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
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
Some 3CNF properties are hard to test
Proceedings of the thirty-fifth 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
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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
2-Transitivity Is Insufficient for Local Testability
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
On the Randomness Complexity of Property Testing
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
On proximity oblivious testing
Proceedings of the forty-first annual ACM symposium on Theory of computing
Green's conjecture and testing linear-invariant properties
Proceedings of the forty-first annual ACM symposium on Theory of computing
Testing Fourier Dimensionality and Sparsity
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Succinct Representation of Codes with Applications to Testing
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Tolerant Linearity Testing and Locally Testable Codes
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Lower bounds for testing triangle-freeness in Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Locally testable vs. locally decodable codes
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Guest column: testing linear properties: some general theme
ACM SIGACT News
Limitation on the rate of families of locally testable codes
Property testing
Invariance in property testing
Property testing
Testing linear-invariant non-linear properties: a short report
Property testing
Optimal Testing of Reed-Muller Code
Property testing
Property testing
Symmetric LDPC codes and local testing
Property testing
Limitation on the rate of families of locally testable codes
Property testing
Invariance in property testing
Property testing
Testing linear-invariant non-linear properties: a short report
Property testing
Optimal Testing of Reed-Muller Code
Property testing
Property testing
Symmetric LDPC codes and local testing
Property testing
Correlation testing for affine invariant properties on Fpn in the high error regime
Proceedings of the forty-third annual ACM symposium on Theory of computing
On the complexity of powering in finite fields
Proceedings of the forty-third annual ACM symposium on Theory of computing
On the average-case complexity of property testing
Studies in complexity and cryptography
Proximity oblivious testing and the role of invariances
Studies in complexity and cryptography
Characterizations of locally testable linear-and affine-invariant families
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
On sums of locally testable affine invariant properties
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Limits on the rate of locally testable affine-invariant codes
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Proximity oblivious testing and the role of invariances
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Hard functions for low-degree polynomials over prime fields
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Characterizations of locally testable linear- and affine-invariant families
Theoretical Computer Science
On Proximity-Oblivious Testing
SIAM Journal on Computing
Testing Fourier Dimensionality and Sparsity
SIAM Journal on Computing
Testing odd-cycle-freeness in Boolean functions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Edge transitive ramanujan graphs and symmetric LDPC good codes
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
SIAM Journal on Discrete Mathematics
Every locally characterized affine-invariant property is testable
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Hard Functions for Low-Degree Polynomials over Prime Fields
ACM Transactions on Computation Theory (TOCT)
An algebraic characterization of testable boolean CSPs
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Testing linear-invariant function isomorphism
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space Kn over a field K to a subfield F. We consider (F-)linear properties that are invariant under linear transformations of the domain and prove that an O(1)-local "characterization" is a necessary and sufficient condition for O(1)-local testability. when |K| = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)-local constraint implies O(1)-local testability. These results generalize and extend the class of algebraic properties, most notably linearity and low-degree-ness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)-locally testable. Our results are proved by introducing a new notion that we term "formal characterizations". Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations essentially imply local testability. We then investigate properties that are linear-invariant and attempt to understand their local formal characterizability. Our results here give coarse upper and lower bounds on the locality of constraints and characterizations for linear-invariant properties in terms of some structural parameters of the property we introduce. The lower bounds rule out any characterization, while the upper bounds give formal characterizations. Combining the two gives a test for all linear-invariant properties with local characterizations. We believe that invariance of properties is a very interesting notion to study in the context of property testing in general and merits a systematic study. In particular, the class of linear-invariant and affine-invariant properties exhibits a rich variety among algebraic properties and offer better intuition about algebraic properties than the more limited class of low-degree functions.