PCPPs: Probabilistically Checkable Proofs of Proximity

• PCPPs is a complexity concept that verifies an input’s closeness to a valid witness using random queries for robust, local testing.
• PCPP systems leverage algebraic and combinatorial methods to achieve constant query complexity with precise completeness and soundness guarantees.
• They play a key role in hardness amplification, reconfiguration problems, and locally decodable codes, guiding advanced proof constructions.

Probabilistically checkable proofs of proximity (PCPPs) are a foundational concept in complexity theory, merging the soundness guarantees of PCPs with a fine-grained test of an input’s proximity to a language. In a PCPP system, a random-query verifier checks—using a small number of queries to both the input and an auxiliary proof—whether the input is a valid witness or far in Hamming distance from any valid witness. Completeness requires that accepting assignments always pass, while soundness mandates rejection (with probability at least the soundness gap) if the input is sufficiently distant from any satisfying assignment. PCPPs arise in contexts ranging from hardness of approximation to locally decodable codes, robust assignment testers, and reconfiguration inapproximability.

1. Formal Definitions and Core Properties

PCPPs are defined by a randomized, non-adaptive verifier which queries $q$ bits of the joined input and proof, using $r$ random bits. For a Boolean circuit $C$ and input $x$, the (proximity) gap $\epsilon$ defines how far $x$ must be from satisfiability, and the soundness error $\delta$ bounds the maximum acceptance probability for proofs paired with far inputs. Specifically, completeness assumes that for $x$ satisfying $C$, there exists $\pi$ such that the acceptance probability is 1. Soundness guarantees that if $x$ is $\epsilon$-far from satisfying the circuit, then for any proof $\pi$, the acceptance probability is at most $\delta$ (Guruswami et al., 1 Jul 2025, Jozeph, 2015). This framework is parameterized by query complexity (number of probes), randomness, proof length, proximity gap, and alphabet size.

A typical formal statement: for $L \subseteq \Sigma_0^n$, a $q$-query PCPP for $L$ with proximity $\eta$ and soundness error $\epsilon$ consists of a structured verifier that accepts iff a local constraint is satisfied, with perfect completeness and tuned soundness as above (Gur et al., 15 Dec 2025).

2. Algebraic, Combinatorial, and Composed Constructions

Recent advances in PCPP research rely on innovative composition and algebraic approaches. The construction in "Ideals, Gröbner Bases, and PCPs" (Amireddy et al., 5 Nov 2025) presents a PCPP system requiring only one composition step (in contrast to previous polynomial or logarithmic steps), built on powerful alternatives to sum-check protocols and Gröbner basis theory for polynomial ideals. The verifier interrogates whether a multivariate polynomial vanishes on a carefully chosen subset $S$ of $\mathbb{F}_q^m$, specifically $S=(\{0,1\}^{m/c}_{\le 1})^c$, a Cartesian product of Hamming balls. This structure enables constant-query proximity tests for polynomial vanishing, leveraging the low Gröbner complexity of such sets:

• Completeness 1, soundness $1/2$
• Randomness $O(n^\varepsilon)$ bits
• Query locality $O(1)$
• Proof length $2^{O(n^\varepsilon)}$
• Alphabet size $$O(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon})$$ (or $O(1)$ for fixed $\varepsilon$)

Sum-check alternatives are constructed using "vanishing-certificate" polynomials, allowing the system to verify polynomial identities in $O(1)$ queries by exploiting Gröbner basis representations. This protocol applies low-degree tests, local corrections, and functional checks using line-samples within the variety $S$. By composing the robust PCPP over $S$ with traditional algebraic encodings, the full PCP theorem is realized in a single composition step, matching both logarithmic randomness and constant alphabet (Amireddy et al., 5 Nov 2025).

3. Query Complexity vs. Soundness and Near-Optimal Constructions

Trade-offs between query complexity and soundness gap are central to both theoretical utility and inapproximability applications. Early robust PCPPs (e.g., Ben-Sasson et al.) achieved constant soundness at poly(1/$\epsilon$) query complexity and proof lengths. Dinur–Reingold assignment testers matched these parameters with randomized local queries. Moshkovitz–Raz and later works managed to push soundness lower for PCPs, but constant-query PCPPs with arbitrarily small soundness remained open until very recently (Guruswami et al., 1 Jul 2025, Gur et al., 15 Dec 2025).

The new 3-query PCPP construction for Circuit-SAT (Gur et al., 15 Dec 2025) achieves quasi-linear proof size, constant alphabet, perfect completeness, and arbitrarily small soundness error $\epsilon$, with proximity parameter $\eta=1-\epsilon^{O(1)}$. The protocol relies on two rounds of composition of decodable PCPs: starting with an HDX-based PCP, the right and left alphabets are reduced, and a local decoder is combined via an extra input query to form the 3-query PCPP verifier. The critical composition lemma ensures that query complexity remains low as alphabet reductions and soundness parameters are maintained. This construction strictly improves upon previous work, especially in the context of relaxed locally decodable codes (RLDCs), demonstrating that PCPP-based RLDCs are strictly stronger than analogous LDCs at the same query complexity and size.

4. Role of Low-Degree and Certificate-Based Tests

PCPPs are deeply linked to locally testable and decodable codes, often implemented using low-degree testing protocols. As shown in Friedl–Sudan (Friedl et al., 2013), refinements to low-degree tests permit PCPP compositions with optimal field sizes ($q = d + 2$ for degree $d$ polynomials) and constant-tolerance error ($\delta < 1/8$ for one-point-on-a-line tests). These ingredients yield locally checkable codes supporting efficient, proximity-sensitive PCPPs with concrete complexity improvements: for every NP language $L$, one attains a constant-query ($Q=O(1)$) PCPP with randomness $O(\log n)$, proof length $n^{2+\epsilon}$, and soundness $1/2$. These techniques directly reduce polynomial blowup in proof size and query complexity found in earlier systems.

5. Constraint Complexity and the APPCPP Dichotomy

A major structural result is the dichotomy for almost perfect PCPPs (APPCPPs), which allows negligible completeness error in exchange for broader applicability. According to Jozeph (Jozeph, 2015), CSP constraint sets $S$ admit an APPCPP if and only if CSP($S$) is NP-hard under Schaefer’s dichotomy. For tractable languages (linear equations, Horn, dual-Horn, 2CNF, 0-valid/1-valid), no APPCPP exists for sufficiently small $\delta$ and large soundness factor $d$, despite any relaxation in verifier completeness. The positive direction follows via gadget reductions from 3SAT to arbitrary NP-hard CSPs, yielding constant-query PCPPs for all NP-hard constraint languages. Conversely, tractable CSPs fail to support proximity-sensitive local tests, as shown by structural impossibility arguments combining block-exclusivity and constraint properties.

This dichotomy is unconditional (assuming P $\neq$ NP) and robust against the choice of allowed local verifier computations, reflecting a conceptual separation between proximity verification and classical membership testing.

6. Applications in Hardness, Reconfiguration, and Codes

PCPPs have broad impact on hardness amplification, reconfiguration problems, and locally decodable coding theory. The direct connection between PCPP parameters and the complexity of approximate CSP reconfiguration is established in (Guruswami et al., 1 Jul 2025), where parallelization frameworks show that PCPPs with $q$ queries and soundness $\delta$ yield PSPACE-hardness for $(q+1)$-query CSP reconfiguration with gap $\delta$. These modular reductions solidify PCPPs as the linchpin in the hardness inapproximability landscape. Advances in constant-query, low-soundness PCPPs immediately strengthen reconfiguration hardness, guide the design of robust assignment testers, and underlie RLDC separations. The duality between PCP composition techniques and code-theoretic constructs has stimulated new research in efficient proof systems, shallow local tests, and explicit gap trade-offs.

7. Research Directions and Open Problems

Current research focuses on further reducing query complexity and soundness gaps, expanding efficient composition frameworks, and enhancing algebraic-geometric characterization of proximity testers, especially leveraging Gröbner basis methods (Amireddy et al., 5 Nov 2025). Open problems include explicit constructions of constant-query PCPPs with subconstant soundness and more refined integration with code theory for optimal RLDC parameters. Further, establishing tight quantitative links between proximity-centric proof systems and the fine-grained hardness of CSP reconfiguration remains active, with parallelization and modular composition playing central roles.

The emerging understanding of the structural limitations of local tests, the algebraic machinery underpinning vanishing checks, and the dynamic interplay with reconfiguration complexity positions PCPPs as a focal area in computational complexity and coding theory research.