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Relaxed locally decodable codes (RLDCs) are error-correcting codes in which individual bits of the message can be recovered by querying only a few bits from a noisy codeword. For uncorrupted codewords, and for every bit, the decoder must decode the bit correctly with high probability. However, for a noisy codeword, a relaxed local decoder is allowed to output a "rejection" symbol, indicating that the decoding failed. We study the power of adaptivity and two-sided error for RLDCs. Our main result is that if the underlying code is linear, adaptivity and two-sided error do not give any power to relaxed local decoding. We construct a reduction from adaptive, two-sided error relaxed local decoders to non-adaptive, one-sided error ones. That is, the reduction produces a relaxed local decoder that never errs or rejects if its input is a valid codeword and makes queries based on its internal randomness (and the requested index to decode), independently of the input. The reduction essentially maintains the query complexity, requiring at most one additional query. For any input, the decoder’s error probability increases at most two-fold. Furthermore, assuming the underlying code is in systematic form, where the original message is embedded as the first bits of its encoding, the reduction also conserves both the code itself and its rate and distance properties We base the reduction on our new notion of additive promise problems. A promise problem is additive if the sum of any two YES-instances is a YES-instance and the sum of any NO-instance and a YES-instance is a NO-instance. This novel framework captures both linear RLDCs and property testing (of linear properties), despite their significant differences. We prove that in general, algorithms for any additive promise problem do not gain power from adaptivity or two-sided error, and obtain the result for RLDCs as a special case. The result also holds for relaxed locally correctable codes (RLCCs), where a codeword bit should be recovered. As an application, we improve the best known lower bound for linear adaptive RLDCs. Specifically, we prove that such codes require block length of n ≥ k^{1+Ω(1/q²)}, where k denotes the message length and q denotes the number of queries.

Suppose we have random sampling access to a huge object, such as a graph or a database. Namely, we can observe the values of random locations in the object, say random records in the database or random edges in the graph. We cannot, however, query locations of our choice. Can we verify complex properties of the object using only this restricted sampling access? In this work, we initiate the study of sample-based proof systems, where the verifier is extremely constrained; Given an input, the verifier can only obtain samples of uniformly random and i.i.d. locations in the input string, together with the values at those locations. The goal is verifying complex properties in sublinear time, using only this restricted access. Following the literature on Property Testing and on Interactive Proofs of Proximity (IPPs), we seek proof systems where the verifier accepts every input that has the property, and with high probability rejects every input that is far from the property. We study both interactive and non-interactive sample-based proof systems, showing: - On the positive side, our main result is that rich families of properties / languages have sub-linear sample-based interactive proofs of proximity (SIPPs). We show that every language in NC has a SIPP, where the sample and communication complexities, as well as the verifier’s running time, are Õ(√n), and with polylog(n) communication rounds. We also show that every language that can be computed in polynomial-time and bounded-polynomial space has a SIPP, where the sample and communication complexities of the protocol, as well as the verifier’s running time are roughly √n, and with a constant number of rounds. This is achieved by constructing a reduction protocol from SIPPs to IPPs. With the aid of an untrusted prover, this reduction enables a restricted, sample-based verifier to simulate an execution of a (query-based) IPP, even though it cannot query the input. Applying the reduction to known query-based IPPs yields SIPPs for the families described above. - We show that every language with an adequate (query-based) property tester has a 1-round SIPP with constant sample complexity and logarithmic communication complexity. One such language is equality testing, for which we give an explicit and simple SIPP. - On the negative side, we show that interaction can be essential: we prove that there is no non-interactive sample-based proof of proximity for equality testing. - Finally, we prove that private coins can dramatically increase the power of SIPPs. We show a strong separation between the power of public-coin SIPPs and private-coin SIPPs for Equality Testing.