Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
On the Robustness of Functional Equations
SIAM Journal on Computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
A Lower Bound for Testing 3-Colorability in Bounded-Degree Graphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Simple analysis of graph tests for linearity and PCP
Random Structures & Algorithms
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Three Theorems Regarding Testing Graph Properties
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Robust pcps of proximity, shorter pcps and applications to coding
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Testing Polynomials over General Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Testing Low-Degree Polynomials over Prime Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Some 3CNF Properties Are Hard to Test
SIAM Journal on Computing
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Sub-constant error low degree test of almost-linear size
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
A combinatorial characterization of the testable graph properties: it's all about regularity
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Graph limits and parameter testing
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
2-Transitivity Is Insufficient for Local Testability
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
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The linear- or affine-invariance is the property of a function family that is closed under linear- or affine- transformations on the domain, and closed under linear combinations of functions, respectively. Both the linear- and affine-invariant families of functions are generalizations of many symmetric families, for instance, the low degree polynomials. Kaufman and Sudan [21] started the study of algebraic properties test by introducing the notions of "constraint" and " characterization" to characterize the locally testable affine- and linear-invariant families of functions over finite fields of constant size. In this article, it is shown that, for any finite field F of size q and characteristic p, and its arbitrary extension field K of size Q, if an affineinvariant family F ⊆ {Kn → F} has a k-local constraint, then it is k′-locally testable for k′= k2Q/p Q2Q/p+4; and that if a linear-invariant family F ⊆ {Kn → F} has a k-local characterization, then it is k′-locally testable for k′= 2k2Q/p Q4(Q/p+1). Consequently, for any prime field F of size q, any positive integer k, we have that for any affine-invariant family F over field F, the four notions of "the constraint", "the characterization", "the formal characterization" and "the local testability" are equivalent modulo a poly(k, q) of the corresponding localities; and that for any linear-invariant family, the notions of "the characterization", "the formal characterization" and "the local testability" are equivalent modulo a poly(k, q) of the corresponding localities. The results significantly improve, and are in contrast to the characterizations in [21], which have locality exponential in Q, even if the field K is prime. In the research above, a missing result is a characterization of linearinvariant function families by the more natural notion of constraint. For this, we show that a single strong local constraint is sufficient to characterize the local testability of a linear-invariant Boolean function family, and that for any finite field F of size q greater than 2, there exists a linear-invariant function family F over F such that it has a strong 2- local constraint, but is not qd/q-1-locally testable. The proof for this result provides an appealing approach towards more negative results in the theme of characterization of locally testable algebraic properties, which is rare, and of course, significant.