# Relating the degree of ambiguity of finite automata to the succinctness of their representation

Session 1 Automata And Formal Languages

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## Abstract

We consider the problem of how the size of a nondeterministic finite automaton (nfa) representing a regular language depends on the degree of ambiguity of the nfa. We obtain results for the unary and bounded inputs, and partial results for the unrestricted inputs. One of the main results of this paper shows that for unrestricted inputs, deterministic, unambiguous and nondeterministic machines form a hierarchy with respect to the number of states, solving an open problem of Stearns and Hunt. We also propose a new approach to the study of the succinctness of representation through regularity preserving closure properties and obtain some results in this direction.

## Keywords

Regular Expression Start State Regular Language Finite Automaton Closure Property
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## References

- [BERM79]Berman, P., A note on sweeping automata,
*ICALP*1979, pp. 91–97.Google Scholar - [CHAN83]Chan, T. and O. Ibarra, On the finite-valuedness problem for sequential machines,
*Theoretical Computer Science*, 1983, pp. 95–101.Google Scholar - [DENE85]Denes, J., K.H. Kim and F.W. Rouch, Automata on one symbol,
*Studies in Pure Mathematics*1985, pp. 127–134.Google Scholar - [EHRE76]Ehrenfeucht, A. and P. Zeiger, Complexity measures for regular expressions,
*Journal of Computer and System Sciences*, 12, 1976, pp. 134–146.Google Scholar - [EILE74]Eilenberg, S.,
*Automata, Languages and machines*, Vol. A, Academic Press, NY 1974.Google Scholar - [HOPC79]Hopcroft J. and J. Ullman,
*Introduction to Automata Theory, Formal Languages and Computation*, Addison-Wesley, 1979.Google Scholar - [IBAR86]Ibarra, O. and B. Ravikumar, On sparseness, ambiguity and other decision problems for acceptors and transducers,
*Third Annual Symposium on Theoretical Aspects of Computer Science*, Orsay, France, 1986, pp. 171–179.Google Scholar - [KINT80]Kintala C. and D. Wotschke, Amounts of nondeterminism in finite automata,
*Acta Informatica*, 1980, pp. 199–204.Google Scholar - [KINT86]Kintala C. and D. Wotschke, Concurrent conciseness of degree, probabilistic, nondeterministic and deterministic finite automata,
*Fourth Symposium on Theoretical Aspects of Computer Science*, 1986, pp. 291–305.Google Scholar - [KRAN85]Kranakis, E.,
*Primality and Cryptography*, Yale University Technical Report, 1985.Google Scholar - [KUIC70]Kuich, W., On the entropy of context-free languages,
*Information and Control*, 1970, pp. 173–200.Google Scholar - [MAND73]Mandl, R., Precise bounds associated with the subset construction on various classes of nondeterministic finite automata,
*Seventh Princeton Conference on Information and System Science*, 1973, pp. 263–267.Google Scholar - [MEYE71]Meyer, A. and M. Fischer, Economy of description by automata, grammars, and formal Systems,
*Proc. of Twelfth IEEE Symposium on Switching and Automata Theory*, 1971, pp. 188–191.Google Scholar - [MOOR71]Moore, F., On the bounds for state-set size in the proofs of equivalence between nondeterministic and two-way automata,
*IEEE Transactions on Computers*, 20, 1971, pp. 1211–1214.Google Scholar - [PAUL79]Paul, W., Kolmogorov complexity and lower bounds,
*2nd International Conference on Fundamentals of Computation Theory*, 1979.Google Scholar - [RABI59]Rabin, M. and D. Scott, Finite automata and their decision problems,
*IBM Journal of Research and Development*, 3, 1959, pp. 114–125.Google Scholar - [SAKO78]Sakoda, W. and M. Sipser, Nondeterminism and the size of two-way finite automata,
*Proceedings of Tenth Annual ACM Symposium on Theory of Computing*, 1978, pp. 275–286.Google Scholar - [SCHM78]Schmidt, E., Succinctness of descriptions of context-free, regular and finite languages, Ph. D. Thesis,
*Cornell University*, 1978.Google Scholar - [SCHM87]Schmidt, E., (private communication).Google Scholar
- [SIPS79]Sipser, M., Lower bounds on the size of sweeping automata,
*Proc. of Eleventh Annual ACM Symposium on Theory of Computing*, 1979, pp. 360–364.Google Scholar - [SIPS80]Sipser, M., Halting space-bounded computations,
*Theoretical Computer Science*, 1980, pp. 335–338.Google Scholar - [STEA85]Stearns, R. and H. Hunt, On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata,
*SIAM Journal of Computing*, 14, 1985, pp. 598–611.CrossRefGoogle Scholar - [WEBE86]Weber, A. and H. Seidl, On the degree of ambiguity of finite automata,
*Proc. of Math. Foundations of Comp. Science*, 1986, pp. 620–629.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1987