Source: J. Symbolic Logic
Volume 76, Issue 4
We give the first systematic study of strong isomorphism reductions,
a notion of reduction more appropriate than polynomial time reduction
when, for example, comparing the computational complexity of the
isomorphim problem for different classes of structures.
We show that the partial ordering of its degrees is
quite rich. We analyze its relationship to a further type of
reduction between classes of structures based on purely
comparing for every n the number
of nonisomorphic structures of cardinality
at most n in both classes. Furthermore,
in a more general setting we address the question of the existence of
a maximal element in the partial ordering of the degrees.
H. U. Besche, B. Eick, and E. A. O'Brien, The groups of order at most 2000, Electronic Research Announcements of the American Mathematical Society, vol. 7(2001), pp. 1–4.
A. Blass and Y. Gurevich, Equivalence relations, invariants, and normal forms, SIAM Journal on Computing, vol. 13 (1984), no. 4, pp. 682–689.
Mathematical Reviews (MathSciNet): MR764172
––––, Equivalence relations, invariants, and normal forms. II, Lecture Notes in Computer Science, vol. 171(1984), pp. 24–42.
R. B. Boppana, J. Håstad, and S. Zachos, Does co-NP have short interactive proofs?, Information Processing Letters, vol. 25 (1987), no. 2, pp. 127–132.
Mathematical Reviews (MathSciNet): MR896155
Y. Chen and J. Flum, On p-optimal proof systems and logics for PTIME, Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP'10), Lecture Notes in Computer Science, vol. 6199, Springer, 2010, pp. 321–332.
H.-D. Ebbinghaus and J. Flum, Finite model theory, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.
L. Fortnow and J. Grochow, Complexity classes of equivalence problems revisited, 2009, arXiv:0907.4775v1 [cs.CC].
H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, Journal of Symbolic Logic, vol. 54 (1989), pp. 894–914.
S. Friedman, Descriptive set theory for finite structures, Lecture at the Kurt Gödel Research Center, 2009, Available at http://www.logic.univie.ac.at/~sdf /papers/wien-spb.pdf.
S. Givant and P. Halmos, Introduction to Boolean algebras, Springer, New York, 2008.
Y. Gurevich, From invariants to canonization, Bulletin of the European Association for Theoretical Computer Science, vol. 63 (1997), pp. 115–119.
J. Hartmanis and L. Hemachandra, Complexity classes without machines: on complete languages for $\rm UP$, Theoretical Computer Science, vol. 58 (1988), pp. 129–142.
Mathematical Reviews (MathSciNet): MR963258
T. Kavitha, Efficient algorithms for abelian group isomorphism and related problems, Proceedings of the 23rd conference on foundations of software technology and theoretical computer science (FSTTCS'02), Lecture Notes in Computer Science, vol. 2914, Springer, Berlin, 2003, pp. 277–288.
W. Kowalczyk, Some connections between presentability of complexity classes and the power of formal systems of reasoning, Proceedings of mathematical foundations of computer science, (MFCS'84), Lecture Notes in Computer Science, vol. 176, Springer, Berlin, 1984, pp. 364–369.
Mathematical Reviews (MathSciNet): MR783466
G. Miller, Isomorphism testing for graphs of bounded genus, Proceedings of the 12th annual ACM Symposium on theory of computing (STOC'80), 1980, pp. 225–235.
T. Thierauf, The computational complexity of equivalence and isomorphism problems, Lecture Notes in Computer Science, vol. 1852, Springer, 2000.