Non-Adaptive Parity Decision Tree Complexity

• Non-adaptive parity decision tree complexity is a measure that computes Boolean functions using fixed XOR queries, capturing parallel query constraints.
• It tightly relates to block sensitivity, certificate complexity, and Fourier spectrum, enabling strong lower bounds and lifting techniques in complexity theory.
• Applications include efficient property testing, lower bounds in communication and circuit complexity, and robust analyses in distributed computing frameworks.

Non-adaptive parity decision tree complexity is a measure of the query complexity of Boolean functions in a model where each query reveals the parity (XOR) of an arbitrary subset of input bits, and all such queries must be fixed in advance—that is, queries are non-adaptive. This model generalizes classical decision trees (which are restricted to single-bit queries) and is central in the paper of property testing, spectral structure of Boolean functions, lower bounds for communication complexity, and the interplay with quantum query algorithms. The paper has deep connections to analytic and combinatorial measures such as block sensitivity, certificate complexity analogues, Fourier spectrum, and the algebraic structure of the function under consideration.

1. Model Definition and Parity Decision Tree Complexity

A non-adaptive parity decision tree (NAPDT) for a function $f : \{0,1\}^n \rightarrow \{0,1\}$ is a model in which a set $$\mathcal{C} = \{c^{(1)}, \dots, c^{(d)}\} \subseteq \{0,1\}^n$$ of parity queries is chosen before seeing the input. For any input $x$, all the parity outcomes $q_i(x) = \langle x, c^{(i)} \rangle$ (over $\mathbb{F}_2$) are computed in parallel, after which a deterministic function of those answers determines $f(x)$. The non-adaptive parity decision tree complexity, denoted $NADT^\oplus(f)$, is the minimum number of such queries required to compute $f$ for all (defined) inputs.

Key points:

• In the adaptive model (classical or parity-based), queries can use information from previous answers to determine the next query.
• In the non-adaptive parity model, all query sets are fixed independently of the answers.

The non-adaptive parity decision tree complexity provides a lower bound on adaptively querying Boolean functions via general linear combinations, and models limitations of parallel, round-free information extraction—a scenario relevant in distributed computing and sublinear-time testing.

2. Relationships to Other Complexity Measures

Non-adaptive parity decision tree complexity is deeply interrelated with various spectral and combinatorial measures:

Complexity Measure	Notation	Parity Analogue/Relation
Block sensitivity	$bs(f)$	Parity block sensitivity $bs_+(f)$, polynomially related
Certificate complexity	$C(f)$	Parity certificate complexity $C_+(f)$, $NADT^\oplus(f) \leq C_+^0(f) \cdot C_+^1(f)$
Deterministic decision tree	$D(f)$	$$NADT^\oplus(f) \leq D_+(f) \leq C_+^0(f) \cdot C_+^1(f)$$
Fourier granularity	$gran(f)$	$NADT^\oplus(f) \geq gran(f) + 1$

The results in (Zhang et al., 2010) establish tight polynomial relations between the minimum parity certificate complexity $C_+^i(f)$ (for $i=0,1$), parity block sensitivity $bs_+(f)$, and deterministic parity decision tree depth $D_+(f)$:

• $D_+(f) \leq C_+^0(f) \cdot C_+^1(f)$
• $bs_+(f) \leq C_+(f) \leq (bs_+(f))^2$

For many natural and symmetric functions, granularity provides a lower bound that refines classical degree or sparsity-based methods: $\mathsf{D}_\oplus(f) \geq gran(f) + 1$ (Chistopolskaya et al., 2018).

3. Communication Complexity and Lifting Equivalences

Non-adaptive parity decision tree complexity enjoys a strong and, for total functions, exact relationship to one-way (and multi-party) communication complexity of XOR-composed functions:

• For a total Boolean function $f$, the deterministic one-way communication complexity of $F(x, y) = f(x \oplus y)$ is $D_{cc}^{\rightarrow}(F) = NADT^\oplus(f)$ (Podolskii et al., 2023, Mande et al., 2021).
• Multiparty communication complexity for $$F(x_1, \ldots, x_k) = f(x_1 \oplus \ldots \oplus x_k)$$ for $k \geq 4$ is polynomially equivalent to $D_+^\oplus(f)$ (Yao, 2015).

For partial functions, this equivalence breaks down. If $f$ is undefined only on at most $O(2^{n-t}/\sqrt{n-t})$ inputs and $D_{cc}^{\rightarrow}(F) = t$, then $NADT^\oplus(f) = t$, but if $f$ is undefined on many more inputs, exponential gaps can occur (Podolskii et al., 2023). The mechanism behind these phenomena is a detailed algebraic analysis of the coset structure induced by non-adaptive parity queries and its interaction with input patterns, covering codes, and partition-induced graphs.

4. Tight Bounds, Property Testing, and Fine-Grained Complexity

The non-adaptive parity decision tree model precisely captures the query complexity of property testing for certain function families, most prominently $k$-parities and $k$-juntas. For testing whether $f:\{0,1\}^n \to \{0,1\}$ is a $k$-parity or far from any $k$-parity, it is established that $\Theta(k \log k)$ non-adaptive queries are both necessary and sufficient (Buhrman et al., 2012). The upper bound leverages randomized influence tests and Chernoff bounds, while the lower bound arises by reductions from communication complexity of $k$-disjointness.

For broader algorithmic problems, reductions show that computing only the parity of a complex property (e.g., all-pairs shortest path parity, median parity) is as hard as the full computation—any NAPDT algorithm must have complexity matching the fine-grained hardness of the base problem (Abboud et al., 2020).

Notably, in network settings with noisy, local transmissions (e.g., wireless sensor networks), computing parity subject to these constraints leads to superlinear lower bounds, again reflecting the combinatorial limitations of non-adaptivity and local access (Dutta et al., 2015).

5. Spectral and Analytical Structure: Granularity and Fourier Inequalities

Fourier-analytic techniques play a central role in lower bounding NAPDT complexity. Granularity captures the minimal denominator required in the (dyadic) rational Fourier coefficients, which translates into query requirements: if the spectrum requires $2^{-k}$ denominators, then at least $k+1$ parity queries are essential (Chistopolskaya et al., 2018).

Further, bounds on the sum of linear Fourier coefficients (level-1 mass) in terms of parity decision tree depth have been established, with generalizations of the O'Donnell–Servedio inequality: for a PDT of depth $d$ computing $f$, $$\sum_{i=1}^n |\hat{f}(i)| \leq \sqrt{4 \ln 2 \cdot \sigma^2 \cdot d}$$ (Blais et al., 2015), where $\sigma^2$ denotes the variance. These inequalities provide both analytic insight and structural lower bounds.

Moreover, upper bounds for Fourier-sparse functions show that $\mathrm{PDT}(f) = O(\sqrt k)$ for functions with Fourier sparsity $k$, achieved via probabilistic sampling techniques for parity selection and supported by the so-called folding property (Mande et al., 2020).

6. Lifting, Lower Bound Techniques, and Limitations

Lifting theorems establish that if a function $f$ has high NAPDT complexity, then certain composed functions (like $f \circ$XOR or $f \circ g$ for appropriate gadgets) inherit high communication or circuit complexity:

• Lifting with XOR gadgets preserves NAPDT complexity exactly (Mande et al., 2021).
• The stifling property for a gadget $g$ (strong setting-forcing property) is sufficient for deterministic decision tree complexity of $f$ to lift to PDT size complexity of $f \circ g$ for constant-size gadgets (Chattopadhyay et al., 2022). This is leveraged to obtain tight size lower bounds and to transfer resolution lower bounds in proof complexity to systems augmented with parity.
• For AND gadgets, such lifting is not tight: exponential gaps between non-adaptive AND decision tree complexity and one-way communication complexity can be constructed (Mande et al., 2021).

Moreover, separation results—such as those between (adaptive) decision tree complexity and subcube partition complexity (Kothari et al., 2015)—demonstrate that lower bound techniques based on partitioning, certificates, or similar decompositions may be insufficient for NAPDT models.

7. Applications and Broader Implications

Non-adaptive parity decision tree complexity forms a foundation for several applications:

• Tight property testing for affine-invariant classes, $k$-juntas, and related function classes (Buhrman et al., 2012).
• Derivation of lower bounds for distributed computation protocols under noise or local communication (Dutta et al., 2015).
• Critical reductions in communication complexity—establishing equivalence to, or polynomial relations with, k-party communication complexity for XOR functions (Yao, 2015).
• Impact on circuit complexity: lower bounds for parity gates or multiplicative circuit complexity for functions such as majority (Chistopolskaya et al., 2018).
• Insights for proof complexity, specifically in translating resolution width or size lower bounds via lifting with parity gadgets (Chattopadhyay et al., 2022).

Recent advances also clarify the limitations: for example, in quantum–classical hybrid circuit models ($\mathsf{AC^0} \circ \mathsf{QNC^0}$), quantum preprocessing cannot significantly reduce the decision tree complexity of parity or provide high correlation for constant-depth classical post-processing, with the tree-decomposition result showing that the depth remains unchanged under such channels (Slote, 2023).

Summary Table of Core Results

Result/Relationship	Formula/Statement	Source(s)
NAPDT upper bound via certificates	$NADT^\oplus(f) \leq C_+^0(f) \cdot C_+^1(f)$	(Zhang et al., 2010)
Parity block sensitivity link	$bs_+(f) \leq C_+(f) \leq (bs_+(f))^2$	(Zhang et al., 2010)
Granularity lower bound	$\mathsf{D}_\oplus(f) \geq gran(f) + 1$	(Chistopolskaya et al., 2018)
Fourier sparsity upper bound	$\mathrm{PDT}(f) = O(\sqrt{k})$ for $k$-sparse $f$	(Mande et al., 2020)
Property testing hardness	$NADT^\oplus(k$-parities$) = \Theta(k \log k)$	(Buhrman et al., 2012)
Lifting to one-way comm. (XOR)	$D_{cc}^{\rightarrow}(F) = NADT^\oplus(f)$	(Podolskii et al., 2023, Mande et al., 2021)
Lifting to PDT size (stifling)	$PDT(f \circ g) \geq DT(f) \cdot k$ if $g$ is k-stifled	(Chattopadhyay et al., 2022)
Exponential gaps for partials	$$\exists f:\ D_{cc}^{\rightarrow}(F) \ll NADT^\oplus(f)$$	(Podolskii et al., 2023)

These relationships and bounds capture the analytic, algebraic, and combinatorial principles governing non-adaptive parity decision tree complexity and its role as a central computational complexity measure.