State Complexity: Recent Results and Future Directions

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State Complexity Recent Results and Future
Directions

• Sheng Yu
• Department of Computer Science
• University of Western Ontario
• London, 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, ngt1, 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, Bzrorowskis 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

• J.-C. Birget, Intersection and union of regular
  languages and state complexity, Information
  Processing Letters 43 (1992) 185-190.
• J.-C. Birget, Partial orders on words, minimal
  elements of regular languages, and state
  complexity, Theoretical Computer Scinece 119
  (1993) 267-291.
• 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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• 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.
• 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.
• M. Domaratzki, State complexity and proportional
  removals Journal of Automata, Languages and
  Combinatorics 7 (2002) 455-468.
• 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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• 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.
• 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.
• 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.
• G. Jiraskova, State complexity of some
  operations on regular languages, DCFS (2003)
  114-125.

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• G. Jiraskova, State complexity of some
  operations on binary regular languages,
  Theoretical Computer Science, to appear.
• 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.
• G.A. Kiraz, Compressed storage of sparse
  finite-state transducers, Proceedings of CIAA
  2001, Springer LNCS 2214, pp. 109-121.
• E. Leiss, Succinct representation of regular
  languages by Boolean automata, Theoretical
  Computer Science 13 (1981) 323-330.

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• 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.
• C. Nicaud, Average state complexity of
  operations on unary automata, MFCS99, LNCS 1672
  (1999) 231-240.
• G. Pighizzini and J. Shallit, Unary language
  operations, state complexity and Jacobsthals
  function, International Journal of Foundations
  of Computer Science Vol.13, No.1 (2002) 145-159.
• M. Rabin and D. Scott, Finite automata and their
  decision problems, IBM J. Res. Dev. 3 (1959)
  114-125.
• A. Salomaa, On the Reducibility of Events
  Represented in Automata, Annales Academiae
  Scientiarum Fennicae, Series A, I. Mathematica
  353, 1964.

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• A. Salomaa, Theorems on the Representation of
  Events in Moore-Automata, Turun Yliopiston
  Julkaisuja Annales Universitatis Turkuensis,
  Series A, 69, 1964.
• A. Salomaa, Theory of Automata, Pergamon Press
  (1969) Oxford.
• A. Salomaa, D. Wood and S. Yu, On the state
  complexity of reversals of regular languages,
  Theoretical Computer Science 320 (2004) 293-313.
• 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.
• 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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• S. Yu, State complexity of regular languages,
  Journal of Automata, Languages and Combinatorics,
  6 (2001) 2, 221-234.
• S. Yu and Q. Zhuang, On the state complexity of
  intersection of regular languages, ACM SIGACT
  News, Vol.22, No.3 (1991) 52-54.
• 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.