Abstract
This paper summarizes some recent results concerned with the extension of formal languages to their corresponding stochastic versions. Weighted grammars and languages are first defined, and stochastic grammars and languages are defined as a special case of weighted grammars and languages. Fuzzy grammars and languages, which have some properties similar to weighted grammars and languages, are also discussed. Stochastic automata are defined from the language recognition viewpoint. Languages accepted by stochastic finite-state and pushdown automata, with and without a cutpoint, are studied. Weighted and stochastic programmed and indexed grammars, and stochastic nested stack automata are defined. Finally, some decidability problems of stochastic (weighted, fuzzy) languages are discussed, and problems for further research are suggested.
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References
A. V. Aho, “Indexed grammars—An extension of context-free grammars,”J. ACM 15:647–671 (1968).
A. V. Aho, “Nested stack automata,”J. ACM 16:383–406 (1969).
A. V. Aho and J. D. Ullman, “The theory of languages,”Mathematical Systems Theory 2:97–125 (1967).
R. B. Banerji, “Phrase structure languages, finite machines, and channel capacity,”Information and Control 6:153–162 (1963).
T. L. Booth, “Probability representation of formal languages,” inIEEE Tenth Annual Symposium on Switching and Automata Theory (1969).
G. D. Bruce and K. S. Fu, “A model for finite-state probabilistic systems,” inProc. of the First Allerton Conference on Circuit and System Theory, University of Illinois (1962).
N. Chomsky and C. A. Miller, “Finite state language,”Information and Control 1:91–112 (1958).
N. Chomsky, “On certain formal properties of grammars,”Information and Control 2:137–167 (1959).
N. Chomsky, “Context-free grammars and pushdown storage,” Quarterly Progress Report No. 65, Research Laboratory of Electronics, Massachusetts Institute of Technology, pp. 187–194 (1962).
C. A. Ellis, “Probabilistic languages and automata,” Report No. 355, Department of Computer Science, University of Illinois (1969).
J. Evey, “The theory and application of pushdown machines,” Doctoral Thesis, Harvard University (1963).
K. S. Fu, “On syntactic pattern recognition and stochastic languages” inProc. International Conference on Frontiers of Pattern Recognition, Honolulu, Hawaii, 1971.
K. S. Fu and T. Li, “On stochastic automata and languages,”Information Sciences 1:403–419 (1969).
K. S. Fu and P. H. Swain, “On syntactic pattern recognition,” inProc. of the Third International Symposium on Computer and Information Science, Bal Harbour, Florida; also published inSoftware Engineering, Vol. II, ed. by J. T. Tou (Academic Press, New York, 1971).
S. Ginsburg and S. A. Greibach, “Deterministic context-free languages,”Information and Control 9:620–648 (1966).
S. A. Greibach, “A new normal form theorem for context-free phrase structure grammars,”J. ACM 12:42–52 (1965).
U. Grenander, “Syntax-controlled probabilities,” Technical Report, Division of Applied Mathematics, Brown University (1967).
T. E. Harris,The Theory of Branching Processes (Springer-Verlag, Berlin, 1963).
J. E. Hopcroft and J. D. Ullman,Formal Languages and Their Relation to Automata (Addison-Wesley, Reading, Mass., 1969).
T. Huang and K. S. Fu, “On stochastic context-free languages,”Information Sciences 3 (no. 3) (1971).
T. Huang and K. S. Fu, “Stochastic languages and their applications to pattern recognition,” Technical Report, Purdue University (1971).
S. E. Hutchins, “Stochastic sources for context-free languages,” Ph.D. Thesis, Department of Applied Physics and Information Science, University of California, San Diego (1970).
M. M. Kherts, “Entropy of languages generated by automata or context-free grammars with a single-valued deduction,”Naucho-Tekhn. Inf. Ser. 2–1 (1968).
E. T. Lee and L. A. Zadeh, “Note on fuzzy languages,”Information Sciences 1:421–434 (1969).
M. Magidor and G. Moran, “Probabilistic tree automata and constext-free languages,”Israel. J. Math. 8:340–348 (1970).
M. Mizumoto, “Fuzzy automata and fuzzy grammars,” Ph.D. Thesis, Faculty of Engineering Sciences, Osaka University (1971).
M. Mizumoto, J. Toyoda, and K. Tanaka, “Some considerations on fuzzy automata,”J. Computer and System Sci. 3:409–422 (1969).
M. Nasu and N. Honda, “Fuzzy events realized by probabilistic automata,”Information and Control 12:284–303 (1966).
M. Nasu and N. Honda, “A context-free language which is not acceptable by a probabilistic automaton,”Information and Control 18:233–236 (1971).
A. Paz, “Some aspects of probabilistic automata,”Information and Control 9:26–60 (1966).
M. O. Rabin, “Probabilistic automata,”Information and Control 6:230–245 (1963).
M. O. Rabin and D. Scott, “Finite automata and their decision problems,”IBM J. Res. Develop. 3:114–125 (1959).
D. J. Rosenkrantz, “Programmed grammars—A new device for generating formal languages,”J. ACM 10:107–131 (1969).
A. Salomaa, “Onm-adic probabilistic automata,”Information and Control 10:215–219 (1967).
A. Salomaa, “On languages accepted by probabilistic and time-variant automata,” inProc. of the Second Annual Princeton Conference on Information Sciences and Systems, 1968, pp. 184–188.
A. Salomaa, “Probabilistic and weighted grammars,”Information and Control 15:529–544 (1969).
E. S. Santos, “Maximin automata,”Information and Control 13:363–377 (1968).
C. R. Souza, “Probabilities in context-free programmed grammars,” TR-A70-2, The Aloha System, University of Hawaii, Honolulu, Hawaii, 1970.
M. P. Schutzenberger, “On context-free languages and pushdown automata,”Information and Control 3:246–264 (1963).
P. H. Swain and K. S. Fu, “Stochastic programmed grammars for syntactic pattern recognition,”Pattern Recognition (Special Issue on Syntactic Pattern Recognition 3(3) (1971).
J. W. Thatcher and J. B. Wright, “Generalized automata theory,”Mathematical Systems Theory 2:57–82 (1968).
P. Turakainen, “On stochastic languages,”Information and Control 12:304–313 (1968).
P. Turakainen, “Some closure properties of the family of stochastic languages,”Information and Control 18:253–256 (1971).
W. G. Wee and K. S. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,”IEEE Trans. Systems Science and Cybernetics SSC-5: 215–223 (1969).
L. A. Zadeh, “Fuzzy set,”Information and Control 8:338–353 (1965).
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This work was supported by the National Science Foundation Grant GK-18225.
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Fu, K.S., Huang, T. Stochastic grammars and languages. International Journal of Computer and Information Sciences 1, 135–170 (1972). https://doi.org/10.1007/BF00995736
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DOI: https://doi.org/10.1007/BF00995736