Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  characterize the set of properties of Boolean-valued functions on a finite domain $\mathcal{X}$ that are testable with a constant number of samples. Specifically,  show that a property $\mathcal{P}$ is testable with a constant number of samples if and only if it is (essentially) a $k$-part symmetric property for some constant $k$, where a property is {\em $k$-part symmetric} if there is a partition $S_1,\ldots,S_k$ of $\mathcal{X}$ such that whether $f:\mathcal{X} \to \{0,1\}$ satisfies the property is determined solely by the densities of $f$ on $S_1,\ldots,S_k$.  use this characterization to obtain a number of corollaries, namely: (i) A graph property $\mathcal{P}$ is testable with a constant number of samples if and only if whether a graph $G$ satisfies $\mathcal{P}$ is (essentially) determined by the edge density of $G$. (ii) An affine-invariant property $\mathcal{P}$ of functions $f:\mathbb{F}_p^n \to \{0,1\}$ is testable with a constant number of samples if and only if whether $f$ satisfies $\mathcal{P}$ is (essentially) determined by the density of $f$. (iii) For every constant $d \geq 1$, monotonicity of functions $f : [n]^d \to \{0, 1\}$ on the $d$-dimensional hypergrid is testable with a constant number of samples.
['Eric Blais', 'Yuichi Yoshida']
I am writing a research paper on a idea A Characterization of Constant-Sample Testable Properties
[{"from": "human", "value": "I am writing a research paper on a idea A Characterization of Constant-Sample Testable Properties"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  characterize the set of properties of Boolean-valued functions on a finite domain $\\mathcal{X}$ that are testable with a constant number of samples. Specifically,  show that a property $\\mathcal{P}$ is testable with a constant number of samples if and only if it is (essentially) a $k$-part symmetric property for some constant $k$, where a property is {\\em $k$-part symmetric} if there is a partition $S_1,\\ldots,S_k$ of $\\mathcal{X}$ such that whether $f:\\mathcal{X} \\to \\{0,1\\}$ satisfies the property is determined solely by the densities of $f$ on $S_1,\\ldots,S_k$.  use this characterization to obtain a number of corollaries, namely: (i) A graph property $\\mathcal{P}$ is testable with a constant number of samples if and only if whether a graph $G$ satisfies $\\mathcal{P}$ is (essentially) determined by the edge density of $G$. (ii) An affine-invariant property $\\mathcal{P}$ of functions $f:\\mathbb{F}_p^n \\to \\{0,1\\}$ is testable with a constant number of samples if and only if whether $f$ satisfies $\\mathcal{P}$ is (essentially) determined by the density of $f$. (iii) For every constant $d \\geq 1$, monotonicity of functions $f : [n]^d \\to \\{0, 1\\}$ on the $d$-dimensional hypergrid is testable with a constant number of samples."}]