Junto-Symmetric Functions, Hypergraph Isomorphism and Crunching

We make a step towards characterizing the boolean functions to which isomorphism can be efficiently tested. Specifically, we prove that isomorphism to any boolean function on {0, 1}n with a polynomial number of distinct permutations can be tested with a number of queries that is independent of n. We also show some partial results in the converse direction, and discuss related problems: testing isomorphism up to linear transformations, and testing isomorphism against a uniform (hyper)graph that is given in advance. Our results regarding the latter topic generalize a theorem of Fischer (SICOMP 2005), and in the process we also provide a simpler proof of his original result which avoids the use of Szemeredi's regularity lemma.

[1]  L. Babai.  On the Order of Uniprimitive Permutation Groups , 1981 .

[2]  Rocco A. Servedio,et al.  Testing for Concise Representations , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[3]  Robert Beals,et al.  Symmetry and complexity , 1992, STOC '92.

[4]  Guy Kindler,et al.  Testing juntas , 2002, J. Comput. Syst. Sci..

[5]  Noga Alon,et al.  Nearly tight bounds for testing function isomorphism , 2011, SODA '11.

[6]  Peter Clote,et al.  Boolean Functions and Computation Models , 2002, Texts in Theoretical Computer Science. An EATCS Series.

[7]  Ryan O'Donnell,et al.  Testing Fourier Dimensionality and Sparsity , 2009, ICALP.

[8]  Eldar Fischer.  The difficulty of testing for isomorphism against a graph that is given in advance , 2004, STOC '04.

[9]  László Babai,et al.  Property Testing of Equivalence under a Permutation Group Action , 2008, Electron. Colloquium Comput. Complex..

[10]  Asaf Shapira,et al.  Approximate Hypergraph Partitioning and Applications , 2010, SIAM J. Comput..

[11]  R. Gregory Taylor.  Review of "Boolean Functions and Computation Models by Peter Clote and Evangelos Kranakis", Springer-Verlag, 2002 , 2004, SIGA.

[12]  Alfred Bochert.  Ueber die Zahl der verschiedenen Werthe, die eine Function gegebener Buchstaben durch Vertauschung derselben erlangen kann , 1889 .

[13]  Dana Ron,et al.  Testing Basic Boolean Formulae , 2002, SIAM J. Discret. Math..

[14]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

[15]  Eldar Fischer,et al.  Testing graph isomorphism , 2006, SODA '06.

[16]  Eric Blais.  Testing juntas nearly optimally , 2009, STOC '09.

[17]  O. Svensson,et al.  Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling , 2007, FOCS 2007.

[18]  Sourav Chakraborty,et al.  Efficient Sample Extractors for Juntas with Applications , 2011, ICALP.

[19]  Peter L. Hammer,et al.  Boolean Functions , 2013, Discrete Applied Mathematics.

[20]  Ryan O'Donnell,et al.  Lower Bounds for Testing Function Isomorphism , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[21]  Michael Clausen.  Almost all Boolean Functions Have no Linear Symmetries , 1992, Inf. Process. Lett..

[22]  Jirí Sgall,et al.  Functions that have read-twice constant width branching programs are not necessarily testable , 2004, Random Struct. Algorithms.

[23]  Toniann Pitassi,et al.  Effectively Polynomial Simulations , 2010, ICS.

[24]  Noga Alon,et al.  Testing Boolean Function Isomorphism , 2010, APPROX-RANDOM.

[25]  Andrzej Kisielewicz,et al.  Symmetry Groups of Boolean Functions and Constructions of Permutation Groups , 1998 .

[26]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[27]  Evangelos Kranakis,et al.  Boolean Functions, Invariance Groups, and Parallel Complexity , 1991, SIAM J. Comput..

[28]  E. Fischer.  THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[29]  Yuichi Yoshida,et al.  Partially Symmetric Functions Are Efficiently Isomorphism-Testable , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[30]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[31]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .