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Erasures vs. Errors in Property Testing and Local List Decoding Sofya Raskhodnikova Boston University Joint work with Noga Ron-Zewi ( Haifa University ) Nithin Varma ( Boston University ) 1

Goal: study of sublinear algorithms resilient to adversarial corruptions in the input Focus: property testing model [Rubinfeld Sudan 96, Goldreich Goldwasser Ron 98]

A Sublinear-Time Algorithm B L A - B L A - B L A - B L A - B L A - B L A - B L A - B L A ? L ? B ? L ? A randomized algorithm approximate answer Resources Quality of • number of queries approximation • running time 3

A Sublinear-Time Algorithm B L A - B L A - B L A - B L A - B L A - B L A - B L A - B L A ? L ? B ? L ? A randomized algorithm approximate answer Is it always reasonable to assume the input is intact? 4

Algorithms Resilient to Erasures (or Errors) ⊥ ⊥ A - B L ⊥ ⊥ B L A - ⊥ L A - B L ⊥ - B L A - B L A - B L A ? L ? B ? L ? randomized algorithm • ≤ 𝜷 fraction of the input is erased (or modified) adversarially before algorithm runs • Algorithm does not know in advance what’s erased (or modified) • Can we still perform computational tasks? 5

Property Testing Property Tester [Rubinfeld Sudan 96, Goldreich Goldwasser Ron 98] randomized far from 𝜁 YES NO algorithm YES Accept with Reject with Don’t probability probability care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 6

Property Testing with Erasures Erasure-Resilient Property Tester [Dixit Property Tester [Rubinfeld Sudan 96, Raskhodnikova Thakurta Varma 16] Goldreich Goldwasser Ron 98] • ≤ 𝛽 fraction of the input is erased adversarially Any completion Can be randomized far from 𝜁 𝜁 is far from YES completed NO NO algorithm YES to YES YES Accept with Accept with Reject with Reject with Don’t Don’t probability probability probability probability care care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 7

Property Testing with Errors Tolerant Property Tester Property Tester [Rubinfeld Sudan 96, [Parnas Ron Rubinfeld 06] Goldreich Goldwasser Ron 98] • ≤ 𝛽 fraction of the input is wrong 𝛽 randomized far from far from 𝜁 𝜁 YES YES NO NO algorithm YES YES Accept with Accept with Reject with Reject with Don’t Don’t probability probability probability probability care care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 8

Property Testing with Errors Tolerant Property Tester Property Tester [Rubinfeld Sudan 96, [Parnas Ron Rubinfeld 06] Goldreich Goldwasser Ron 98] • ≤ 𝛽 fraction of the input is wrong 𝛽 randomized far from far from 𝜁 𝜁 YES YES NO NO algorithm YES YES Accept with Accept with Reject with Reject with Don’t Don’t probability probability probability probability care care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 9

Relationships Between Models Containments are strict: • [Fischer Fortnow 05]: standard vs. tolerant • [Dixit Raskhodnikova Thakurta Varma 16]: standard vs. erasure-resilient • new : erasure-resilient vs. tolerant ε -testable 𝛃 -erasure-resiliently ε -testable (𝛃, ε ) -tolerantly testable 10

Our Separation Separation Theorem There is a property of 𝒐 -bit strings that • can be 𝜷 -resiliently 𝜻 -tested with constant query complexity, • but requires 𝒐 𝛁 𝟐 queries for tolerant testing. Most of the talk: constant vs. 𝛁 𝐦𝐩𝐡 𝒐 separation. 11

Main Tool: Locally List Erasure-Decodeable Codes • Locally list decodable codes have been extensively studied [Goldreich Levin 89, Sudan Trevisan Vadhan 01, Gutfreund Rothblum 08, Gopalan Klivans Zuckerman 08, Ben-Aroya Efremenko Ta-Shma 10, Kopparty Saraf 13, Kopparty 15, Hemenway Ron-Zewi Wootters 17, Goi Kopparty Oliveira Ron-Zewi Saraf 17, Kopparty Ron-Zewi Saraf Wootters 18] • Only errors, not erasures were previously considered – Not the case without the locality restriction [Guruswami 03, Guruswami Indyk 05] Can locally list decodable codes perform better with erasures than with errors? 12

A Locally List Erasure-Decodable Code • An error-correcting code 𝓓 𝑜 : Σ 𝑜 → Σ 𝑂 • Parameters: 𝜷 fraction of erasures, list size ℓ and 𝒓 queries. 𝑥 : ⊥ ⊥ 0 0 0 1 ⊥ ⊥ 0 1 0 0 0 1 1 1 ⊥ 1 1 1 0 1 1 1 0 1 ⊥ 1 0 1 1 (𝛃, ℓ, 𝒓) -local list 𝐵 ℓ 𝐵 2 𝐵 1 Output ...... erasure-decoder – the fraction of erased bits in w is at most 𝜷 , – the decoder makes at most 𝒓 queries to 𝑥, – w.p. ≥ 2/3 , for every 𝑦 ∈ Σ 𝑜 with encoding 𝓓 𝑜 (𝑦) that agrees with 𝑥 on all non-erased bits, one of the algorithms 𝐵 𝑘 , given oracle access to 𝑥, implicitly computes 𝑦 ( that is, 𝐵 𝑘 𝑗 = 𝑦 𝑗 ) ; – each algorithm 𝐵 𝑘 makes at most 𝒓 queries to 𝑥 . 13

Hadamard Code • Hadamard: 0,1 𝑙 → 0,1 2 𝑙 ; Hadamard 𝑦 = 𝑦, 𝑧 𝑧∈ 0,1 𝑙 • Impossible to decode when fraction of errors 𝜷 ≥ 𝟐/𝟑. Type of Corruption List size, Number of Upper Lower bound corruptions tolerance 𝜷 ℓ queries, 𝑟 bound 1 1 [Goldreich [Blinovsky 86, Θ Θ Levin 89] Guruswami 𝟐 2 2 Errors 1 1 𝛽 ∈ 0, Vadhan 10, 2 − 𝛽 2 − 𝛽 𝟑 Grinberg Shaltiel Viola 18] new Implicit in 1 1 Erasures 𝛽 ∈ (0,1) O Θ [Grinberg Shaltiel 1 − 𝛽 1 − 𝛽 Viola 18] An improvement in dependence on 𝛽 was suggested by Venkat Guruswami 14

How does separating erasures from errors in local list decoding help with separating them in property testing?

3CNF Properties: Hard to Test, Easy to Decide • Formula 𝜚 𝑜 : 3CNF formula on 𝑜 variables, 𝜄(𝑜) clauses 0,1 𝑜 : set of satisfying assignments to 𝜚 𝑜 • Property 𝑄 𝜚 𝑜 ⊆ Theorem [Ben-Sasson Harsha Raskhodnikova 05] For sufficiently small ε , ε -testing 𝑄 𝜚 𝑜 requires 𝛁 𝒐 queries. • 𝑄 𝜚 𝑜 decidable by an 𝐏(𝒐) -size circuit. 16

Testing with Advice: PCPs of Proximity (PCPPs) [Ergun Kumar Rubinfeld 99, Ben-Sasson Goldreich Harsha Sudan Vadhan 06, Dinur Reingold 06] 𝑦 proof 𝜌(𝑦) ? ? PCPP Verifier • If 𝑦 has the property, then ∃𝜌(𝑦) for which verifier accepts. • If 𝑦 is 𝜁 -far, then ∀𝜌(𝑦) verifier rejects with probability ≥ 2/3 . Theorem Every property decidable with a circuit of size 𝒏 has PCPP with proof length 𝑷(𝒏) and constant query complexity. 17

Testing 3CNF Properties with/without a Proof 𝑦 Need Ω(𝑜) ? queries to test ε without a proof Tester for 𝑆 𝜚 𝑜 𝑦 proof 𝜌(𝑦) ? ? ε Constant query PCPP Verifier complexity with a proof of length for 𝑆 𝜚 𝑜 𝑃(𝑜) 18

Separating Property 𝑦 r Enc( 𝑦 ∘ 𝜌(𝑦) ) • 𝑦 satisfies the hard 3CNF property • 𝑠 is the number of repetitions (to balance the lengths of 2 parts) • 𝜌(𝑦) is the proof on which the PCPP verifier accepts 𝑦 • Enc uses a locally list erasure-decodable error-correcting code – E.g., Hadamard; – Codes with a better rate imply a stronger separation. 19

Separating Property: Erasure-Resilient Testing 𝑦 r Hadamard( 𝑦 ∘ 𝜌(𝑦) ) Idea: If a constant fraction (say, 1/4) of the encoding is preserved, we can locally list erasure-decode. Erasure-Resilient Tester 1. Locally list erasure-decode Hadamard to get a list of algorithms. 2. For each algorithm, check if: the plain part is 𝑦 𝑠 by comparing u.r. bits with the • corresponding bits of the decoding of 𝑦 PCPP verifier accepts 𝑦 ∘ 𝜌(𝑦) • 3. Accept if, for some algorithm on the list, both checks pass. Constant query complexity. 20

Separating Property: Hardness of Tolerant Testing 𝑦 r Hadamard( 𝑦 ∘ 𝜌(𝑦) ) Idea: Reduce standard testing of 3CNF property to tolerant testing of the separating property. • Given a string 𝑦 , we can simulate access to 𝑦 r 00000 … 00000 • All-zero string is Hadamard( 𝑦 ∘ 𝜌(𝑦) ) with 1/2 of the encoding bits corrupted! • Testing 3CNF property requires Ω 𝑜 queries, where 𝑜 = 𝑦 . The input length for separating property is 𝑂 ≈ 2 𝑑𝑜 . Ω 𝑜 ≈ Ω log 𝑂 queries are needed. 21

What We Proved The separating property is • erasure-resiliently testable with a constant number of queries, • but requires Ω(log 𝑂) queries to tolerantly test. Tolerant testing is harder than erasure-resilient testing in general. 22

Strengthening the Separation: Challenges If there exists a code that is locally list decodable from an 𝛽 < 1 fraction of erasures with • list size ℓ and number of queries 𝑟 that only depend on 𝛽 • inverse polynomial rate then there is a stronger separation: constant vs. 𝑂 𝑑 . The existence of such a code is an open question. The corresponding question for the case of errors is the holy grail of research on local decoding. 23