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V. Vazirani, and V. V. Vazirani, Matching is as easy as matrix inversion, Combinatorica, 7, 1 (1987), 105-113. 34. S. Chari, P. Rohatgi, and A. Srinivasan, Randomness-optimal element isolation, with applications to perfect matching and related problems, SIAM Journal on Computing, 24, 5 (1995), 1036-1050. 35. H. K. Zachos, Two remarks on the power of counting, Technical Report MIT/LCS/TM-228, Massachusetts Institute of Technology, Laboratory for Computer Science, August 1982. O. edu Science USA For this volume, I have combined columns I wrote for the June, 1997 and October, 1998 Bulletin.

Szelepcsenyi, The method of forced enumeration for nondeterministic automata, Acta Information,, 26 (1988), 279-284. 47. J. Toran, On the hardness of graph isomorphism, submitted for publication. 48. V. Vinay, Rudiments of complexity theory for scientists and engineers, Sadhana, 19 (1994), 985-994. 49. P. Young, J. Hartmanis, Fundamental contributions to isomorphism problems, in Complexity Theory Retrospective, A. Selman, ed. (Springer-Verlag, New York, 1990). edu Toronto Proof complexity, the study of the lengths of proofs in propositional logic, is an area of study that is fundamentally connected both to major open questions of computational complexity theory and to practical properties of automated theorem provers.

The constants will be slightly different, but the flavor of the theorem remains unchanged. It is important to note that essentially any asymptotic lower bound can be analyzed in this way to obtain concrete bounds for a given computing technology. However, it is almost never done. In fact, I am not aware of any other instance where concrete bounds of this sort were obtained. ) Why are results of this sort so rare? The answer, of course, is that this approach first requires an asymptotic lower bound, and we don't have many of those, either!

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