Direct product testing Irit Dinur, Inbal Livni Navon Weizmann Institute

Direct product operation 𝑓 ⊗𝑘 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑘 =𝑓 𝑥 1 ,𝑓 𝑥 2 ,…𝑓 𝑥 𝑘 Direct product operation Basic operation on functions Useful for hardness amplification “one new instance is as hard as k”

Testing direct products Given a function, 𝑓: 𝑛 𝑘 → 0,1 𝑘 is it a direct product? Related to parallel repetition: the honest players’ strategy is a direct product; if we knew that the player strategy is a DP then perfect parallel repetition is immediate, with perfectly exponential decay Natural test:

Testing direct products How good is this test? Suppose it succeeds with probability 99%, how close is f to DP ? This is a property testing question. Hope: f is 99% close to DP Warmup: for each 𝑧∈[𝑛] choose the most popular value An averaging argument shows that for 90% of tuples ( 𝑧 1 ,…, 𝑧 𝑘 ), f outputs the popular value on about 90% of the coordinates Aside: [D-Steurer 14] showed a stronger result: for 99% of the tuples, f agrees with m on every entry

Testing direct products Studied and used for some PCP constructions; initially suggested as a combinatorial alternative to “low degree test” [Goldreich-Safra, D-Reingold, D-Goldenberg, Impagliazzon-Kabanets- Wigderson, D-Steurer] Main interest: the 1% regime, aka “small soundness” or “list decoding“ regime

The 1% regime Suppose f passes the test with probability 1%, can anything be said about its structure? [unlike standard property testing set-up, we cannot afford boosting success by repetition] …cannot expect a conclusion of the form “there is a single DP g that approximates f”. we must allow a mix of several valid DP functions (aka a “list”) Examples of properties that have 1% tests: Low degree tests Long code tests Gowers norm and additive combinatorics (these are related to low degree tests) Direct product tests For what properties do we have tests that imply meaningful structure even if they pass with low “1%” probability? (Can be viewed as a stronger form of property testing) For which value of “1%” can structure be found? Ultimately= anything above random

The 1% regime for Direct Product tests D-Goldenberg: if f passes the V test with probability 𝜖 > 1/𝑝𝑜𝑙𝑦(𝑘), then it is 𝑝𝑜𝑙𝑦(𝜖) correlated to a DP Cannot expect such structure for 𝜖 below 1/𝑝𝑜𝑙𝑦(𝑘) : There is a function f that is not close to any DP, but passes the test w prob 𝜖 Impagliazzo Kabanets Wigderson: add a third query, and the counter-example goes away IKW theorem: for any 𝜖 > exp⁡(−√𝑘), if f passes the Z test with probability ≥𝜖, it is 𝑝𝑜𝑙𝑦(𝜖) correlated to a DP New (almost a decade later): same holds for all 𝜖 > exp⁡(−𝑘),

Main Result If 𝑓: 𝑛 𝑘 → 0,1 𝑘 passes the Z-test with probability 𝜖≥ exp −𝑘 , then it is 𝑝𝑜𝑙𝑦(𝜖) close to a direct product function The result is “ballpark” tight Number of queries: it is impossible with 2 queries, as per [DG] example; so 3 is it Soundness: cannot go below exp⁡(−𝑘) a random function where 𝑓(𝑥) is chosen independently for each 𝑥∈ 𝑛 𝑘 passes the test with probability exp −𝑘 Open: can we get soundness test down to the randomness threshold? ( 2 −𝑘 +𝛿 and not only (1.001) −𝑘 ) “close to DP” : there is approximate closeness, and exact closeness,

Proof Given 𝑓: 𝑛 𝑘 → 0,1 𝑘 that passes the Z-test with probability 𝜖 ≥ exp −𝑘 The proof will construct a DP that is close to f. How? Taking a popularity vote on a restricted set of x’s Choose a random restriction A= 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 ,∗, ∗,∗,∗ Choose a fixed answer 𝛼 ∈ 0,1 𝐴 , e.g. “1001”, Consider only 𝑥’s that 𝑓 𝑥 𝐴 =𝛼 Take a majority vote on these x’s, (assume for now that f is invariant for permutations) 𝑔 𝑦 =𝑝𝑜𝑝𝑢𝑙𝑎𝑟 𝑓 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 ,𝑦,∗,∗,∗ Need to prove that inside this restriction, indeed f is a DP, i.e. the value of f on the i-th coordinate, depends only on the i-th coordinate.

Proof IKW: used sampling properties of k-sets and t-sets, which hold as long as 𝜖≥ exp − 𝑘 𝑡≤ 𝑘 When 𝜖≈ exp −𝑘 this fails. Instead, we perform “densification”

Inside the restriction: We have partial function 𝑓: 𝑛 𝑘 ′ → 0,1 𝑘 ′ ∪{⊥}, that is defined on some 𝜖≥exp −𝑘′ fraction of the inputs We know that whenever the function is defined on two “intersecting” inputs, the answers agree whp “densify” the function by using majority on small balls Prove that the densified function has similar properties to the original function, (using: reverse hyper-contractivity to reason about expansion of small sets in the test graph) With the dense function we can finish like before

Restriction decoding Assuming success 99%  Unique decoding List decoding ? Restriction decoding and “zoom ins” Third query – brings us back to list decoding

(possibly: some form of high dimensional expansion) Agreement tests The question of DP testing belongs to a family of testing questions called “agreement testing” Given a collection of partial views of a domain, that typically pairwise agree, is there a global view they all agree on? E.g. plane vs. plane low degree test E.g. tuple vs. tuple tests that we saw today Such agreement theorems are an important ingredient in all PCPs What is the connection between these questions and the geometry of the set of “local views” ? (possibly: some form of high dimensional expansion)