Abstract
Many difficult open problems in theoretical computer science center around nondeterminism. We study the fundamental problem of converting a given deterministic finite automaton (DFA) to a minimal nondeterministic finite automaton (NFA). Despite extensive work on finite automata, this fundamental problem has remained open. Recently, in [Ji91] we studied this problem and showed that this (and related) problems are computationally hard. Here we study the restriction of this problem to the case when the input DFA is over a one-letter alphabet. Even in this restricted case the problem is computationally hard even though our evidence of hardness is different from (and is weaker than) the standard ones such as NP-hardness. Let A → B denote the problem of converting a finite automaton of type A to a minimal finite automaton of type B. Our main result is that DFA → NFA, when the input is a unary cyclic DFA (a DFA whose graph is a simple cycle), is in NP but not in P unless NP ⊑ DTIME(n O(log n)). Our work was also motivated by the problem of finding structurally simple ‘normal forms’ of NFA's over a unary alphabet. We present some normal forms for minimal NFA's over a unary alphabet and present an application to lower bounds on the size complexity of an NFA. In fact, the normal form result is used in a nontrivial manner to show the NP membership result stated above.
This work was supported in part by a grant from SERB, McMaster University, and NSERC Operating Grant OGP 0046613.
This work was supported in part by a summer faculty fellowship of the URI Foundation.
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References
Adleman, L. and K. Manders, ‘Reducibility, randomness and intractability', Proc. of 9th Annual ACM symp. on Theory of Computing, 1977, 151–163.
Chrobak, M., ‘Finite automata and unary languages', Theoretical Computer Science 47, 1986, 149–158.
Cook, S., ‘Complexity of theorem proving procedures', Proc. of 3rd Annual ACM Symp. on Theory of Computing, 1971, 151–158.
Ehrenfeucht, A. and P. Zeiger, ‘Complexity measures for regular expressions', Journal of Computer and System Sciences 12, 1976, 134–146.
Garey, M. and D. Johnson, Computers and Intractability: A Guide to NP-completeness, 1978, Freeman, San Fransisco.
Graham, R., D. Knuth and O. Patashnik, Concrete Mathematics, 1989, Addison-Wesley, Reading, Massachusetts.
Hopcroft, J. and J. Ullman, Introduction to Automata Theory, Languages and Computation, 1979, Addison-Wesley, Reading, Mass.
Hunt, H., ‘On the time and tape complexity of languages', Proc. of 5th Annual ACM Symp. on Theory of Computing, 1973, 10–19.
Hunt, H. and D. Rosenkrantz, ‘Computational Parallels between regular and context-free languages', Proc. of 6th Annual ACM Symp. on Theory of Computing, 1974, 64–74.
Hunt H., D. Rosenkrantz and T.Szymanski, ‘On the equivalence, containment and covering problems for the regular and context-free languages', Jl. of Comp. and Sys. Sciences 12, 1976, 222–268.
Jiang, T. and B. Ravikumar, ‘Minimal NFA problems are hard', Proc. 18th International Colloquium on Automata, Languages, and Programming, 1991, Lecture Notes in Computer Science 510, Springer-Verlag, 629–640.
Karp, R., ‘Reducibility among combinatorial problems', in R.E. Miller and J.W. Thatcher (eds.,), Complexity of Computer Computations, 1972, Plenum Press, NY, 85–103.
Kozen, D., ‘Lower bounds for natural proof systems', Proc. of 18th Annual IEEE Symp. on FOCS, 1977, 254–266.
Meyer, A. and M. Fischer, ‘Economy of description by automata, grammars and formal systems', Proc. of 12th Annual IEEE Symp. on Switching and Automata Theory, 1971, IEEE Computer Society, Washington D.C., 188–191.
Meyer, A. and L. Stockmeyer, ‘The equivalence problem for regular expressions with squaring requires exponential space', Proc. of 13th Annual IEEE Symp. on Switching and Automata Theory, 1972, IEEE Computer Society, 125–129.
Plaisted, D., ‘Sparse complex polynomials and polynomial reducibility', Jl. of Comp. and Sys. Sciences 14, 1977, 210–221.
Rabin, M. and D. Scott, ‘Finite automata and their decision problems', IBM Journal of Res. and Development 3, 1959, 114–125.
Ravikumar, B. and O. Ibarra, ‘Relating the type of ambiguity to the succinctness of their representations', SIAM Journal on Computing 18, 1989, 1263–1282.
Stockmeyer, L. and A. Meyer, ‘Word problems requiring exponential time’ (prelim, report), Proc. of 5th Annual ACM symposium on Theory of Computing, 1973, 1–9.
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Jiang, T., McDowell, E., Ravikumar, B. (1991). The structure and complexity of minimal NFA's over a unary alphabet. In: Biswas, S., Nori, K.V. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1991. Lecture Notes in Computer Science, vol 560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54967-6_67
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DOI: https://doi.org/10.1007/3-540-54967-6_67
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