-state NFA is limited to 2n. Later in 1971, F. Moore proved that the bound is tight.

Arto Salomaa studied several state complexity issues in the 1960s.

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From 1950s to early 1990s (continue)

E. Leiss studied succinct representation of regular language in early 1980s.

J.C. Birget studied the state complexity of multiple intersection and union of regular languages in early 1990s.

Some other scattered results concerning state complexity have been obtained during this period of time.

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From early 1990s to now

In 1994, we systematically studied the state complexity problems of basic operations on regular languages over a general alphabet as well as over a one-letter alphabet.

We later studied the state complexity of basic operations on finite languages.

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State complexity of basic operations on regular languages

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State complexity of basic operations on finite languages

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From early 1990s to Now (continue)

Pighizzini and Shallit solved the state complexity problems of unary language operations.

Nicaud investigated the average state complexity of operations on unary languages.

Holzer and Kutrib studied the state complexity of nondeterministic finite automata.

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From early 1990s to now (continue)

Domaratzki studied the state complexity on propotional removals of regular languages.

Campeanu, Salomaa and Yu obtained state complexity of shuffle of regular languages.

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From early 1990s to now (continue)

Jiraskova (one paper with Szabari) had several results on state complexity issues including the catenation and complementation operations of finite automata.

Many other results have been obtained in this period of time.

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Why so many state complexity problems are not solved earlier?

Motivation of the study

Help of computer programs

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Motivation of the study

In the 60s and 70s, the number of states of finite automata used in applications were usually small. There was no strong motivation from the practice to study the state complexity issues in general then.

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Motivation of the study (continue)

In recent years, there have been many new applications of finite automata, e.g.,in natural language and speech processing, software engineering, and image generation and encoding. A large number of states are needed in a finite automaton in many new applications.

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Motivation of the study (continue)

For example, in natural language and speech processing, the Bell Labs multilingual TTS system need 26.6 mbytes for German, 30.0 for French, and 39.0 for Mandarin.

The study of state complexity problems is strongly motivated by practical applications

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Help of computer programs

In the last ten to twenty years, a number of software systems have been developed for the manipulation of finite automata and formal language objects, e.g., Grail+, Automate, and FireLite.

Many state complexity results were obtained with the help of those computer software systems.

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What are the next possible topics in state complexity research

State complexity of combinations of multiple operations

Conditions for cases that are not the worst case

Average state complexity

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State complexity of combinations of multiple operations

The state complexity of a combination of operations usually is not equal to the combination of the state complexities of individual operations.

Many interesting and useful combinations of operations on regular languages can be found in applications

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Conditions for cases that are not the worst case

The state complexity is a worst-case complexity. However, in many applications, the worst case situation may not happen. It is useful and desirable to know under what conditions the worst case will not happen.

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Conditions for cases that are not the worst case (continue)

For example, Bzrorowski’s DFA minimization algorithm uses two reversals of a DFA. The worst case time complexity of the algorithm is exponential. However, the algorithm is quite fast, observed by many people. Then we have the question: under what conditions the state complexity of a reversal of a DFA will not have an exponential explosion?

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Average state complexity

Average state complexity has not been studied except in the paper by Nicaud.

Average state complexity is clearly a useful topic. However, it also appears to be very difficult.

Experimental results for average state complexity may also be useful.

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Conclusion

State complexity questions are both practically motivated and theoretically interesting.

There have been many new results in recent years.

Computer software has been a factor in solving many problems.

There are still many open problems that need to be solved in this area.

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References

1. J.-C. Birget, “Intersection and union of regular languages and state complexity”, Information Processing Letters 43 (1992) 185-190.

2. J.-C. Birget, “Partial orders on words, minimal elements of regular languages, and state complexity”, Theoretical Computer Scinece 119 (1993) 267-291.

3. C. Campeanu, K. Culik, K. Salomaa, S. Yu, “State complexity of basic operations on finite languages”, Proceedings of the Fouth International Workshop on Implementing Automata VIII 1-11, 1999, LNCS 2214, pp. 60-70.

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4. C. Campeanu, K. Salomaa, S. Yu, “Tight lower bound for the state complexity of shuffle of regular languages”, Journal of Automata, Languages and Combinatorics, 7 (2002) 3, 303-310.

5. C. Campeanu, K. Salomaa, S. Yu, Chapter 5: “State complexity of regular languages: finite versus infinite”, in Finite vs Infinite – Contributions to an Eternal Dilemma, edited by C. Calude and G. Paun, Springer 2000, pp. 53-73.

6. M. Domaratzki, “State complexity and proportional removals” Journal of Automata, Languages and Combinatorics 7 (2002) 455-468.

7. M. Holzer and M. Kutrib, “State complexity of basic operations on nondeterministic finite automata”, CIAA 2002, Springer LNCS 2608, pp.148-157.

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8. M.Holzer and M. Kutrib, “Unary language operations and their nondeterministic state complexity”, Developments in Language Theory (DLT 2002), Springer LNCS 2450, pp. 162-172.

9. M. Holzer, K. Salomaa, S. Yu, “On the state complexity of K-entry deterministic finite automata”, Journal of Automata, Languages and Combinatorics 6 (2001) 4, 453-466.

10. K. Iwama, Y. Kambayashi and K. Takaki, “Tight bounds on the number of states of DFAs that equivalent to n-state NFAs”, Theoretical Computer Science 237 (2000) 485-494.

11. G. Jiraskova, “State complexity of some operations on regular languages”, DCFS (2003) 114-125.

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12. G. Jiraskova, “State complexity of some operations on binary regular languages”, Theoretical Computer Science, to appear.

13. J.Jirasek, G. Jiraskova and A. Szabari, “State complexity of the concatenation and complementation of regular languages”, CIAA 2004, Springer LNCS 3317, pp. 178-189.

14. G.A. Kiraz, “Compressed storage of sparse finite-state transducers”, Proceedings of CIAA 2001, Springer LNCS 2214, pp. 109-121.

15. E. Leiss, “Succinct representation of regular languages by Boolean automata”, Theoretical Computer Science 13 (1981) 323-330.

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16. F. Moore, “On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata”, IEEE Trans. Comput. C-20 (1971) 1211-1214.

17. C. Nicaud, “Average state complexity of operations on unary automata”, MFCS”99, LNCS 1672 (1999) 231-240.

18. G. Pighizzini and J. Shallit, “Unary language operations, state complexity and Jacobsthal’s function”, International Journal of Foundations of Computer Science Vol.13, No.1 (2002) 145-159.

19. M. Rabin and D. Scott, “Finite automata and their decision problems”, IBM J. Res. Dev. 3 (1959) 114-125.

20. A. Salomaa, On the Reducibility of Events Represented in Automata, Annales Academiae Scientiarum Fennicae, Series A, I. Mathematica 353, 1964.

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21. A. Salomaa, Theorems on the Representation of Events in Moore-Automata, Turun Yliopiston Julkaisuja Annales Universitatis Turkuensis, Series A, 69, 1964.

22. A. Salomaa, Theory of Automata, Pergamon Press (1969) Oxford.

23. A. Salomaa, D. Wood and S. Yu, “On the state complexity of reversals of regular languages”, Theoretical Computer Science 320 (2004) 293-313.

24. K. Salomaa and S. Yu, “NFA to DFA transformation for finite languages over arbitrary alphabets”, Journal of Automata, Languages and Combinatorics, 2 (1997) 3, 177-186.

25. S. Yu, “Chapter 2: Regular Languages”, in Handbook of Formal Languages, edited by G. Rozenber and A. Salomaa, Springer 1998, pp. 41-110.

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26. S. Yu, “State complexity of regular languages”, Journal of Automata, Languages and Combinatorics, 6 (2001) 2, 221-234.

27. S. Yu and Q. Zhuang, “On the state complexity of intersection of regular languages”, ACM SIGACT News, Vol.22, No.3 (1991) 52-54.

28. S. Yu, Q. Zuang and K. Salomaa, “On the state complexity of some basic operations on regular languages”, Theoretical Computer Science 125 (1994) 315-328.