Abstract
Regular languages and finite automata are among the oldest topics in formal language theory. The formal study of regular languages and finite automata can be traced back to the early forties, when finite state machines were used to model neuron nets by McCulloch and Pitts [83]. Since then, regular languages have been extensively studied. Results of early investigations are, for example, Kleeneās theorem establishing the equivalence of regular expressions and finite automata [69], the introduction of automata with output by Mealy [86] and Moore [88], the introduction of nondeterministic finite automata by Rabin and Scott [99], and the characterization of regular languages by congruences of finite index by Myhill [90] and Nerode [91].
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References
A. V. Aho and J. D. Ullman, The Theory of Parsing, Translation,and Compiling, Vol. 1, Prentice-Hall, Englewood Cliffs, N.J., (1972).
A. V. Aho, R. Sethi, and J. D. Ullman, Compilers - Principles,Techniques, and Tools, Addison-Wesley, Reading, (1986).
J. C. M. Baeten and W. P. Weijland, Process Algebra, Cambridge University Press, Cambridge, (1990).
J. L. BalcĆ”zar, J. Diaz, and J. GabarrĆ³, Structured Complexity I, II, EATCS Monagraphs on Theoretical Computer Science, vol. 11 and 22, Springer-Verlag, Berlin 1988 and 1990.
Y. Bar-Hillel, M. Perles, and E. Shamir, āOn formal properties of simple phrase structure grammarsā, Z. Phonetik. Sprachwiss. Kommunikationsforsch. 14 (1961) 143ā172.
J.-C. Birget, āState-Complexity of Finite-State Devices, State Compressibility and Incompressibilityā, Mathematical Systems Theory 26 (1993) 237ā269.
G. Berry and R. Sethi, āFrom Regular Expressions to Deterministic Automataā, Theoretical Computer Science 48 (1986) 117ā126.
J. Berstel, Transductions and Context-Free Languages, Teubner, Stuttgart, (1979).
J. Berstel and M. Morcrette, āCompact representation of patterns by finite automataā, Pixim 89: LāImage NumĆ©rique Ć Paris, AndrĆ© Gagalowicz, ed., Hermes, Paris, (1989), pp.387ā395.
J. Berstel and C. Reutenauer, Rational Series and Their Languages, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin (1988).
W. Brauer, Automatentheorie, Teubner, Stuttgart, (1984).
W. Brauer, āOn Minimizing Finite Automataā, EATCS Bulletin 35 (1988) 113ā116.
A. BrĆ¼ggemann-Klein, āRegular expressions into finite automataā, Theoretical Computer Science 120 (1993) 197ā213.
A. BrĆ¼ggemann-Klein and D. Wood, āDeterministic Regular Languagesā, Proceedings of STACSā92, Lecture Notes in Computer Science 577, A. Finkel and M. Jantzen (eds.), Springer-Verlag, Berlin (1992) 173ā184.
J. A. Brzozowski, āCanonical regular expressions and minimal state graphs for definite eventsā, Mathematical Theory of Automata, vol. 12 of MRI Symposia Series, Polytechnic Press, NY, (1962), 529ā561.
J. A. Brzozowski, āDerivatives of Regular Expressionsā, Journal of the ACM 11:4 (1964) 481ā494.
J. A. Brzozowski, āDevelopments in the theory of regular languagesā, Information Processing 80, S. H. Lavington edited, Proceedings of IFIP Congress 80, North-Holland, Amsterdam (1980) 29ā40.
J. A. Brzozowski, āOpen problems about regular languagesā, Formal Language Theory - Perspectives and open problems, R. V. Book (ed.), Academic Press, New York, (1980), pp.23ā47.
J. A. Brzozowski and E. Leiss, āOn Equations for Regular Languages, Finite Automata, and Sequential Networksā, Theoretical Computer Science 10 (1980) 19ā35.
J. A. Brzozowski and I. Simon, āCharacterization of locally testable eventsā, Discrete Mathematics 4 (1973) 243ā271.
J. A. Brzozowski and M. Yoeli, Digital Networks, Prentice-Hall, Englewood Cliffs, N. J., (1976).
A. K. Chandra and L. J. Stockmeyer, āAlternationā, FOCS 17 (1976) 98ā108.
A. K. Chandra, D. C. Kozen, L. J. Stockmeyer, āAlternationā, Journal of the ACM 28 (1981) 114ā133.
J. H. Chang, O. H. Ibarra and B. Ravikumar, āSome observations concerning alternating Turing machines using small spaceā, Inform. Process. Lett. 25 (1987) 1ā9.
C.-H. Chang and R. Paige, āFrom Regular Expressions to DFAās Using Compressed NFAāsā, Proceedings of the Third Symposium on Combinatorial Pattern Matching (1992) 90ā110.
S. Cho and D. Huynh, āThe parallel complexity of finite state automata problemsā, Technical Report UTDCS-22ā88, University of Texas at Dallas, (1988).
D. I. A. Cohen, Introduction to Computer Theory, Wiley, New York, (1991).
K. Culik II and S. Dube, āRational and Affine Expressions for Image Descriptionā, Discrete Applied Mathematics 41 (1993) 85ā120.
K. Culik II and S. Dube, āAffine Automata and Related Techniques for Generation of Complex Imagesā, Theoretical Computer Science 116 (1993) 373ā398.
K. Culik II, F. E. Fich and A. Salomaa, āA Homomorphic Characterization of Regular Languagesā, Discrete Applied Mathematics 4 (1982)149ā152.
K. Culik II and T. Harju, āSplicing semigroups of dominoes and DNAā, Discrete Applied Mathematics 31 (1991) 261ā277.
K. Culik II and J. KarhumƤki, āThe equivalence problem for single-valued two-way transducers (on NPDTOL languages) is decidableā, SIAM Journal on Computing, vol. 16, no. 2 (1987) 221ā230.
K. Culik II and J. Kari, āImage Compression Using Weighted Finite Automataā, Computer and Graphics, vol. 17, 3, (1993) 305ā313.
J. Dassow, G. PƤun, A. Salomaa, āOn Thinness and Slenderness of L Languagesā, EATCS Bulletin 49 (1993) 152ā158.
F. Dejean and M. P. SchĆ¼tzenberger, āOn a question of Egganā, Information and Control 9 (1966) 23ā25.
A. de Luca and S. Varricchio, āOn noncounting regular classesā, Theoretical Computer Science 100 (1992) 67ā104.
V. Diekert and G. Rozenberg (ed.), The Book of Traces, World Scientific, Singapore, (1995).
D. Drusinsky and D. Harel, āOn the power of bounded concurrency I: Finite automataā, Journal of the ACM 41 (1994) 517ā539.
L. C. Eggan, āTransition graphs and the star height of regular eventsā, Michigan Math. J. 10 (1963) 385ā397.
A. Ehrenfeucht, R. Parikh, and G. Rozenberg, āPumping Lemmas for Regular Setsā, SIAM Journal on Computing vol. 10, no. 3 (1981) 536ā541.
S. Eilenberg, Automata, Languages,and Machines, Vol. A, Academic Press, New York, (1974).
S. Eilenberg, Automata, Languages,and Machines, Vol. B, Academic Press, New York, (1974)
C. C. Elgot and J. D. Rutledge, āOperations on finite automataā, Proc. AIEE Second Ann. Symp. on Switching Theory and Logical Design, Detroit, (1961).
A. Fellah, Alternating Finite Automata and Related Problems, PhD Dissertation, Dept. of Math. and Computer Sci., Kent State University, (1991).
A. Fellah, H. JĆ¼rgensen, S. Yu, āConstructions for alternating finite automataā, Intern. J. Computer Math. 35 (1990) 117ā132.
M. R. Carey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, (1979.)
S. Ginsburg, Algebraic and automata-theoretic properties of formal languages, North-Holland, Amsterdam, (1975).
V. M. Glushkov, āThe abstract theory of automataā, Russian Mathematics Surveys 16 (1961) 1ā53.
D. Gries, āDescribing an Algorithm by Hoperoftā, Acta Informatica 2 (1973) 97ā109.
L. Guo, K. Salomaa, and S. Yu, āSynchronization Expressions and Languagesā, Proceedings of the Sixth IEEE Symposium on Parallel and Distributed Processing (1994) 257ā264.
M. A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, Reading, (1978).
K. Hashiguchi, āAlgorithms for Determining Relative Star Height and Star Heightā, Information and Computation 78 (1988) 124ā169.
T. Head, āFormal language theory and DNA: An analysis of the generative capacity of specific recombinant behaviorsā, Bull. Math. Biol. 49 (1987) 737ā759.
F. C. Hennie, Finite-State Models for Logical Machines, Wiley, New York, (1968).
T. Hirst and D. Harel, āOn the power of bounded concurrency II: Pushdown automataā, Journal of the ACM 41 (1994), 540ā554.
J. E. Hoperoft, āAn n log n algorithm for minimizing states in a finite automatonā, in Theory of Machines and Computations, Z. Kohavi (ed.), Academic Press, New York, (1971).
J. E. Hoperoft and J. D. Ullman, Introduction to Automata Theory,Languages, and Computation, Addison-Wesley, Reading, (1979), 189ā196.
J. M. Howie, Automata and Languages, Oxford University Press, Oxford, (1991).
H. B. Hunt, D. J. Rosenkrantz, and T. G. Szymanski, āOn the Equivalence, Containment, and Covering Problems for the Regular and Context-Free Languagesā, Journal of Computer and System Sciences 12 (1976) 222ā268.
O. Ibarra, āThe unsolvability of the equivalence problem for epsilon-free NGSMās with unary input (output) alphabet and applicationsā, SIAM Journal on Computing 4 (1978) 524ā532.
K. Inoue, I. Takanami, and H. Tanaguchi, āTwo-Dimensional alternating Turing machinesā, Proc. 14th Ann. ACM Symp. On Theory of Computing (1982) 37ā46.
K. Inoue, I. Takanami, and H. Tanaguchi, āA note on alternating on-line Turing machinesā, Information Processing Letters 15:4 (1982) 164ā168.
J. Jaffe, āA necessary and sufficient pumping lemma for regular languagesā, SIGACT News (1978) 48ā49.
T. Jiang and B. Ravikumar, āA note on the space complexity of some decision problems for finite automataā, Information Processing Letters 40 (1991) 25ā31.
T. Jiang and B. Ravikumar, āMinimal NFA Problems are Hardā, SIAM Journal on Computing 22 (1993), 1117ā1141. Proceedings of 18th ICALP, Lecture Notes in Computer Science 510, Springer-Verlag, Berlin (1991) 629ā640.
N. Jones, āSpace-bounded reducibility among combinatorial problemsā, Journal of Computer and System Sciences 11 (1975) 68ā85.
T. Kameda and P. Weiner, āOn the state minimization of nondeterministic finite automataā, IEEE Trans. on Computers C-19 (1970) 617ā627.
D. Kelley, Automata and Formal Languages - An Introduction, Prentice-Hall, Englewood Cliffs, N. J., (1995).
S. C. Kleene, āRepresentation of events in nerve nets and finite automataā, Automata Studies, Princeton Univ. Press, Princeton, N. J., (1996), pp.2ā42.
D. E. Knuth, J. H. Morris, and V. R. Pratt, āFast pattern matching in stringsā, SIAM Journal on Computing, vol.6, no.2 (1977) 323ā350.
D. Kozen, āOn parallelism in Turing machinesā, Proceedings of 17th FOCS (1976) 89ā97.
R. E. Ladner, R. J. Lipton and L. J. Stockmeyer, āAlternating pushdown automataā, Proc. 19th IEEE Symp. on Foundations of Computer Science, Ann Arbor, MI, (1978) 92ā106.
E. Leiss, āSuccinct representation of regular languages by boolean automataā, Theoretical Computer Science 13 (1981) 323ā330.
E. Leiss, āOn generalized language equationsā, Theoretical Computer Science 14 (1981) 63ā77.
E. Leiss, āSuccinct representation of regular languages by boolean automata IIā, Theoretical Computer Science 38 (1985) 133ā136.
E. Leiss, āLanguage equations over a one-letter alphabet with union, concatenation and star: a complete solutionā, Theoretical Computer Science 131 (1994) 311ā330.
E. Leiss, āUnrestricted complementation in language equations over a one-letter alphabetā, Theoretical Computer Science 132 (1994) 71ā84.
H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, Englewood Cliffs, N. J., (1981).
P. A. Lindsay, āAlternation and w-type Turing acceptorsā, Theoretical Computer Science 43 (1986) 107ā115.
P. Linz, An Introduction to Formal Languages and Automata, D. C. Heath and Company, Lexington, (1990).
O. B. Lupanow, āĆ¼ber den Vergleich zweier Typen endlicher Quellenā, Prob-lerne der Kybernetik 6 (1966) 328ā335, and Problemy Kibernetiki 6 (1962) (Russian original).
A. Mateescu, āScattered deletion and commutativityā, Theoretical Computer Science 125 (1994) 361ā371.
W. S. McCulloch and W. Pitts, āA logical calculus of the ideas immanent in nervous activityā, Bull. Math. Biophysics 5 (1943) 115ā133.
R. McNaughton, Counter-Free Automata, MIT Press, Cambridge, (1971).
R. McNaughton and H. Yamada, āRegular Expressions and State Graphs for Automataā, Trans. IRS EC-9 (1960) 39ā47. Also in Sequential Machines - Selected Papers, E. F. Moore ed., Addison-Wesley, Reading, (1964), 157ā174.
G. H. Mealy, āA method for synthesizing sequential circuitsā, Bell System Technical J. 34: 5 (1955), 1045ā1079.
A. R. Meyer and M. J. Fischer. āEconomy of description by automata, grammars, and formal systemsā, FOCS 12 (1971) 188ā191.
E. F. Moore, āGedanken experiments on sequential machinesā, Automata Studies, Princeton Univ. Press, Princeton, N.J., (1966), pp.129ā153.
F. R. Moore, āOn the Bounds for State-Set Size in the Proofs of Equivalence Between Deterministic, Nondeterministic, and Two-Way Finite Automataā, IEEE Trans. Computers 20 (1971), 1211ā1214.
J. Myhill, āFinite automata and the representation of eventsā, WADD TR-57624, Wright Patterson AFB, Ohio, (1957), 112ā137.
A. Nerode, āLinear automata transformationā, Proceedings of AMS 9 (1958) 541ā544.
O. Nierstrasz, āRegular Types for Active Objectsā, OOPSLAā93, 1ā15.
M. Nivat, āTransductions des langages de Chomskyā, Ann. Inst. Fourier, Grenoble 18 (1968) 339ā456.
W. J. Paul, E. J. Prauss and R. Reischuck, āOn Alternationā, Acta Inform. 14 (1980) 243ā255.
G. PĆ”un and A. Salomaa, āDecision problems concerning the thinness of DOL languagesā, EATCS Bulletin 46 (1992) 171ā181.
G. PĆ”un and A. Salomaa, āClosure properties of slender languagesā, Theoretical Computer Science 120 (1993) 293ā301.
G. Patin and A. Salomaa, āThin and slender languagesā, Discrete Applied Mathematics 61 (1995) 257ā270.
D. Perrin, (Chapter 1) Finite Automata, Handbook of Theoretical Computer Science, Vol. B, J. van Leeuwen (ed.), MIT Press, (1990).
M. O. Rabin and D. Scott, āFinite automata and their decision problemsā, IBM J. Res. 3: 2 (1959) 115ā125.
G. N. Raney, āSequential functionsā, Journal of the ACM 5 (1958) 177.
B. Ravikumar, āSome applications of a technique of Sakoda and Sipserā, SIGACT News, 21:4 (1990) 73ā77.
B. Ravikumar and O. H. Ibarra, āRelating the type of ambiguity of finite automata to the succinctness of their representationā, SIAM Journal on Computing vol. 18, no. 6 (1989), 1263ā1282.
W. L. Ruzzo, āTree-size bounded alternationā, Journal of Computer and System Sciences 21 (1980) 218ā235.
A. Salomaa, On the Reducibility of Events Represented in Automata, Annales Academiae Scientiarum Fennicae, Series A, I. Mathematica 353, (1964).
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, Oxford, (1969).
A. Salomaa, Jewels of Formal Language Theory, Computer Science Press, Rockville, Maryland, (1981).
A. Salomaa, Computation and Automata, Cambridge University Press, Cambridge, (1985).
A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag, New York, (1978).
K. Salomaa and S. Yu, āLoop-Free Alternating Finite Automataā, Technical Report 482, Department of Computer Science, Univ. of Western Ontatio, (1996).
K. Salomaa, S. Yu, Q. Zhuang, āThe state complexities of some basic operations on regular languagesā, Theoretical Computer Science 125 (1994) 315ā328.
M. P. SchĆ¼tzenberger, āFinite Counting Automataā, Information and Control 5 (1962) 91ā107.
M. P. SchĆ¼tzenberger, āOn finite monoids having only trivial subgroupsā, Information and Control 8 (1965) 190ā194.
M.P. SchĆ¼tzenberger, āSur les relations rationellesā, in Proc. 2nd GI Conference, Automata Theory and Formal languages, H. Braklage (ed.), Lecture Notes in Computer Science 33, Springer-Verlag, Berlin (1975) 209ā213.
J. Shallit, āNumeration systems, linear recurrences, and regular setsā, Information and Computation 113 (1994) 331ā347.
J. Shallit and J. Stolfi, āTwo methods for generating fractalsā, Computers & Graphics 13 (1989) 185ā191.
P. W. Shor, āA Counterexample to the triangle conjectureā, J. Combinatorial Theory, Series A (1985) 110ā112.
J. L. A. van de Snepscheut, What Computing Is All About, Springer-Verlag, New York, (1993).
L. Stockmeyer and A. Meyer, āWord problems requiring exponential time (preliminary report)ā, Proceedings of the 5th ACM Symposium on Theory of Computing, (1973) 1ā9.
A. Szilard, S. Yu, K. Zhang, and J. Shallit, āCharacterizing Regular Languages with Polynomial Densitiesā, Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 629 Springer-Verlag, Berlin (1992) 494ā503.
K. Thompson, āRegular Expression Search Algorithmā, Communications of the ACM 11:6 (1968) 410ā422.
B. W. Watson, Taxonomies and Toolkits of Regular Language Algorithms, PhD Dissertation, Department of Mathematics and Computing Science, Eindhoven University of Technology, (1995).
D. Wood, Theory of Computation, Wiley, New York, (1987).
S. Yu and Q. Zhuang, āOn the State Complexity of Intersection of Regular Languagesā, ACM SIGACT News, vol. 22, no. 3, (1991) 52ā54.
Y. Zalcstein, āLocally testable languagesā, Journal of Computer and System Sciences 6 (1972) 151ā167.
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Yu, S. (1997). Regular Languages. In: Rozenberg, G., Salomaa, A. (eds) Handbook of Formal Languages. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59136-5_2
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