Abstract
We study the query complexity of one-sided \(\epsilon \)-testing the class of Boolean functions \(f:\mathcal{F}^n\rightarrow \{0,1\}\) that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where \(\mathcal{F}\) is any finite field. We give a polynomial-time \(\epsilon \)-testers that ask \(\tilde{O}(1/\epsilon )\) queries. This improves the query complexity \(\tilde{O}(|\mathcal{F}|/\epsilon )\) in [11].
We then show that any one-sided \(\epsilon \)-tester with proximity parameter \(\epsilon <1/|\mathcal{F}|^d\) for the class of Boolean functions that describe \((n-d)\)-dimensional affine subspaces and Boolean functions that describe axis-parallel \((n-d)\)-dimensional affine subspaces must make at least \(\varOmega (1/\epsilon +|\mathcal{F}|^{d-1}\log n)\) and \(\varOmega (1/\epsilon +|\mathcal{F}|^{d-1}n)\) queries, respectively. This improves the lower bound \(\varOmega (\log n/\log \log n)\) that is proved in [11] for \(\mathcal{F}=\mathrm{{GF}}(2)\). We also give testers for those classes with query complexity that almost match the lower bounds. (See the definitions of the classes in the introduction and many other results in Figs. 1 and 2).
Center for Theoretical Sciences, Guangdong Technion, (GTIIT), China.
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Notes
- 1.
In the literature, this class is defined as conjunction of d (non-negated) variables. Testability of f for this class is equivalent to testability of \(f(x+1^n)\) of d-Monomial as defined in this paper. The same applies to the classes \((\le d)\)-Monomial and Monomial.
- 2.
A literal is a variable or its negation.
- 3.
They also gave a tester for \((\le d)\)-AS\(\cup \{z(x)\}\) and \((\le d)\)-APAS\(\cup \{z(x)\}\) with the same query complexity where z(x) is the zero function.
- 4.
Goldreich and Ron algorithm and our algorithm run in time linear in the number of queries.
- 5.
d is known to the tester.
- 6.
In coding theory for \(f\in d\)-WSLS, \(f^{-1}(1)\) is called systematic code.
- 7.
The generator matrix of a linear subspace \(L\subseteq \mathcal{F}^n\), is a matrix with a minimum number of rows where the span of its rows is L.
- 8.
It is easy to show that if \(f_d(x)=f(xM)\) is \(\epsilon \)-close to \(d^*\)-LS if and only if f(x) is \(\epsilon \)-close to \(d^*\)-LS.
- 9.
If \(d=D\), then by property \(P_1(d)\), it follows that f is \(\epsilon \)-close to the zero function.
- 10.
If in every iteration \(f_d\) is not \(\epsilon \)-far from \(P_1(d)\) and \(P_3(d)\), then it gets to iteration D, and therefore f is \(\epsilon \)-close to LS. A contradiction.
- 11.
if \(|\mathcal{F}|^d\epsilon >2\), the tester accepts. This is because any function in d-R is \(|\mathcal{F}|^{n-d}/|\mathcal{F}|^n\le 1/|\mathcal{F}|^d\le \epsilon /2\) close to any function in d-F. Therefore, if f is \(\epsilon /2\)-close to d-R, and \(|\mathcal{F}|^d\epsilon >2\) then it is \(\epsilon \)-close to d-F.
- 12.
This is true since \(\Pr [f\not =0]\ge \Pr [f\not =h]-\Pr [h\not =0]\ge \epsilon -1/|\mathcal{F}|^n.\) Now we may assume that \(\epsilon \ge 2/|\mathcal{F}|^n\) because, otherwise, we can query f in all the points using \(O(|\mathcal{F}|^n)=O(1/\epsilon )\) queries.
- 13.
If f is \(\epsilon \)-far from AS, then it is \(\epsilon \)-far from the function h(x) that satisfies \(h^{-1}(1)=\{0^n\}\). Therefore, whp, some point a satisfies \(f(a)=1\). This is not true for \((\le d)\)-AS because \(h\not \in (\le d)\)-AS.
- 14.
By f(a, b), we mean the following: If \(a=(a_1,\ldots ,a_{n-d})\) and \(b=(b_1,\ldots ,b_d)\), then \(f(a,b)=f(a_1,\ldots ,a_{n-d},b_1,\ldots ,b_d)\).
- 15.
For \(f\in C\) and access to a black-box to f, the algorithm returns a function equivalent to f.
- 16.
For a random uniform \(g\in d'\)-APLS, we have \(\textbf{E}_s[\textbf{E}_g[T(g)]]=\textbf{E}_g[\textbf{E}_s[T(g)]]\ge 2/3\) where s is the random seed of T. Then there is \(s_0\) such that \(\textbf{E}_g[T(g)]\ge 2/3\).
- 17.
Here, we assume that \(d\ll n\). For large d, we can replace step 4 in the learning algorithm that makes at most d queries with the algorithm in [16] that makes \(d'\log (d/d')-O(d')\) queries. This changes \(\log |C|-d\) to \(\log |C|-d'\log (d/d')-O(d')\), and we get the lower bound for any d.
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Bshouty, N.H. (2023). On One-Sided Testing Affine Subspaces. In: Mavronicolas, M. (eds) Algorithms and Complexity. CIAC 2023. Lecture Notes in Computer Science, vol 13898. Springer, Cham. https://doi.org/10.1007/978-3-031-30448-4_12
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