Gatis Midrijanis
We will show that if there exists a quantum query algorithm that exactly computes some total Boolean function f by making T queries, then there is a classical deterministic algorithm A that exactly computes f making O(T^3) queries. The best know bound previously was O(T^4) due to Beals et al.
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Polynomial representation of general partial Boolean functions with a single quantum query
该研究针对单次量子查询下一般部分布尔函数刻画复杂的问题，建立了其与一次多项式简单部分布尔函数的等价关系，明确4比特下仅10个非平凡实例，并给出量子1-查询算法可计算函数的构造方法，推动相关基础研究与量子优势函数的挖掘。
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with quantum query algorithm complexity lower than the deterministic one. Relation between the degree of representing polynomial and potentially possible quantum algorithm complexity has been already described; unfortunately, there are few examples of quantum algorithms to illustrate theoretical evaluation of the complexity. Work in this direction was aimed (1) t…
Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean function is affine, then one more query to the oracle (the two-query algorithm) is required to identify the affinity of the function with certainty. The second aim of the paper is to show that if the provided Boolean function is incompletely defined, then the one-query and the two-query algorithms can be used as bounded-error quantum polynomial algorithms to identi…
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.
It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures the correct lower bound is Omega(log n / loglog n), and we exhibit quantum algorithms for two functions where this bound is achieved.
Classical and Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
本文研究布尔函数在不确定因果顺序下的经典与量子查询复杂度，证明经典广义模型中原有下界仍成立，构造出可通过因果不确定性降低查询复杂度的布尔函数并实现多项式分离，还给出量子查询复杂度可降低的布尔函数。
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant…
We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs …
We study the average case approximation of the Boolean mean by quantum algorithms. We prove general query lower bounds for classes of probability measures on the set of inputs. We pay special attention to two probabilities, where we show specific query and error lower bounds and the algorithms that achieve them. We also study the worst expected error and the average expected error of quantum algorithms and show the respective query lower bounds. Our results extend the optimality of the algorithm of Brassard et al.
The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an Ω(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide a…
This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the "black-box" approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even in the unbounded error setting. We also study a se…
We study the probability of making an error if, by querying an oracle a fixed number of times, we declare constant a randomly chosen n-bit Boolean function. We compare the classical and the quantum case, and we determine for how many oracle-queries k and for how many bits n one querying procedure is more efficient than the other.
Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
该研究将量子查询复杂度框架推广至量子超映射，证明带因果序量子控制的量子电路无法降低查询复杂度，但部分函数用两次查询时因果不定超映射的计算误差更低。
Near-Optimal Separations of Certificate Complexity from Randomized and Quantum Query Complexity
构造全布尔函数，证明量子查询复杂度与证书复杂度存在近四次分离，解决长期开放问题，结果在对数因子内最优。
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed b…
Closed-Form Optimal Quantum Circuits for Single-Query Identification of Boolean Functions
该研究提出了一种单次查询识别布尔函数的闭式最优量子电路，通过显式的量子态制备和测量单元ary实现Helstrom最优成功概率3/4，展示了在最小设置中无需纠缠的低深度电路，并探讨了更广泛查询情况下的构造性方法。
Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact, zero-error, bounded-error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a fresh notion in addition to already studied nondeterministic algorithms and introduce dual nondeterministic quantum query algorithms. We examine properties of such algorithms and prove relations with exact and nondeterministic quantum query …
We give and prove an optimal exact quantum query algorithm with complexity $k+1$ for computing the promise problem (i.e., symmetric and partial Boolean function) $DJ_n^k$ defined as: $DJ_n^k(x)=1$ for $|x|=n/2$, $DJ_n^k(x)=0$ for $|x|$ in the set $\{0, 1,\ldots, k, n-k, n-k+1,\ldots,n\}$, and it is undefined for the rest cases, where $n$ is even, $|x|$ is the Hamming weight of $x$. The case of $k=0$ is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmet…
We study randomized and quantum query (a.k.a. decision tree) complexity for all total Boolean functions, with emphasis to derandomization and dequantization (removing quantumness from algorithms). Firstly, we show that $D(f) = O(Q_1(f)^3)$ for any total function $f$, where $D(f)$ is the minimal number of queries made by a deterministic query algorithm and $Q_1(f)$ is the number of queries made by any quantum query algorithm (decision tree analog in quantum case) with one-sided constant error; both algorithms compute function $f$. Secondly, we show that for all total Boolean functions $f$ holds…
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
Testing Boolean Functions Properties
本文利用Deutsch-Jozsa算法和振幅放大技术，提出了测试布尔函数同一性、相关性和平衡性的最优量子算法，并给出了对应的经典算法复杂度。
Our problem is to evaluate a multi-valued Boolean function $F$ through oracle calls. If $F$ is one-to-one and the size of its domain and range is the same, then our problem can be formulated as follows: Given an oracle $f(a,x): \{0,1\}^n\times\{0,1\}^n \to \{0,1\}$ and a fixed (but hidden) value $a_0$, we wish to obtain the value of $a_0$ by querying the oracle $f(a_0,x)$. Our goal is to minimize the number of such oracle calls (the query complexity) using a quantum mechanism. Two popular oracles are the EQ-oracle defined as $f(a,x)=1$ iff $x=a$ and the IP-oracle defined as $f(a,x)= a\cdot x \…
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be ti…
The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coin…
Improved Quantum Query Bounds for Boolean Matrix Product Verification
本文首次给出布尔矩阵乘积验证的量子查询上界Õ(n^{17/12})，并证明Ω(n^{5/4})下界，通过等价性分析实现半径与直径判定问题的多项式分离。
In this paper we study the separation between the deterministic (classical) query complexity ($D$) and the exact quantum query complexity ($Q_E$) of several Boolean function classes using the parity decision tree method. We first define the Query Friendly (QF) functions on $n$ variables as the ones with minimum deterministic query complexity $(D(f))$. We observe that for each $n$, there exists a non-separable class of QF functions such that $D(f)=Q_E(f)$. Further, we show that for some values of $n$, all the QF functions are non-separable. Then we present QF functions for certain other values …
We construct a total Boolean function $f$ satisfying $R(f)=\tildeΩ(Q(f)^{5/2})$, refuting the long-standing conjecture that $R(f)=O(Q(f)^2)$ for all total Boolean functions. Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to $R(f)=\tildeΩ(Q(f)^3)$. Our construction is motivated by the Göös-Pitassi-Watson function but does not use it.
Bias-Class Discrimination of Universal QRAM Boolean Memories
研究通过通用量子随机存取存储器接口鉴别布尔记忆的偏倚类别，揭示相位偏倚幅度对鉴别结果的影响，扩展了德斯楚-乔兹萨算法的应用范围。
An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree
本文证明部分布尔函数的精确多项式次数与近似量子查询复杂度之间存在指数级分离，并在无界字母表下实现常数与多项式分离。
We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice.…
We establish a lower bound of $Ω{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $Ω(\sqrt{D(f)})$ is a lower bound for all Boolean functions. Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' of initially coherently superposed inputs in the query register, having different values of the function. The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pair…
The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)$. Belovs gave a modified version of the negative weight adversary method, $\mathrm{ADV}_{rel}^\pm(f)$, that characterizes the bounded-error quantum query complexity of a relation $f \subseteq \{0,1\}^n \times [K]$, provided the relation is efficiently verifiable. A relation is efficiently verifiable if $\mathrm{ADV}^\pm(f_…
We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes some total boolean function depending on n variables must make Omega(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using O(k log m) queries to solve the …
Quantum advantage and lower bounds in parallel query complexity
本文研究并行查询复杂度中的量子优势，通过构造全函数展示了量子相对随机化的无界并行优势及多项式分离，并提出从顺序上界推导并行量子下界的新技术，否定了既有猜想并界定了布尔对称函数等情形的优势范围。
The oracle identification problem (OIP) is, given a set $S$ of $M$ Boolean oracles out of $2^{N}$ ones, to determine which oracle in $S$ is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to $S$. The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper and lower bounds. They are quite general and mostly optimal: (i) The query complexity of OIP is $O(\sqrt{N\log M \log N}\log\log M)$ for {\it…
The Goldreich-Levin algorithm was originally proposed for a cryptographic purpose and then applied to learning. The algorithm is to find some larger Walsh coefficients of an $n$ variable Boolean function. Roughly speaking, it takes a $poly(n,\frac{1}ε\log\frac{1}δ)$ time to output the vectors $w$ with Walsh coefficients $S(w)\geqε$ with probability at least $1-δ$. However, in this paper, a quantum algorithm for this problem is given with query complexity $O(\frac{\log\frac{1}δ}{ε^4})$, which is independent of $n$. Furthermore, the quantum algorithm is generalized to apply for an $n$ variable $…
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of $O(1)$ vs. $\sqrt{N}$ between quantum and randomized query complexities remain the state-of-the-art (where $N$ is the input length), leaving open the question of whether $O(1)$ vs. $N^{1/2+Ω(1)}$ separations are possible? We answer this question in …
We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain speed-ups over the best known quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an $n$-input $m$-output boolean function is balanced or $ε$-far from balanced; and (c) give a sub-lin…
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondetermini…
The quantum query models is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm and others. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be exactly computed by a one-query quantum algorithm if and only if $f(x)=x_{i_1}$ or ${x_{i_1} \oplus x_{i_2} }$ (up to isomorphism). Note that unlike most work in the literature bas…
We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Briët and Palazuelos for characterizing the quantum query complexity of Boolean functions. Combined with their results, we obtain a new characterization of the quantum query complexity through semidefinite programming. Using the duality theory of semidefinite programming we obtain a new type of certificates for large query complexity. We show that our class of certificates …
On the Fine-Grained Query Complexity of Symmetric Functions
本文探讨了 Watrous 猜想的细粒度版本，证明了在成功概率任意接近 1/2 时，对称布尔函数的量子与随机查询复杂度存在多项式甚至二次方的等价关系，并确立了多种复杂度度量间的多项式等价性。
We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the Bernstein-Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. The same quantum algorithm can also be used to learn which input variables other types of Boolean functions depend on, with a success probability that depends on the form of the Boolean function that is tested,…
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box, but the aim is to compute function value for arbitrary input using as few queries as possible. In this paper we concentrate on quantum query algorithm designing tasks. The main aim of research was to find new efficient algorithms and develop general algorithm designing techniques. We present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next we introduce algorithm transformation methods that allow significant enlarging …
In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $Ω(n/\log(n))$ while its zero-error randomized query complexity is $\tilde O(\sqrt{n})$. We further show that the quantum query complexity of the same fun…
We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f o h o g) = Omega(R(f) R(h) R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f o g) >= RS(f) RS(g), and a composition theo…
Robust and Space-Efficient Dual Adversary Quantum Query Algorithms
本文提出了一种鲁棒的对偶对抗者量子查询算法，能够处理近似满足的约束条件，并证明了对于多项式个1值输入的布尔函数存在使用对数量子比特的查询最优算法，同时开发了相关压缩工具和开源代码。
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted, unweighted methods of Ambainis, and the spectral method of Barnum, Saks and Szegedy. As an immediate consequence of our main theorem, adversary methods can only prove lower bounds for boolean functions $f$ in $O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))$, where $C_0, C_1$ is the certificate complexity, and $n$ is the size of the input. We also…
We give a new upper bound on the quantum query complexity of deciding $st$-connectivity on certain classes of planar graphs, and show the bound is sometimes exponentially better than previous results. We then show Boolean formula evaluation reduces to deciding connectivity on just such a class of graphs. Applying the algorithm for $st$-connectivity to Boolean formula evaluation problems, we match the $O(\sqrt{N})$ bound on the quantum query complexity of evaluating formulas on $N$ variables, give a quadratic speed-up over the classical query complexity of a certain class of promise Boolean for…
Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. A repetition code is an error detection scheme that repeats each bit of the original message r times. After a message with redundant bits is transmitted via a communication channel, it must be verified. If the received message consists of r-size blocks of equal bits, the conclusion is that there were no errors. The verification procedure can be interpreted as an application of…
We describe a method to upper bound the quantum query complexity of Boolean formula evaluation problems, using fundamental theorems about the general adversary bound. This nonconstructive method can give an upper bound on query complexity without producing an algorithm. For example, we describe an oracle problem which we prove (non-constructively) can be solved in $O(1)$ queries, where the previous best quantum algorithm uses a polylogarithmic number of queries. We then give an explicit $O(1)$-query algorithm for this problem based on span programs.
Quantum Query-Space Lower Bounds Using Branching Programs
本研究通过分析受限广义量子分支程序（GQBP），推导出ORₙ等问题的量子查询-空间下界Q²s=Ω(n²)，并给出对称布尔函数查询复杂度下界的新证法。
We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs ~sqrt(N)/log(N) queries (improving an ~N^{1/4} lower bound of Aaronson). Conversely, we show that this 1 versus ~sqrt(N) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated …
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $ε<1/2$ iff $f$ can be approximated by a degree-2 polynomial with error bounded by $ε'<1/2$. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis (STOC'2015, arxiv:1411.5729). We also show two results for polynomials of h…
We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than th…
Strong Selective and List-Decoding Direct Product Theorems for Quantum Query Complexity
本文为量子查询复杂度证明了所有函数的强选择性直接积定理，并推广了列表解码直接积定理至部分布尔函数，通过新的乘法对手表述统一了相关结果。
On query complexity measures and their relations for symmetric functions
本文探讨了对称函数类中量子查询复杂性下界与其他基于决策树的复杂性度量之间的关系，为正对手方法等给出了显式构造，并刻画了Gap Majority的量子查询复杂性及紧分离。
Quantum and Classical Communication Complexity of Permutation-Invariant Functions
本文刻画了置换不变布尔函数的量子通信复杂度，证明其量子与随机化通信复杂度二次等价，并将该对称性结论推广至通信复杂度，验证了相关秩猜想。
Randomized query complexity can beat certificate complexity
本文构造了一个全布尔函数，证明其随机化查询复杂度可远低于证书复杂度，达到平方根级别的最优，同时量子查询复杂度也接近最优。
A Quantum Algorithm for the Classification of Patterns of Boolean Functions
本文提出一种量子算法，通过单次预言机查询以概率1分类不平衡布尔函数的层级类别，并详述了可扩展的设计方法论。
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