Toward Probabilistic Checking against Non-Signaling Strategies with Constant Locality
aa r X i v : . [ c s . CC ] S e p Toward Probabilistic Checking against Non-Signaling Strategieswith Constant Locality
Mohammad Mahdi Jahanara [email protected]
Simon Fraser University
Sajin Koroth sajin [email protected]
Simon Fraser University
Igor Shinkar [email protected]
Simon Fraser University
September 11, 2020
Abstract
Non-signaling strategies are a generalization of quantum strategies that have been studiedin physics over the past three decades. Recently, they have found applications in theoreticalcomputer science, including to proving inapproximability results for linear programming andto constructing protocols for delegating computation. A central tool for these applications isprobabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies .In this paper we show, assuming a certain geometrical hypothesis about noise robustness ofnon-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali-Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound againstnon-signaling strategies with constant locality .Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test ) in the non-signaling setting.
Keywords : direct product testing; linearity testing; non-signaling strategies; parallel repetition; prob-abilistically checkable proofs ontents t -nsPCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 t -repeated k -non-signaling function . . . . . . . . . . . . . . . . . 207.3 Self-correction is almost linear and almost consistent . . . . . . . . . . . . . . . . . . 217.3.1 d F ( t ) is permutation folded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3.2 d F ( t ) is almost linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3.3 d F ( t ) is almost consistent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Introduction
Probabilistically Checkable Proofs (PCPs) [BFLS91, FGL +
96, AS98, ALM +
98] are proofs thatcan be verified by a probabilistic verifier that queries only a few locations of the proof. PCPshave been a powerful tool in the theory of computing, with applications in diverse areas such ashardness of approximation [FGL +
96] and delegation of computation [Kil92, Mic00]. A seminalresult of [AS98, ALM + T ( n ) has a PCP system which allows to check if a giveninput of length n is in the language by using O (log( T ( n ))) random bits and making only O (1)queries to the given proof.Recall that in the classical setting of PCPs the two standard requirements are completeness andsoundness. Completeness requires that if a given input is in the language, then there is some proofthat convinces the prover with probability 1.
Soundness requirement states that if the input is notin the language, the prover rejects any proof with some significant probability. In this paper westudy PCP systems that are sound against non-signalling proofs or non-signalling strategies , i.e.,we require the prover to reject any non-signalling proof with some significant probability.Non-signalling strategies are a certain restricted class of probabilistic oracles. When such oracleis given a set of queries, the response to the queries is sampled from a distribution such that theanswer to each query may depend on all queries. More precisely, a non signalling strategy withlocality k is a collection F = { F S } S ⊆ D, | S |≤ k , where each F S is a distribution over Σ S (i.e., overfunctions f : S → Σ), and for any two subsets
S, T ⊆ D of size at most k , the restrictions of F S and F T to S ∩ T are equal as distributions. This setting stands in contrast to the standard notionof a classical proof, where the answer to each query is deterministic. Note that if the locality isthe maximum possible, i.e., k = | D | , then F is a distribution over functions, which is (essentially)equivalent to the classical notion of a proof.We note that one may think about k -non-signalling functions, equivalently, as the class of allfeasible solutions to the linear program arising from the k ’th level relaxation of the Sherali-Adamshierarchy [SA90]. This implies that computing the maximum acceptance probability of an nsPCPverifier that uses r random bits, where the maximum is taken over all k -non-signaling proofs, reducesto a linear program with 2 O ( r · k ) · Σ O ( k ) variables and constraints. In particular, if a language L hasa PCP verifier that on an input of length n uses r = O (log( n )) random bits, and is sound against O (1)-non-signaling proofs over an alphabet of constant size, then L is decidable in time poly ( n ).Non-signaling strategies have been studied in physics since 1980’s [Ras85, KT85, PR94] in orderto better understand quantum entanglement. Indeed, these strategies strictly generalize quantumstrategies and capture minimal requirements on “non-local” correlations that rule out instantaneouscommunication.PCP systems that are sound against non-signalling proofs have recently found numerous appli-cations in theoretical computer science, including schemes for 1-round delegation of computationfrom cryptographic assumptions [KRR13, KRR14], and hardness of approximation for linear pro-gramming [KRR16]. However, as opposed to the well studied setting of the classical PCP theorem,where there are many constructions achieving best parameters possible, in the non-signalling settingmany parameters of the known PCP constructions appear to be far from optimal.One of the most important parameters associated with a non-signalling proof is the localityparameter, denoted by k . Indeed, [KRR13, KRR14] have studied the related notion of multi-proverinteractive proofs that are sound against non-signaling strategies (nsMIPs). They have shown thatnsPCPs are essentially equivalent to nsMIPs where k , the locality of the proof in the nsPCP setting,3xactly corresponds to the number of provers in the nsMIP setting.Despite the importance of the locality parameter, the exact complexity of languages admittingnsPCPs that are sound against k -non-signaling proofs is still open for most k ’s. Note that as thelocality of the proof decreases, there are fewer constraints imposed on the proof, and hence the taskof the verifier becomes more challenging. The seminal result of Kalai, Raz, and Rothblum [KRR13,KRR14] showed that every language in DTIME ( T ) has an nsPCP verifier that uses polylog ( T )random bits, makes polylog ( T ) queries to a proof of length poly ( T ), and is sound against polylog ( T ) -non-signaling proofs. In particular, every language in EXP is captured by a nsMIP with a polynomialtime randomized verifier who communicates with poly ( n ) non-signaling provers. For the limitationsof nsPCPs, Ito [Ito10] proved that for k = 2 the corresponding linear program is solvable in PSPACE ,which is tight by the result of [IKM09], and hence the class
PSPACE is captured by PCPs that aresounds against 2-non-signaling proofs. Much less is known about the power of PCP systems thatare sound against k -non-signaling proofs for k >
2. Recently, Holden and Kalai [HK20] proved that o ( p log( n ))-prover non-signalling proofs with negligible soundness is contained in PSPACE .All these results give rise to the following question, raised in [CMS19], asking for the non-signaling analogue of the PCP theorem.
Question 1.1.
Is it true that every language in
DTIME ( T ) has an nsPCP verifier that uses O (log( T )) random bits, makes O (1) queries to the proof, and is sound against O (1) -non-signallingfunctions? Motivated by this problem, Chiesa et. al [CMS19, CMS20] started a systematic study of non-signalling PCPs. They proposed studying the classical (algebraic) PCP constructions and theirbuilding blocks (which are very well understood in the classical setting), and adapting each of thebuilding blocks to the non-signaling setting. In particular, focusing on the PCP construction of[ALM +
98] they made an appropriate definition of linear non-signalling functions and analyzed thelinearity test of [BLR93] against non-signalling strategies [CMS20]. Then, building on the linearitytest, they proved in [CMS19] that the classical exponential length O (1)-query PCP of [ALM + O (log ( N ))-non-signalling proofs. We emphasize, that even for exponential lengthnsPCPs (corresponding to nsPCPs with r = poly ( n ) randomness), there are no known constructionsthat are sound against O (1)-non-signaling proofs. Given this state of affairs, it is natural to askthe following question, that is simpler than Question 1.1 Question 1.2.
Is it true that every language in
DTIME ( T ( n )) has an nsPCP verifier that uses O ( poly ( T ( n )) random bits, makes O (1) queries to the proof, and is sound against O (1) -non-signallingfunctions? One must be careful with the precise formulation of Question 1.2. Note that if the verifier usesmore than T ( n ) random bits, the runtime spent on reading the randomness is more than T ( n ),which is the time complexity of the problem. To recover a nontrivial question, we require theverifier to be input oblivious . That is, in order to decide whether an instance x belongs to thegiven language L ∈ DTIME ( T ( n )), the verifier generates the queries based only on the length ofthe input x and its randomness (but not the input itself), and then rules according to an o ( T )-timedecision predicate (where the predicate does depend on x ). Indeed, the [ALM +
98] verifier studiedin [CMS19] is input oblivious.In this work we build on the work of [CMS19] and provide a positive answer to Question 1.2 assuming a certain geometric hypothesis . Specifically, we construct an input oblivious nsPCP4erifier for any language L ∈ DTIME ( T ( n )) that uses poly ( T ( n )) random bits, makes O (1) queriesto a given proof, and is sound against O (1)-non-signalling functions, with two caveats .1. The first is that the alphabet of the nsPCP system is Σ = { , } polylog ( T ( n )) , instead of the binaryalphabet employed by [CMS19, KRR14, ALM + polylog ( T ( n )) bits from the proof, which makes our result non-trivial. Also, recall thatin the classical setting, we have the alphabet reduction technique using proof composition, andit is plausible that we can apply similar ideas also in the non-signaling setting. Indeed, proofcomposition is an important building block in the classical PCP literature, and we believe it willalso be an important step toward resolving Question 1.1.2. The second caveat is that our result depends on a certain quantitative geometric hypothesis aboutproximity between almost non-signaling proofs and exactly non-signaling proofs. Equivalently,the hypothesis says that every feasible solution for the noisy Sherali-Adams LP is close (insome precise, rather weak, sense) to a feasible solution for the (exact) Sherali-Adams LP. SeeHypothesis 2 for details, and the discussion in Appendix A.Our work follows the general philosophy of [CMS19, CMS20], who proposed building modularanalogues of tools and techniques from the classical PCP literature. A classical tool used in theconstruction of PCPs is parallel repetition [Raz98, Hol09]. In the classical setting of 2-query PCP,parallel repetition is used to reduce the soundness error. In this work we use parallel repetitionfor non-signalling proofs to reduce the locality to O (1), while the soundness stays in the “high-probability acceptance regime”. In addition to parallel repetition, we study additional tools fromthe PCP literature. Specifically, we use the modular approach that is typical for the classicalsetting. Specifically, we show first that the parallel repetition of the [ALM +
98] verifier is soundagainst “nicely structured” proofs. Then, we use linearity test and direct product test , and claimthat proofs that satisfy both tests with high probability must be nicely structured, and hence weessentially reduce the analysis to the structured case.Another interesting feature of our proof is the reduction from the parallel repetition of the [ALM + +
98] verifier. Specifically, we show that if for some input x , theparallel repetition of the [ALM +
98] verifier accepts a proof with high probability, and the proof is“nicely structured”, then it is possible to “flatten” the repeated proof into a proof over the binaryalphabet, that satisfies the (non-repeated) [ALM +
98] verifier with high probability. Therefore, byapplying the result of [CMS19] about the soundness of the [ALM +
98] verifier, we conclude that theinput x is in the language. Below we discuss the main result of the paper. Our result depends on an hypothesis about approx-imating almost non-signaling functions using exactly non-signaling functions.
Hypothesis 1 (Informal) . Any almost linear, almost non-signaling function F : { , } n → { , } can be well approximated by some non-signaling function F ′ : { , } n → { , } of slightly lowerlocality.Equivalently, any solution to the noisy Sherali-Adams LP can be well approximated by a solutionto the (exact) Sherali-Adams LP of slightly lower level in the hierarchy. non-signaling and almost non-signaling functions (or, equivalently, the related notions of noisy Sherali-Adams LP ),as well as the appropriate definitions of distance. For the formal statement of the hypothesis seeHypothesis 2 following the required definitions in Section 2.We are now ready to state our main theorem. Theorem 1 (Main theorem - informal) . Assuming Hypothesis 1 every language L ∈ DTIME ( T ) hasan input oblivious nsPCP verifier that on an input of length n uses e O ( T ) random bits, makes O (1) queries to proofs over the alphabet Σ = { , } polylog ( T ) , and is sound against O (1) -non-signalingproofs. The query sampler runs in time e O ( T ) , and the decision predicate runs in time O ( n · polylog ( T )) . To the best our knowledge, this is the first result that constructs a PCP system that is soundagainst non-signaling proofs with constant locality.
The rest of the paper is organized as follows. In Section 2 we formally define the notions that weutilize throughout this work, and use them to formally state our hypothesis and the main theorem inSection 3. In Section 4 we recall the ALMSS verifier, and define our variant of its parallel repetition.In Section 5 we provide an overview of the soundness proof. In Section 6 we prove soundness ofour verifier against structured proofs. In Section 7 we discuss our local testing and self-correction,which enables us to reduce soundness against general proofs to soundness against structured proofs.Finally, in Section 8 we prove the main result.
We start with the definition of Probabilistically Checkable Proofs (PCPs). Recall that a classicalPCP verifier for a language L is given an input x , and an oracle access to a proof. The verifierreads the input, uses randomness, queries the proof in a small number of coordinates, and basedon the answers to the queries decides whether to accept or reject. Completeness requires that if x ∈ L , then there exists a proof that makes the verifier always accept. Soundness requires that if x L , then for any proof the verifier will reject with high probability.In the non-signaling setting, a non-signaling PCP verifier is a verifier, whose soundness isfurther required to hold against any non-signaling proof of prescribed locality. More precisely, annsPCP verifier V for a language L gets an input x and an oracle access to a non-signaling function F : D → Σ. The verifier reads the input x , uses random bits to decide on a subset S ⊆ D on which F is queried. Then, based on the answer F ( S ) ∈ Σ S it decides to accept or reject. Definition 2.1. A nsPCP verifier for a language L ⊆ { , } ∗ is a randomized algorithm V thatgets an input x ∈ { , } n and oracle access to a k -non-signaling proof F : D → Σ . The verifieruses randomness to decide on a subset S ⊆ D of size | S | ≤ k , and queries F on S . Then, based onthe answer F ( S ) ∈ Σ S it decides to accept or reject. We say that V has perfect completeness andsoundness error γ against k -non-signaling proofs if the following holds. Completeness:
For all x ∈ L there exists a (classical) proof π such that Pr[ V π ( x ) = 1] = 1 . oundness: If x / ∈ L , then for all k -non-signaling proofs F it holds that Pr[ V F ( x ) = 1] ≤ γ .We say that verifier V is input oblivious if the choice of the query set S depends only on the inputlength n , the randomness of the verifier, but is independent of x . Remark 2.2.
Note that in the non-signaling setting the locality parameter k upper bounds thenumber of queries made by the verifier, and it is possible that the actual predicate used by theverifier depends on significantly less than k coordinates of the proof. For example, [CMS19] provedthat the 11-queries verifier of [ALM +
98] is sound against O (log ( n ))-non-signaling proofs, and it isnot known whether the verifier is sound against O (1)-non-signaling proofs, or even o (log ( n ))-non-signaling proofs. In the classical setting a proof is assumed to be a string, or equivalently, a static function π : D → Σcommitted by the prover. A t -parallel repetition of a proof π is a mapping π t : D t → Σ t that allowsaccessing t locations of the (supposed) proof by making only 1 query to a (longer) proof over a largeralphabet. That is, the intended proof π ( t ) corresponds to some “base” proof π : D → Σ defined as π ( t ) (( x , . . . , x t )) = ( π ( x ) , . . . , π ( x t )). Analogously, given a verifier V , a t -repeated verifier whichis denoted by V ( t ) , runs t parallel independent instances of V and accepts if and only if all instancesaccept.The original motivation for using parallel repetition was to reduce the soundness error of a proofsystem, while keeping the number of queries fixed. In the classical setting, if the repeated proofis indeed a parallel repetition of some base proof π , then it is not hard to see that the soundnesserror of V ( t ) π t is exponentially smaller than the soundness error of V π . The soundness analysis of therepeated proof need not be based on this comparison to the soundness error of the base proof, andanalyzing such proofs in both classical and non-signalling settings has been a subject of a long lineof research [Ver96, Raz98, Hol09, DS14a, BG15, LW16, HY19].In this work, we use parallel repetition to improve the minimum locality parameter of non-signaling proofs required for the soundness of the verifier, rather than its soundness error. Next,we formally define non-signaling proofs, and some properties of such proofs that we will need inthe paper. In this work we consider PCP verifiers that are sound against non-signaling proofs. Below, weformally define the notion of non-signaling functions, and introduce some notation we will use inthe paper. Throughout the paper we will use terms non-signaling function , non-signaling proof ,and non-signaling strategy interchangeably. Definition 2.3.
Fix a domain D , an alphabet Σ , and a parameter k ∈ N . A k -non-signalingfunction F : D → Σ is a collection F = {F S } S ⊆ D, | S |≤ k , where each F S is a distribution overassignments f S : S → Σ , such that for every two subsets S, T ⊆ D each of size at most k , themarginal distributions of F S and F T restricted to S ∩ T are equal. Unlike a classic function, we can use a k -non-signaling function only once in the sense that onehas to present the set of at most k queries all at once. In other words, it is not possible to use thenon-signaling function adaptively. 7 emark 2.4. Throughout the paper we will consider non-signaling functions of two types: • functions over the domain D = { , } N for some N ∈ N and alphabet Σ = { , } ; • functions over the domain D = ( { , } N ) t and alphabet Σ = { , } t for some parameters N, t ∈ N .Next, we define a relaxed notion of non-signaling functions, that allows the marginal distribu-tions induced by different query sets to be only statistically close rather equal on the intersection.This relaxation arises in our analysis. It has also appeared naturally in other works in this area,especially in cryptographic applications [ABOR00, DLN +
04, KRR13, KRR14].
Definition 2.5.
Fix a domain D , an alphabet Σ , and parameters k ∈ N and ε ∈ [0 , . A ( ε, k ) -non-signaling function over a domain D and an alphabet Σ , is a collection F = {F S } S ⊆ D, | S |≤ k ,where each F S is a distribution over assignments f S : S → Σ , such that for every two subsets S, T ⊆ D each of size at most k , the marginal distributions of F S and F T restricted to S ∩ T are ε -close with respect to total variation distance, i.e., max E ⊆ Σ S ∩ T (cid:12)(cid:12)(cid:12)(cid:12) Pr F S [ F S | S ∩ T ∈ E ] − Pr F T [ F T | S ∩ T ∈ E ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . In particular, a ( ε = 0 , k ) -non-signaling-function coincides with the definition of k -non-signalingfunction from Definition 2.3. Next we define non-signaling and almost non-signaling counterpart of parallel repeated functions.
Definition 2.6.
Fix a domain D , an alphabet Σ , and parameters k, t ∈ N . A t -repeated ( δ, k ) -non-signaling function is an ( δ, k ) -non-signaling function F ( t ) : D t → Σ t . Namely, a t -repeated ( δ, k ) -non-signaling function F ( t ) : D t → Σ t is a collection F ( t ) = {F ( t ) S } S ⊆ D t , | S |≤ k , where each F ( t ) S is a distribution over assignments f ( t ) S : S → Σ , such that for every two subsets S, T ⊆ D t eachof size at most k , the marginal distributions of F ( t ) S and F ( t ) T restricted to S ∩ T are δ -close withrespect to total variation distance. We will also need the definition of distance between non-signaling or almost non-signalingfunctions.
Definition 2.7 (Statistical distance) . Let F , F ′ : D → Σ be two non-signaling or almost non-signaling functions with locality k . For ℓ ≤ k the ∆ ℓ -distance between F and F ′ is defined as ∆ ℓ ( F , F ′ ) = max S ⊆ D, | S |≤ ℓ ∆( F S , F ′ S ) , where ∆( F S , F ′ S ) = max E ⊆ Σ S | Pr[ F S ∈ E ] − Pr[ F ′ S ∈ E ] | is the total variation distance between F S and F ′ S .We say that F and F ′ are ε -close in the ∆ ℓ -distance if ∆ ℓ ( F , F ′ ) ≤ ε , and say that they are ε -far otherwise. Folding is a technique used to impose some structure on the given proof without really making extraqueries. The idea of using folded proofs was first introduced by [BGS98]. We formally define the permutation folding property, and then explain why we can impose this property without makingextra queries. 8 efinition 2.8.
Let Q = ( q , . . . , q t ) ∈ D t be a D -values vector, and let π ∈ S t be a permutationof the indices [ t ] . Define π ( Q ) = ( q π (1) , . . . , q π ( t ) ) to be the vector obtained from Q by permuting thecoordinates according to π .Let F ( t ) : ( D n ) t → Σ t be a t -repeated k -non-signaling function. F ( t ) is said to be permutationfolded or permutation invariant if for any S = { Q , . . . , Q ℓ } ⊆ ( D n ) t with ≤ ℓ ≤ k , for any T = { π ( Q ) , . . . , π ℓ ( Q ℓ ) } for some permutations π , . . . π ℓ ∈ S t , and for any b , . . . , b ℓ ∈ Σ t it holdsthat Pr h ∀ i ∈ [ ℓ ] F ( t ) S ( Q i ) = b i i = Pr h ∀ i ∈ [ ℓ ] F ( t ) T ( π i ( Q i )) = π i ( b i ) i . Observation 2.9.
It is important to note that we can fold any given t -repeated k -non-signalingfunction F ( t ) : D t → Σ t by partitioning D t into equivalence classes, where Q and Q ′ belong to thesame class if Q ′ = π ( Q ) for some permutation π .We defined the folding of F ( t ) , denoted by F ( t ) as follows. For any query Q to F ( t ) , let π ∈ S t be a uniformly random permutation, and define the distribution of F ( t ) ( Q ) as the distribution of π − ( F ( t ) ( π ( Q ))) .It is easy to see that F ( t ) is indeed k -non-signaling and permutation folded. Furthermore, notethat if F ( t ) is permutation folded, then F ( t ) = F ( t ) . In this part, we define linear t -repeated non-signaling functions. Linear non-signaling boolean functions have been studied in [CMS20, CMS19], and played a key role in the proving that thePCP verifier of [ALM +
98] is sound against non-signaling proofs. We also use such structurednon-signaling proofs in this paper. See Section 4 for details.
Definition 2.10 (Linear t -repeated functions) . Let L ( t ) : ( { , } n ) t → { , } t be a t -repeated ( ε, k ) -non-signaling function. We say that L ( t ) is linear if for all X, Y ∈ ( { , } n ) t , and X + Y ∈ ( { , } n ) t defined by the coordinate-wise addition modulo 2, and for all S ⊆ ( { , } n ) t containing X, Y, X + Y of size at most | S | ≤ k , it holds that Pr L ( t ) S h L ( t ) ( X ) + L ( t ) ( Y ) = L ( t ) ( X + Y ) i = 1 . Remark 2.11.
Note that in the degenerate case of t = 1 if a (non-repeated) k -non-signalingfunction F satisfies the linearity condition in Definition 2.10 then Pr [ F ( x ) + F ( y ) = F ( x + y )] = 1for all x, y ∈ { , } n , i.e., F satisfies the linearity test of [BLR93] with probability 1. Non-signalingfunctions satisfying this property have been the subject of work on linearity testing in the non-signaling setting [CMS20].Next we extend Definition 2.10 by introducing the notion of an almost linear t -repeated non-signalling function. Definition 2.12 (Almost linear t -repeated functions) . Let L ( t ) : ( { , } n ) t → { , } t be a t -repeated ( δ, k ) -non-signaling function. We say that L ( t ) is (1 − ε ) - linear if for all X, Y ∈ ( { , } n ) t , and X + Y ∈ ( { , } n ) t defined by the coordinate-wise addition modulo 2, and for all S ⊆ ( { , } n ) t containing X, Y, X + Y of size at most | S | ≤ k , it holds that Pr L ( t ) S h L ( t ) ( X ) + L ( t ) ( Y ) = L ( t ) ( X + Y ) i ≥ − ε .
9e will allow ourselves to use the informal term almost linear , when referring to a non-signalingfunction L ( t ) that is (1 − ε )-linear for some small ε . In this part, we define the notion of consistency for t -repeated k -non-signaling function. Definition 2.13 (Consistent t -repeated functions) . Let C ( t ) : D t → Σ t be a t -repeated k -non-signaling function. We say that C ( t ) is consistent , if for any Q, Q ′ ∈ D t it holds that Pr C ( t ) h C ( t ) ( Q ) j = C ( t ) ( Q ′ ) j ∀ j ∈ [ t ] such that Q j = Q ′ j i = 1 . Similarly to the almost linear property, we define the relaxed notion of almost consistent non-signalling function.
Definition 2.14 (Almost consistent t -repeated functions) . Let C ( t ) : D t → Σ t be a t -repeated k -non-signaling function. We say that C ( t ) is (1 − ε ) - consistent , if for any Q, Q ′ ∈ D t Pr C ( t ) h C ( t ) ( Q ) j = C ( t ) ( Q ′ ) j ∀ j ∈ [ t ] such that Q j = Q ′ j i ≥ − ε . We will allow ourselves to use the informal term almost consistent , when referring to a non-signaling function C ( t ) that is (1 − ε )-consistent for some small ε . Claim 2.15.
Let C ( t ) : D t → Σ t be a t -repeated k -non-signaling function for k ≥ , and supposethat C ( t ) is (1 − ε ) - consistent . Fix Q, Q ′ ∈ D t and let J = { j ∈ [ t ] : Q j = Q ′ j } . Then, for any event E ⊆ Σ J it holds that (cid:12)(cid:12)(cid:12) Pr[ C ( t ) ( Q ) J ∈ E ] − Pr[ C ( t ) ( Q ′ ) J ∈ E ] (cid:12)(cid:12)(cid:12) ≤ ε . Proof.
Note that Pr[ C ( t ) ( Q ) | J ∈ E ] ≥ Pr[ C ( t ) ( Q ) | J ∈ E ∧ C ( t ) ( Q ) | J = C ( t ) ( Q ′ ) | J ]= Pr[ C ( t ) ( Q ′ ) | J ∈ E ∧ C ( t ) ( Q ) | J = C ( t ) ( Q ′ ) | J ] ≥ Pr[ C ( t ) ( Q ′ ) | J ∈ E ] − ε , where the last inequality is by the assumption that C ( t ) is (1 − ε )-consistent. By symmetry, we alsoget the inequality in the other direction, and the claim follows.We observe that for D = { , } n and Σ = { , } (almost) linearity implies (almost) consistency.Specifically, we prove the following claim. Claim 2.16.
Let L ( t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function, and supposethat (i) L ( t ) is (1 − ε ) -linear, and (ii) Pr (cid:2) L ( t ) ( Q ) j = 0 ∀ j ∈ [ t ] such that Q j = 0 n (cid:3) > − ε for all Q ∈ ( { , } n ) t . Then, L ( t ) is (1 − ε ) -consistent when treated as a ( k − -non-signaling function.Proof. Let S ∈ ( { , } n ) t be a set of queries of size | S | ≤ k −
1. Let
Q, Q ′ ∈ S , and let J = { j ∈ [ t ] : Q j = Q ′ j . We show below thatPr C ( t ) Q,Q ′ h C ( t ) ( Q ) j = C ( t ) ( Q ′ ) j ∀ j ∈ J i ≥ − ε . S ′ = S ∪ { Q ′′ } , where Q ′′ = Q + Q ′ . In particular, Q ′′ j = 0 n for all j ∈ J .By the assumption of the claim we get that Pr (cid:2) L ( t ) ( Q ′′ ) j = 0 ∀ j ∈ J (cid:3) ≥ − ε . Therefore, usingthe assumption that L ( t ) is (1 − ε )-linear it follows thatPr h C ( t ) ( Q ) j = C ( t ) ( Q ′ ) j ∀ j ∈ J i ≥ Pr h L ( t ) ( Q ) j + L ( t ) ( Q ′ ) j = L ( t ) ( Q ′′ ) j ∧ L ( t ) ( Q ′′ ) j = 0 ∀ j ∈ J i ≥ − ε . Therefore, L ( t ) is (1 − ε )-consistent, as required. t -nsPCP Below we define the flattening operation, which transforms a given t -repeated proof into a non-repeated proof in the natural way. Namely, given a query set S to the non-repeated proof, wecreate a vector Q S containing all the elements of S , query the repeated proof on Q S , and respondaccording to the received answer. Definition 2.17.
Let F ( t ) : D t → Σ t be a k -non-signaling t -repeated proof. Define the flatteningof F ( t ) , denoted by e F = Flat [ F ( t ) ] : D → Σ as follows. For a query set S = { q , . . . , q s } ⊆ D of size s ≤ t , define a vector Q S whose first s entries are ( q , . . . , q s ) and the rest are set arbitrarily, query F ( t ) on the single query Q S , and let the distribution of e F ( S ) be e F ( S ) = ( F ( t ) ( Q S ) , . . . , F ( t ) ( Q S ) s ) . Claim 2.18.
Let C ( t ) : D t → Σ t be a k -non-signaling function that is permutation folded and (1 − ε ) -consistent for k ≥ . Then e F = Flat [ C ( t ) ] is a ( ε, t ) -non-signaling function.Furthermore, fix a query Q = ( w , . . . , w t ) ∈ D t for C ( t ) , a query set S ⊆ D of size s for F , alsolet ≤ ℓ ≤ t such that w , . . . , w ℓ are distinct and w j ∈ S for all j ∈ [ ℓ ] . Then, the distribution of F S ( { w , . . . , w ℓ } ) and ( C ( t ) ( Q ) , . . . , C ( t ) ( Q ) ℓ ) are ε -close in total variation distance.Proof. To prove that e F is ( ε, t )-non-signaling function let S, T ∈ D be two sets of queries, andsuppose S ∩ T = { w , . . . , w ℓ } . We want to show that for any event E ⊆ Σ S ∩ T it holds that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pr e F S [ e F S | S ∩ T ∈ E ] − Pr e F T [ e F T | S ∩ T ∈ E ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (1)Define Q S , Q T ∈ D t as in Definition 2.17, let π, π ′ ∈ S t be permutations such that for all j ∈ [ ℓ ]it holds that π ( Q S ) j = π ′ ( Q T ) j = w j . By non-signaling and permutation invariance of C ( t ) , if wequery it on { π ( Q S ) , π ′ ( Q T ) } we have:Pr e F S [ e F S | S ∩ T ∈ E ] = Pr h(cid:16) C ( t ) ( π ( Q S )) , . . . , C ( t ) ( π ( Q S )) ℓ (cid:17) ∈ E i Pr e F T [ e F T | S ∩ T ∈ E ] = Pr h(cid:16) C ( t ) ( π ′ ( Q T )) , . . . , C ( t ) ( π ′ ( Q T )) ℓ (cid:17) ∈ E i . Then, by Claim 2.15 we get the following: (cid:12)(cid:12)(cid:12) Pr h(cid:16) C ( t ) ( π ( Q S )) , . . . , C ( t ) ( π ( Q S )) ℓ (cid:17) ∈ E i − Pr h(cid:16) C ( t ) ( π ′ ( Q T )) , . . . , C ( t ) ( π ′ ( Q T )) ℓ (cid:17) ∈ E i(cid:12)(cid:12)(cid:12) ≤ ε e F is a ( ε, t )-non-signaling function.Next we prove the second part of the claim. Given S , define Q S ∈ D t as in Definition 2.17,and consider the query set { Q S , Q } to C ( t ) . Since C ( t ) is permutation folded, we may assume that Q j = Q Sj = w j for all j ∈ [ ℓ ]. Therefore, for any E ⊆ Σ ℓ we have: (cid:12)(cid:12)(cid:12) Pr h(cid:16) e F S ( w ) , . . . , e F S ( w ℓ ) (cid:17) ∈ E i − Pr h(cid:16) C ( t ) ( Q ) , . . . , C ( t ) ( Q ) ℓ (cid:17) ∈ E i(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Pr h(cid:16) C ( t ) ( Q S ) , . . . , C ( t ) ( Q S ) ℓ (cid:17) ∈ E i − Pr h(cid:16) C ( t ) ( Q ) , . . . , C ( t ) ( Q ) ℓ (cid:17) ∈ E i(cid:12)(cid:12)(cid:12) , which is upper bounded by ε by Claim 2.15. This complets the proof of Claim 2.18The following claim is follows rather immediately from Claim 2.18 above. Claim 2.19.
Let k ≥ , and let L ( t ) : ( { , } n ) t → { , } t be a k -non-signaling function that ispermutation folded, (1 − ε ) -linear, and (1 − ε ) -consistent. Then e L = Flat [ L ( t ) ] is a (non-repeated) ( ε , t ) -non-signaling (1 − ε − ε ) -linear function.Proof. By applying Claim 2.18 on L ( t ) , we get that e L = Flat [ L ( t ) ] is a ( ε , t )-non-signaling function.Next we prove that e L is (1 − ( ε + 3 ε ))-linear. Fix x, y ∈ { , } n , and let S ⊆ { , } n be a queryset for e L such that { x, y, x + y } ⊆ S . We want to prove thatPr[ e L ( x ) + e L ( y ) = e L ( x + y )] ≥ − ε − ε . (2)Let Q S be as in Definition 2.17. By the permutation folding property of L ( t ) we may assume thatthe first three coordinates of Q S are x, y, x + y . That is Q S = x, Q S = y , and Q S = x + y .By definition of Q S we have Pr[ e L ( x )+ e L ( y ) = e L ( x + y )] = Pr[ L ( t ) ( Q S ) + L ( t ) ( Q S ) = L ( t ) ( Q S ) ].Consider now the vectors Q x = ( x, n , n , . . . , n ), Q y = ( y, n , n , . . . , n ), and Q x + y = ( x + y, n , n , . . . , n ). Since L ( t ) is (1 − ε )-linear, we get that Pr[ L ( t ) ( Q x ) + L ( t ) ( Q y ) = L ( t ) ( Q x + y )] ≥ − ε . Since L ( t ) is (1 − ε )-consistent, it follows thatPr[ e L ( x ) + e L ( y ) = e L ( x + y )] = Pr[ L ( t ) ( Q S ) + L ( t ) ( Q S ) = L ( t ) ( Q S ) ] ≥ Pr[ L ( t ) ( Q x ) + L ( t ) ( Q y ) = L ( t ) ( Q x + y )] − ε ≥ − ε − ε , as required. In this section we formally state the main result of the paper. In order to describe the result weneed to first state the hypothesis conditioned on which our main theorem holds.
Hypothesis 2.
Fix integers n and k ≤ n , and let ε ∈ (0 , . For any ( ε, k ) -almost non-signaling function F : { , } n → { , } that is (1 − ε ) -linear there exists a k ′ -non-signaling function F ′ : { , } n → { , } such that ∆ ( F , F ′ ) ≤ ε ′ , where k ′ ≥ k e hyp for some positive absolute constant e hyp > , and ε ′ = ε ′ hyp ( ε ) is some function that depends only on ε such that ε ′ hyp ( ε ) → as ε → . Remark 3.1.
We make two remarks regarding the hypothesis.12
A statement analogous to Hypothesis 2 has been proven in [CMS20], showing that there exist a k -non-signaling function F ′ : { , } n → { , } such that ∆ k ( F , F ′ ) ≤ O (4 k · ε ). The multiplicativefactor of 4 k is too large, which makes it insufficient for our applications. • For our applications, we need a much weaker version of Hypothesis 2. We elaborate more on thehypothesis in Appendix A.For a computable function N : N → N we denote by SIZE ( N ) the complexity class of all lan-guages L having a uniform family of boolean circuits ( C n : { , } n → { , } ) n ∈ N of maximum fan-in2 with AND, OR, and NOT gates, such that C n has at most N ( n ) wires for all n ∈ N . Theorem 2 (Main theorem) . Assuming Hypothesis 2 every language L ∈ SIZE ( N ) has an inputoblivious nsPCP verifier that on input of length n uses e O ( N ) random bits, makes O (1) queries toproofs over the alphabet Σ = { , } polylog ( N ) , and is sound against O (1) -non-signaling proofs. Thequery sampler runs in time e O ( N ) , and the decision predicate runs in time O ( n · polylog ( N )) . Thatis, SIZE ( N ) ⊆ nsPCP soundness error: − Ω(1) randomness: e O ( N ) proof length: e O ( N ) query complexity: locality: O (1) query sampler time: e O ( N ) decision predicate time: O ( n · polylog ( N )) . It is clear that Theorem 1 follows from Theorem 2 since
DTIME ( T ) ⊆ SIZE ( O ( T log( T ))). In this section we formally describe our PCP construction. In one sentence, the PCP verifier getsa permutation invariant proof F ( t ) : ( { , } N ) t → { , } t , runs on it linearity test, direct producttest, and the parallel repetition of the ALMSS verifier, and accepts if and only if all tests accepts.We start by recalling the setting of the PCP verifier of [ALM +
98] (the “linear ALMSS verifier”).Let L ∈ SIZE ( N ) be a language, and let { C n } n ∈ N be a uniform family of boolean circuits with N = N ( n ) wires that decides L . That is, for all inputs x ∈ { , } n of length n it holds that C n ( x ) = 1 if and only if x ∈ L .For a given length n let C := C n be the circuit corresponding to the computation on in-puts of length n . The computation of C on the input x is viewed as a system of M := N + 1constraints { P j ( w ) = c j } j ∈ [ M ] over N boolean variables w = ( w , . . . , w N ) ∈ { , } N , where P , . . . , P M : { , } N → { , } are quadratic polynomials (each involving at most three variables in w ) and c , . . . , c M are boolean constants. Each variable represents the value of one of the wires of C during the computation on the input x . In particular, the first n variables, w , . . . , w n , correspondto the n input wires, and the variable w N corresponds to the output wire. The constraints are ofthree types: Input consistency:
For every j ∈ { , . . . , n } , P j ( w ) := w i and c j := x j . Note that our complexity measure for the size of a circuit is the number of wires, (and not the number of gates,which is more standard) as this measure directly affects the complexity of the PCP construction. However, for circuitswith bounded fan-in, the two quantities are equal up to a multiplicative constant factor. ate consistency: For every j ∈ { n + 1 , . . . , N } , • If the wire represented by the variable w j is an output of an AND gate g , where the inputsto g are given by w j , w j , then P j ( w ) := w j − w j · w j and c j := 0. • If the wire represented by the variable w j is an output of an OR gate g , where the inputsto g are given by w j , w j , then P j ( w ) := w j − w j − w j + w j · w j and c j := 0. • If the wire represented by the variable w j is an output of a NOT gate g , where the inputto g is given by w j , then P j ( w ) := w j − w j and c j := 1. Accepting output: P M ( w ) := w N and c M := 1.We overload notation, and use P j to also denote the upper triangular matrix in { , } N with P j ( w ) = h P j , w ⊗ w i That is, if P j ( w ) = P Ni =1 a i w i + P ≤ i
ALMSS verifier of [ALM +
98] is defined as follows.
Algorithm 1:
The linear ALMSS verifier
Exlpicit input :
A circuit C : { , } n → { , } with N wires, and input x ∈ { , } n to C . Oracle access : A k -non-signaling linear function L ( t ) : { , } N → { , } . Use the circuit C and input x to construct the matrices P , . . . , P M ∈ { , } N and constants c , . . . , c M ∈ { , } representing the computation of C on x . Sample u, v ∈ { , } N and s ∈ { , } M uniformly and independently at random. Query the oracle L on the 4-element set S = { D u , D v , u ⊗ v, P Mj =1 s j P j } . return ACCEPT if and only if L ( D u ) L ( D v ) = L ( u ⊗ v ) and L ( P Mj =1 s j P j ) = P Mj =1 s j c j .That is, the verifier makes 4 queries to a linear proof L : { , } N → { , } (of exponential length). Completeness.
Completeness of the ALMSS verifier is the same as in the classical setting. Indeed, C ( x ) = 1, then the classical proof defined by the design is accepted with probability 1. Soundness.
For soundness, Chiesa et al. [CMS19] proved that this construction is indeed soundagainst linear O (log N )-non-signaling proofs with soundness error bounded below 1. Theorem 4.1 (Theorem 6 in [CMS19]) . For any language L ∈ SIZE ( N ) there is an input obliviousPCP system, where the verifier gets as an explicit input a circuit C of size N = N ( n ) deciding L and an input x ∈ { , } n , and an oracle access to a linear proof π { , } O ( N ) → { , } . The verifieruses O ( N ) random coins, makes queries to the proof that are independent of x . If x ∈ L , thenthere exists a (classical) proof that causes the verifier to accepts with probability . If x L , thenfor any O (log( N )) -non-signaling linear proof the verifier to accepts with probability at most / . hat is, we have SIZE ( N ) ⊆ nsLPCP soundness error: / randomness: O ( N ) proof length: O ( N ) query complexity: locality: O (log N ) query sampler time: O ( N ) decision predicate time: O ( n ) . Next, we consider the t -repeated parallel repetition of the linear ALMSS verifier. Specifically, theverifier samples t independent sets of queries, makes 4 queries to the PCP over the alphabet { , } t ,and accepts if and only if all t sets of answers satisfy the basic linear ALMSS verifier. Formally,the t -repetition of the linear ALMSS verifier is defined as follows. Algorithm 2:
The t -repeated linear ALMSS verifier Exlpicit input :
A circuit C : { , } n → { , } with N wires, and input x ∈ { , } n to C . Oracle access : A t -repeated k -non-signaling linear function L ( t ) : ( { , } N ) t → { , } t . Construct the matrices P , . . . , P M ∈ { , } N and constants c , . . . , c M ∈ { , } , representingthe computation of C on x . Sample u (1) , . . . , u ( t ) , v (1) , . . . , v ( t ) ∈ { , } N and s (1) , . . . , s ( t ) ∈ { , } M independently anduniformly at random. Let Q = ( D u ( i ) ) i ∈ [ t ] ; Q = ( D v ( i ) ) i ∈ [ t ] ; Q = ( u ( i ) ⊗ v ( i ) ) i ∈ [ t ] ; Q = ( P Mj =1 s ( i ) j P j ) i ∈ [ t ] . Query the oracle L ( t ) on the 4-element set S = { Q , Q , Q , Q } . Check that L ( t ) ( Q ) i · L ( t ) ( Q ) i = L ( t ) ( Q ) i ∀ i ∈ [ t ]. Check that and L ( t ) ( Q ) i = P Mj =1 s ( i ) j c j ∀ i ∈ [ t ]. return ACCEPT if and only if in the two previous steps all equalities hold.Here, just as in the previous case, the verifier makes 4 queries to a linear proof. However, now theproof is over the alphabet { , } t . Completeness.
Completeness of the repeated linear ALMSS verifier is clear. Indeed, if C ( x ) = 1,then we can take the parallel repetition of the intended classical linear proof, and it will satisfy therepeated linear ALMSS verifier with probability 1. Soundness.
For soundness we prove in Section 6 that if t ≥ O (log( N )), then the verifier is soundagainst O (1)-non-signaling proofs that are linear and consistent . The proof works by reducing tothe soundness of the non-repeated linear ALMSS verifier, Specifically, we consider a circuit C andan input x to C , and consider a t -repeated k -non-signaling proof that is accepted with probabilityat least γ . We show that if the proof is linear and consistent, then its flattening is a t -non-signaling(non-repeated) linear proof that satisfies the non-repeated ALMSS verifier with the same probability.Therefore, if γ ≥ /
40, then by Theorem 4.1 we conclude that C ( x ) = 1. So far we have assumed that the given non-signaling proof is linear and consistent. Below we showhow to discard this assumption, and prove Theorem 2 by constructing a PCP verifier that is sound15gainst arbitrary proofs. This is done by running (the parallel repetition of) the linearity test, theconsistency test, and then feeding (the self-corrected version of) the proof to the linear repeatedALMSS verifier from Algorithm 2. We describe the verifier formally below.
Algorithm 3:
The 2 t -repeated ALMSS verifier + consistency test Exlpicit input :
A circuit C : { , } n → { , } with N wires, and input x ∈ { , } n to C Oracle access :
A 2 t -repeated k -non-signaling linear function F (2 t ) : ( { , } N ) t → { , } t Sample uniformly random
X, Y ∈ ( { , } N ) t . Sample uniformly random
W, Z , Z ∈ ( { , } N ) t . Sample the four queries Q , Q , Q , Q ∈ ( { , } N ) t of the t -repeated linear ALMSS verifierfrom Algorithm 2. and let D LIN : ( { , } t ) → { ACCEP T, REJ ECT } be the correspondingpredicate. Define d F ( t ) : ( { , } N ) t → { , } t as in Definition 7.3, which makes two queries to F (2 t ) forevery query to d F ( t ) . Sample an input S ⊆ ( { , } N ) t to F (2 t ) corresponding to querying d F ( t ) on { Q , Q , Q , Q } . Query F (2 t ) on the set S ∪ { X, Y, Z + Y } ∪ { [ W ; Z ] , [ W ; Z ] } . Linearity test:
Check that F (2 t ) ( X ) + F (2 t ) ( Y ) = F (2 t ) ( X + Y ). Consistency test:
Check that F (2 t ) ([ W ; Z ]) | W = F (2 t ) ([ W ; Z ]) | W . Linear PCP verifier:
Interpret F (2 t ) ( S ) as the answers of d F ( t ) on the query set( { Q , Q , Q , Q } ), and check that d F ( t ) ( { Q , Q , Q , Q } ) satisfies D LIN . return ACCEPT if and only if all three steps above accept.That is, the verifier is almost the parallel repetition of the classical ALMSS verifier. The onlydifference is that our verifier makes 2 additional queries for the consistency test.
Completeness.
Completeness of the repeated ALMSS verifier is clear, as by design the expectedproof is linear, and hence F ( t ) satisfies the linearity constraint with probability 1. Furthermore, itfollows that d F ( t ) is equal to F ( t ) , and thus the predicate D LIN is also satisfied with probability 1.
Soundness.
We prove soundness of the PCP system in Algorithm 3 in Section 8. Specifically,we use Hypothesis 2,and prove that for t ≥ polylog ( N ) and k ≥ O (1), the verifier is sound against O (1)-non-signaling linear proofs. The proof works, again, by reducing to the soundness of the non-repeated linear ALMSS verifier. Specifically, we consider a circuit C and an input x to C , and provethat if the PCP verifier accepts a t -repeated k -non-signaling proof 1 − ε , then its flattening (or ratherthe flattening of its self-correction) is a ( ε, t )-non-signaling (non-repeated) proof F : { , } N →{ , } that is (1 − ε )-linear, and it satisfies the non-repeated linear ALMSS verifier with highprobability. By applying Hypothesis 2 we obtain a t Ω(1) -non-signaling (non-repeated) proof thatis (1 − ε ′ )-linear and satisfies the non-repeated linear ALMSS verifier from Algorithm 1 with highprobability. By applying a result from [CMS20] we conclude that F is close to a linear t Ω(1) -non-signaling (non-repeated) proof b F that also satisfies the linear ALMSS verifier with high probability,and thus, by Theorem 4.1 it follows that C ( x ) = 1. See Section 8 for details.16 Proof overview: Soundness
In this section we give an overview of the soundness analysis of the parallel repetition of the PCPverifier from Algorithm 3. Before describing the actual proof, we first consider soundness against structured proofs. Indeed, this is a common approach in the analysis of PCP systems. Specifically,we show first that the PCP verifier from Algorithm 2 is sound against such structured proofs. Thenwe use local testing and self-correction to show the proofs that are accepted by the verifier withhigh probability satisfy the desired properties.
Soundness of the linear t -repeated ALMSS verifier. Fix a circuit C : { , } N → { , } andlet x ∈ { , } N be an input to C . Consider a 2 t -repeated linear ALMSS verifier for C ( x ), andsuppose that L (2 t ) : ( { , } N ) t → { , } t is a 2 t -repeated linear and consistent proof such that2 t -repeated linear ALMSS verifier the accepts L (2 t ) with high probability. According to Claim 2.19it follows that we can “flatten” L (2 t ) into a t -non-signaling linear proof L : { , } N → { , } . Weshow that since L (2 t ) is acepted with high probability by the 2 t -repeated linear verifier, it followsthat the (non-repeated) linear ALMSS verifier accepts L with high probability. Therefore, byapplying the result of [CMS19] it follows that if t > c log( N ) for some (sufficiently large) constant c , then C ( x ) = 1. See Section 6 for details. General proofs - forcing consistency using extended linearity test.
Next we prove sound-ness of the general (i.e., non-linear) parallel repetition PCP against arbitrary O (1)-non-signalingproofs. The general approach is analogous to the approach used for analyzing PCPs, specifically,we first run a test that “forces” the proof to be (close to) linear and consistent, and then apply theanalysis of the linear proof in the previous paragraph.More concretely, we fix a circuit C : { , } N → { , } and an input x ∈ { , } N to C , and considera 2 t -repeated (non-linear) ALMSS verifier for C ( x ). Suppose that F (2 t ) : ( { , } N ) t → { , } t isa 2 t -repeated proof such that 2 t -repeated ALMSS verifier accepts F (2 t ) with high probability. Ourgoal is to prove that C ( x ) = 1, and our high level strategy to show it is the following:1. First we assume that the proof is permutation invariant as in Definition 2.8.2. Suppose the repeated ALMSS verifier accepts F (2 t ) with high probability 1 − ε . In particular,this implies that F (2 t ) passes linearity test with at least same probability.3. By applying the self-correction procedure, we obtain the self-correction of F (2 t ) . The self-correction of F (2 t ) , denoted by d F ( t ) , is a t -repeated b k -non-signaling proof for b k = Ω( k ) suchthat in order to make one query to d F ( t ) we make O (1) queries to F (2 t ) . We prove that d F ( t ) satisfies the following two properties.(a) d F ( t ) is (1 − O ( ε ))-linear, i.e., for all X, Y ∈ ( { , } N ) t it holds that Pr[ d F ( t ) ( X ) + d F ( t ) ( Y ) = d F ( t ) ( X + Y )]. That is, the self-correction transforms an average-case guarantee aboutlinearity testing F (2 t ) into a guarantee that d F ( t ) satisfies the linearity constraints for all X, Y, X + Y .(b) d F ( t ) is (1 − O ( ε ))-consistent, i.e., for all Q, Q ′ ∈ ( { , } N ) t with high probability d F ( t ) ( Q ) j = d F ( t ) ( Q ′ ) j for all j ∈ [ t ] such that Q j = Q ′ j . Here also, the average-case guarantee ofthe consistency test is converted into the worst-case guarantee holding for all Q, Q ′ ∈ ( { , } N ) t . 17. Next, we let e F = Flat [ d F ( t ) ] be the flattening of d F ( t ) . By Claim 2.18, e F : { , } N → { , } is aalmost linear ( O ( ε ) , t )-no-signaling function. Furthermore, using the fact that d F ( t ) is (1 − O ( ε ))-linear and is accepted by the repeated ALMSS verifier from Algorithm 2 with high probability,we prove that e F is accepted by the (non-repeated) ALMSS verifier from Algorithm 1 with highprobability.At this point we would like to apply Theorem 4.1 on d F ( t ) , and say that since e F is acceptedby the AMLSS verifier with high probability, it follows that C ( x ) = 1. However, the difficulty inapplying Theorem 4.1 is that e F is not necessarily non-signaling, but only almost non-signaling (seeDefinition 2.5 for reference). In order to still apply this result we use Hypothesis 2 to “round” e F into a non-signaling proof, and then apply Theorem 4.1 to conclude that C ( x ) = 1. Specifically,we do the following.5. Assuming Hypothesis 2, there exist a t ′ -no-signaling function F , which is close to e F . In partic-ular, F is an almost linear non-signaling function.6. Using the result of [CMS19] on linearity testing we get that for some ¯ k = Ω( √ t ′ ) there exists a¯ k -non-signaling linear proof L that is O ( qε )-close to F ′ on queries sets of size at most q ≤ ¯ k .7. By our choice of parameters, the locality of L is ¯ k = Ω( √ t ′ ) > C log( N ), and by the previousitem the linear ALMSS verifier accepts L with high probability. Therefore, using Theorem 4.1we conclude that C ( x ) = 1.This completes the overview of the proof. Below we describe each step in detail. In this section we prove that the t -repeated linear PCP verifier from Algorithm 2 is sound againstlinear consistent proofs. Specifically, we prove the following theorem. Theorem 3.
Fix a circuit C : { , } n → { , } with N wires, and input x ∈ { , } n to C . Let k ≥ , and t be positive integers such that t ≥ K log( N ) for some sufficiently large constant K > .Let L ( t ) : ( { , } N ) t → { , } t be a k -non-signaling t -repeated linear consistent proof, and supposethat the t -repeated ALMSS verifier accepts L ( t ) with probability ≥ / . Then C ( x ) = 1 .Proof. Let L ( t ) : ( { , } N ) t → { , } t be a k -non-signaling t -repeated linear consistent proof, andsuppose that the t -repeated ALMSS verifier accepts L ( t ) with probability ≥ / e L = Flat [ L ( t ) ] be the flattening of L ( t ) as per Definition 2.17. Since L ( t ) is linear andconsistent, it follows by Claim 2.19 that e L is a t -non-signaling linear function.Next we show that the non-repeated linear ALMSS verifier accepts e L with probability > / C ( x ) = 1.Indeed, consider the random choices of u, v ∈ { , } N and s ∈ { , } M in Algorithm 1, andlet Q ∗ ∈ ( { , } N ) t be a query to L ( t ) that contains the four queries Q ALMSS = { D u , D v , u ⊗ v, P Mj =1 s j P j } in its first 4 coordinates. That is, Q ∗ = D u , Q ∗ = D v , Q ∗ = u ⊗ v , and Q ∗ = P Mj =1 s j P j . Denoting by D LIN the predicated in Algorithm 1, by definition of the flattening opera-tion we havePr[ D LIN ( e L ( Q ALMSS ))] = Pr[ D LIN ( L ( t ) ( Q ∗ ) , L ( t ) ( Q ∗ ) , L ( t ) ( Q ∗ ) , L ( t ) ( Q ∗ ) ) = 1] . (3)18ext, consider the random choices of u (1) , . . . , u ( t ) , v (1) , . . . , v ( t ) ∈ { , } N and s (1) , . . . , s ( t ) ∈{ , } M in Algorithm 2, and let Q = ( D u ( i ) ) i ∈ [ t ] , Q = ( D v ( i ) ) i ∈ [ t ] , Q = ( u ( i ) ⊗ v ( i ) ) i ∈ [ t ] , and Q = ( P Mj =1 s ( i ) j P j ) i ∈ [ t ] be the queries made by the repeated ALMSS verifier.Since u (1) , v (1) and s (1) are distributed identically to the random choices of u, v ∈ { , } N and s ∈ { , } M in Algorithm 1, it follows by consistency of L ( t ) that Pr[ D LIN ( L ( t ) ( Q ∗ ) |{ , , , } ) = 1] isequal to Pr[ D LIN ( L ( t ) ( Q ) , L ( t ) ( Q ) , L ( t ) ( Q ) , L ( t ) ( Q ) ) = 1] , i.e., to the probability that D LIN accepts the responses of L ( t ) in the first coordinate of the parallelrepetition in Algorithm 2. However, since the verifier in Algorithm 2 accepts L ( t ) with probability ≥ /
40, it follows in particular, that the first coordinate is accepted with probability ≥ / D LIN ( e L ( Q ALMSS ))] ≥ / , and hence, by Theorem 4.1 we have C ( x ) = 1. This completes the proof of Theorem 3. As shown in Section 6, it is rather straightforward to construct a PCP system that is sound againstrepeated non-signaling proofs that are consistent and linear. Therefore, we would like to makesure that the given proof satisfies these properties. We “enforce” these properties in Algorithm 3by first running linearity test and consistency test on a given t -repeated non-signaling proof, andthen run the linear PCP on the self-correction of the given proof. Next, we show that if the testsaccept a given proof with high probability, then its self-correction (almost) satisfies the desiredproperties, hence reducing the problem to the structured case. In this section we analyze the testsand prove guarantees about the self-correction of any non-signaling function that passes the testwith high probability. Then, in Section 8 we use these results on testing and self-correction in orderto analyze the PCP system from Algorithm 3. Testing linearity.
Linearity test is a randomized algorithm that given an input function f ,queries it on 3 inputs and wishes decides whether f is linear or far from linear. The test wasfirst analyzed in [BLR93]. Bellare et al. in [BCH +
96] simplified the analysis and proved forany boolean function f , the probability that it passes the test is at most 1 − ∆( f ), where ∆( f )is the normalized Hamming distance of f to the closest linear function. Extension of [BLR93]linearity test to general groups and many other closely related problems have been studied sincethen [AHRS01, SW04, BCLR08, BKS +
10, DDG + F ( t ) : ( { , } n ) t → { , } t the test is as follows. Definition 7.1 (Linearity test [BLR93]) . Let F ( t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function. Linearity test works by uniformly sampling X, Y ∈ ( { , } n ) t , querying F ( t ) onset { X, Y, X + Y } , and checking that F ( t ) ( X ) + F ( t ) ( Y ) = F ( t ) ( X + Y ) , i.e., that for all j ∈ [ t ] itholds that F ( t ) ( X ) j + F ( t ) ( Y ) j = F ( t ) ( X + Y ) j .
19n the non-signaling setting, linearity test was analyzed by [CMS20] for boolean functions. Theyproved that any k -non-signaling boolean function F that passes the linearity test with probability1 − ε can be self-corrected to a ⌊ k/ ⌋ -non-signaling function b F that is 2 O ( k ) ε -close to a linear ⌊ k/ ⌋ -non-signaling function L . However, we cannot directly apply their result to our setting, asour functions are not boolean. Furthermore, adapting the approach of [CMS20] will give a linearnon-signaling function with the guarantee that the distance between b F and a truly linear function L is at most 2 O ( tk ) ε , which is too large for our application. Testing consistency.
Next we consider consistency test , whose goal is to check that a given t -repeated non-signaling function is (close to) consistent as per Definition 2.13. The test works asfollows. Definition 7.2 (Consistency test) . Let F (2 t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signalingfunction for an integer t . Consistency test chooses W, Z , Z ∈ ( { , } n ) t uniformly at random,queries F ( t ) on { [ W ; Z ] , [ W ; Z ] } , and checks that F ( t ) ([ W ; Z ]) | W = F ( t ) ([ W ; Z ]) | W . Similar tests have been studied in the literature in the context of
Direct product testing in along series of work [DR04, IKW12, DS14b, DN17, GCS19].We prove below that if a 2 t -repeated k -non-signaling proof F (2 t ) passes both the linearity testand the consistency test with probability 1 − ε , then its self-correction d F ( t ) is (1 − O ( ε ))-linearand (1 − O ( ε ))-consistent. That is, d F ( t ) is close to having the properties we need in order to provesoundness against repeated non-signaling proofs. Next, we discuss the notion of self-correction, andprove if F (2 t ) passes the tests with high probability, then d F ( t ) satisfies the desired properties. t -repeated k -non-signaling function Below we define the self-correction of a given t -repeated k -non-signaling function F ( t ) . Observethat if F ( t ) passes the linearity test with high probability 1 − ε , it does not necessarily imply thatit satisfies all linearity constraints with high probability, i.e., it does not imply that F ( t ) is (1 − ε ′ )-linear. As a simple example, one may consider the case when F ( t ) is a deterministic function that isobtained from a linear function by changing some small fraction of its outputs. The same applies tothe consistency test, i.e., satisfying the consistency constraints on average as opposed to satisfyingeach consistency constraint with high probability.A standard approach to transform the “average-case” guarantee of the tests into a “point-wise”guarantee is by employing the idea of self-correction . Next we define the notion of self-correctionsuitable for our tests. Definition 7.3.
Let F (2 t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function. The self-correction of F (2 t ) , is a t -repeated b k -non-signaling function d F ( t ) : ( { , } n ) t → { , } t , for b k ≤ k defined as follows.Given a query Q ∈ ( { , } n ) t , in order to sample d F ( t ) ( Q ) we uniformly choose R, W ∈ ( { , } n ) t ,query F (2 t ) on the set { [ R ; W ] , [ R + Q ; W ] } , and output the first half of ( F (2 t ) ([ R ; W ]) + F (2 t ) ([ R + Q ; W ]) .More generally, for a query set b S = { Q , . . . , Q s } of size s ≤ b k we sample R i , W i ∈ ( { , } n ) t independently, uniformly at random for each i ∈ [ s ] , query F (2 t ) on the set S = s [ i =1 { [ R i ; W i ] , [ Q i + R i ; W i ] } , nd output d F ( t ) ( Q i ) j := ( F (2 t ) ([ R i ; W i ]) + F (2 t ) ([ Q i + R i ; W i ])) j ∀ j ∈ [ t ] . for all i ∈ [ s ] . Observe that the self-correction of F (2 t ) is indeed a non-signaling function with the appropriatelocality parameter. Indeed, this follows immediately from the assumption that F (2 t ) is k -non-signaling and the fact that the R i , W i ’s are uniformly random and independent. Next we show that if F (2 t ) passes both the linearity test and the agreement test with high probabilitythen its self-correction d F ( t ) is almost linear and almost consistent. Indeed, this average-to-worst-case is a standard step in the analysis of non-signaling PCPs [KRR14, CMS19]. Theorem 4.
Let F (2 t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function, and sup-pose that F (2 t ) is permutation folded. If F (2 t ) passes both the linearity and consistency tests withprobability at least − ε , then d F ( t ) is b k -non-signaling function that is permutation folded, (1 − ε ) -linear, and (1 − ε ) -consistent, for for b k = k/ − . The rest of this section is devoted to the proof of Theorem 4 d F ( t ) is permutation folded We first prove that if F (2 t ) is permutation folded, then d F ( t ) is also permutation-folded. (RecallDefinition 2.8 for the definition of the permutation folded property and the application of permu-tations on vectors.) Lemma 7.4.
Assuming F (2 t ) is permutation-folded, d F ( t ) is also permutation-folded.Proof. Fix S = { Q , . . . , Q ℓ } ⊆ ( { , } n ) t with 1 ≤ ℓ ≤ k , and let T = { π ( Q ) , . . . , π ℓ ( Q ℓ ) } forsome permutations π , . . . π ℓ ∈ S t . By definition of d F ( t ) for any b , . . . , b ℓ ∈ { , } t it holds thatPr h ∀ i ∈ [ ℓ ] d F ( t ) S ( Q i ) = b i i = Pr R i ,W i h ∀ i ∈ [ ℓ ] F (2 t ) ([ R i ; W i ]) + F (2 t ) ([ Q i + R i ; W i ]) = b i i = Pr R i ,W i h ∀ i ∈ [ ℓ ] F (2 t ) ([ π ( R i ) i ; W i ]) + F (2 t ) ([ π i ( Q i + R i ); W i ]) = b i i = Pr h ∀ i ∈ [ ℓ ] d F ( t ) T ( π i ( Q i )) = π i ( b i ) i , as required. d F ( t ) is almost linear Next, we show that if F ( t ) passes the linearity test with high probability, then its self-correction isalmost linear as per Definition 2.12. 21 emma 7.5. Let F (2 t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function such that k ≥ . If F (2 t ) passes the linearity test with probability at least − ε , then d F ( t ) is (1 − ε ) -linear. That is, for any query set b S = { X, Y, X + Y } ⊆ ( { , } n ) t we have Pr[ d F ( t ) ( X ) + d F ( t ) ( Y ) = d F ( t ) ( X + Y )] ≥ − ε . The proof is almost the same as in [CMS20] Theorem 12 (1 = ⇒ Proof.
For
X, Y ∈ ( { , } n ) t define Z = X + Y , and sample R X , R Y , R Z , W X , W Y , W Z ∈ ( { , } n ) t uniformly at random independently of each other. By definition of d F ( t ) we havePr[ d F ( t ) ( X ) + d F ( t ) ( Y ) = d F ( t ) ( X + Y )] ≥ Pr (cid:20) F (2 t ) ([ R X ; W X ]) + F (2 t ) ([ X + R X ; W X ])+ F (2 t ) ([ R Y ; W Y ]) + F (2 t ) ([ Y + R Y ; W Y ])= F (2 t ) ([ R Z ; W Z ]) + F (2 t ) ([ Z + R Z ; W Z ]) (cid:21) . Define S := { [ R X ; W X ] , [ R Y ; W Y ] , [ R Z ; W Z ] , [ X + R X ; W X ] , [ Y + R Y ; W Y ] , [ X + Y + R Z ; W Z ] } ,S := { [ R X ; W X ] , [ R Z ; W Z ] , [ X + R X + R Y ; W X + W Y ] , [ Y + R Y ; W Y ] , [ X + Y + R Z ; W Z ] } ,S := { [ R X ; W X ] , [ X + R X + R Y ; W X + W Y ] , [ Y + R Y + R Z ; W Y + W Z ] , [ X + Y + R Z ; W Z ] } ,S := { [ X + R X + R Y ; W X + W Y ] , [ Y + R Y + R Z ; W Y + W Z ] , [ X + Y + R X + R Z ; W X + W Z ] } . Note that | S i ∪ S i +1 | ≤ ≤ k for i = 1 , , add ( · ) be the addition function, and consider the sets S and S . ThenPr[ add ( F (2 t ) ( S )) = add ( F (2 t ) ( S ))]= Pr[ F (2 t ) ([ X + R X + R Y ; W X + W Y ] + F (2 t ) ([ X + R X ; W X ]) = F (2 t ) ([ R Y ; W Y ])] . Observing that the distribution on the right hand side is exactly as in the linearity test, we getthat Pr[ add ( F (2 t ) ( S )) = add ( F (2 t ) ( S ))] ≥ − ε . Similarly, we havePr[ add ( F (2 t ) ( S )) = add ( F (2 t ) ( S ))]= Pr[ F (2 t ) ([ R Z ; W Z ] + F (2 t ) ([ Y + R Y ; W Y ]) = F (2 t ) ([ Y + R Y + R Z ; W Y + W Z ])] ≥ − ε , and Pr[ add ( F (2 t ) ( S )) = add ( F (2 t ) ( S ))] 22 Pr[ F (2 t ) ([ R X ; W X ] + F (2 t ) ([ X + R Z ; W Z ]) = F (2 t ) ([ X + Y + R X + R Z ; W X + W Z ])] ≥ − ε . Therefore, (cid:12)(cid:12)(cid:12)
Pr[ add ( F (2 t ) ( S )) = 0] − Pr[ add ( F (2 t ) ( S )) = 0] (cid:12)(cid:12)(cid:12) ≤ X i =1 (cid:12)(cid:12)(cid:12) Pr[ add ( F (2 t ) ( S i )) = 0] − Pr[ add ( F (2 t ) ( S i +1 )) = 0] (cid:12)(cid:12)(cid:12) ≤ ε . Finally, note that Pr[ add ( F (2 t ) ( S )) = 0] ≥ − ε , because the distribution of S is equal to the distribution of a three tuple used for linearity testing.Therefore, Pr[ d F ( t ) ( X ) + d F ( t ) ( Y ) = d F ( t ) ( X + Y )] ≥ Pr[ add ( F (2 t ) ( S )) = 0] ≥ − ε , as required. d F ( t ) is almost consistent Finally, we prove in Lemma 7.7 that if F ( t ) passes the consistency test with high probability, thenits self-correction is almost consistent. Before proving it we need the following claim. Claim 7.6.
Let F (2 t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function such that k ≥ . Suppose that F (2 t ) passes both linearity and consistency tests with probability at least − ε .Then for any Q ∈ ( { , } n ) t it holds that Pr h d F ( t ) ( Q ) j = 0 ∀ j ∈ [ t ] such that Q j = 0 n i > − ε Proof.
The key observation here is that for a uniformly random
R, W ∈ ( { , } n ) t it holds thatPr h F (2 t ) ([ Q + R ; W ]) j = F (2 t ) ([ R ; W ])) j ∀ j = t + 1 , . . . , t i ≥ − ε . (4)(Note that Eq. (4) does not follow from consistency testing since R and Q + R are not independent.)Indeed, let R ′ , R ′′ , W ′ ∈ ( { , } n ) t be sampled uniformly at random, independently of all otherrandom variables. Then, since F (2 t ) passes linearity test with probability at least 1 − ε , it followsthat with probability at least 1 − ε the following equalities hold: F (2 t ) ([ Q + R ; W ]) = F (2 t ) ([ Q + R ′′ ; W ′ ]) + F (2 t ) ([ R + R ′′ ; W + W ′ ]) F (2 t ) ([ R ; W ]) = F (2 t ) ([ R ′ ; W ′ ]) + F (2 t ) ([ R + R ′ ; W + W ′ ])If these two equalities hold, then F (2 t ) ([ Q + R ; W ]) + F (2 t ) ([ R ; W ]) = F (2 t ) ([ Q + R ′′ ; W ′ ]) + F (2 t ) ([ R ′ ; W ′ ])+ F (2 t ) ([ R + R ′′ ; W + W ′ ]) + F (2 t ) ([ R + R ′ ; W + W ′ ]) . { [ Q + R ′′ ; W ′ ] , [ R ′ ; W ′ ] } are distributed as in the consistency test, it followsthat Pr h F (2 t ) ([ Q + R ′′ ; W ′ ]) j = F (2 t ) ([ R ′ ; W ′ ])) j ∀ j = t + 1 , . . . , t i ≥ − ε . (5)By the same argument we havePr h F (2 t ) ([ R + R ′′ ; W + W ′ ]) j = F (2 t ) ([ R + R ′ ; W + W ′ ])) j ∀ j = t + 1 , . . . , t i ≥ − ε . (6)These immediately imply Eq. (4).In order to complete the proof let π ∈ S t be an arbitrary permutation such that for all j ∈ [ t ], π ( j ) ∈ { t + 1 , . . . , t } . Then,Pr h d F ( t ) ( Q ) j = 0 ∀ j ∈ [ t ] such that Q j = 0 n i = Pr h F (2 t ) ([ Q + R ; W ]) j = F (2 t ) ([ R ; W ])) j ∀ j ∈ [ t ] such that Q j = 0 n i = Pr h F (2 t ) ( π ([ Q + R ; W ])) π ( j ) = F (2 t ) ( π ([ R ; W ]))) π ( j ) ∀ j ∈ [ t ] such that Q j = 0 n i ≥ − ε , where the last inequality follows from Eq. (4) together with the permutation invariance of F (2 t ) .The following lemma, saying that d F ( t ) is (1 − O ( ε ))-consistent, follows almost immediately fromClaim 7.6. Lemma 7.7.
Let F ( t ) : ( { , } n ) t → { , } t be a t -repeated k -non-signaling function such that k ≥ , and suppose that F (2 t ) is permutation folded.If F (2 t ) passes both linearity and consistency tests with probability − ε , then d F ( t ) is (1 − ε ) -consistent. That is, for any two queries X, Y ∈ ( { , } n ) t to d F ( t ) it holds that Pr h d F ( t ) ( X ) j = d F ( t ) ( Y ) j ∀ j ∈ [ t ] such that X j = Y j i ≥ − ε . Proof.
Let J = { j ∈ [ t ] : X j = Y j } . Consider the query set S = { X, Y, Z = X + Y } , and note that Z j = 0 n for all j ∈ J . Therefore, by Claim 7.6 it follows that Pr h d F ( t ) ( Z ) j = 0 ∀ j ∈ J i > − ε .By applying Lemma 7.5 we have Pr[ d F ( t ) ( X )+ d F ( t ) ( Y ) = d F ( t ) ( Z )] ≥ − ε . Therefore, by the unionbound we conclude that Pr h d F ( t ) ( X ) j = d F ( t ) ( Y ) j ∀ j ∈ J i ≥ − ε , thus concluding the proof ofLemma 7.7.Theorem 4 is an immediate conclusion from Lemma 7.4, Lemma 7.5, and Lemma 7.7. Below we prove Theorem 2. Specifically, we show that assuming Hypothesis 2 the PCP constructionin Algorithm 3 is sound against O (1)-non-signaling proofs. Theorem 2 follows immediately fromthe following statement. 24 heorem 8.1. Fix a circuit C : { , } n → { , } with N wires, and input x ∈ { , } n to C . Let k ≥ be a sufficiently large positive constant, and t be a positive integer such that t ≥ K log / e hyp ( N ) for some sufficiently large constant K > . Let F (2 t ) : ( { , } N ) t → { , } t be a k -non-signaling t -repeated linear consistent proof, and suppose that F (2 t ) is permutation invariant. If t -repeatedALMSS verifier from Algorithm 3 accepts F (2 t ) with probability ≥ − ε for some sufficiently small ε , then C ( x ) = 1 . The proof follows the steps outlined in Section 5.
Proof.
Fix a 2 t -repeated k -non-signaling proof F (2 t ) that satisfies the repeated ALMSS verifier fromAlgorithm 3 with probability at least 1 − ε . In particular, F (2 t ) passes the linearity test and theconsistency test with probability at least 1 − ε .By applying Theorem 4 we conclude that d F ( t ) , the self-correction of F (2 t ) , is a 4-no-signalingfunction that is (1 − ε )-linear and (1 − ε )-consistent. Furthermore, d F ( t ) satisfies the linear t -repeated verifier from Algorithm 2 with probability at least 1 − ε .Define e F = Flat [ d F ( t ) ] to be the flattening of d F ( t ) , as per Definition 2.17. Then, by Claim 2.19the function e F : { , } N → { , } is (1 − (4 + 3 · ε )-linear (8 ε, t )-non-signaling. Furthermore, since d F ( t ) is (1 − ε )-consistent, and satisfies the linear repeated verifier from Algorithm 2 with probabilityat least 1 − ε , it follows that e F satisfies the (non-repeated) linear verifier from Algorithm 1 withprobability at least 1 − ε .Next, we use Hypothesis 2 to round e F to an exactly non-signaling function F close to it.Specifically, since e F is (1 − ε )-linear (8 ε, t )-non-signaling, by Hypothesis 2 there exist t ′ -non-signaling function F : { , } N → { , } for t ′ ≥ t e hyp = K ′ log ( N ), such that ∆ ( e F , F ) ≤ ε ′ , where ε ′ = ε ′ hyp (28 ε ). In particular, since e F is (1 − ε )-linear, it follows that F is (1 − ε − ε ′ )-linear,and satisfies the PCP verifier from Algorithm 1 with probability at least 1 − ε − ε ′ .Next, we apply the following theorem on almost linear non-signaling functions from [CMS19].The theorem says that any almost linear function F can be “rounded” into an exactly non-signalingfunction L , such that the two are close to each other on predicates that depend on a small numberof coordinates. Theorem 8.2 (Theorem 7 in [CMS19]) . Let t ′ , ¯ k ∈ N and ε ∈ (0 , / be such that t ′ = Ω( ¯ kε · (¯ k + log ε )) . Suppose that F : { , } n → { , } is a t ′ -non-signaling function such that for all x, y ∈ { , } n it holds that Pr[ F ( x ) + F ( y ) = F ( x + y )] ≥ − ε . Then there exists a linear ¯ k -non-signaling function L : { , } n → { , } such that for all query sets Q ⊆ { , } n of size | Q | ≤ ¯ k andfor all events E ⊆ { , } Q it holds that | Pr[ F ( Q ) ∈ E ] − Pr[ L ( Q ) ∈ E ] | ≤ (6 | Q | + 3) √ ε . Remark 8.3.
Actually, Theorem 7 in [CMS19] assumes that linearity test accepts F with highprobability, and the conclusion of the theorem holds for its self-correction b F . However, if we makethe stronger assumption that Pr[ F ( x ) + F ( y ) = F ( x + y )] ≥ − ε holds for all x, y ∈ { , } n , thenby following the proof, it is easy to see that the conclusion holds for F , without the self-correction.By applying Theorem 8.2 on F , and using it for all 4-ary predicates used by Algorithm 1, itfollows that there exists a linear ¯ k -non-signaling function L : { , } n → { , } that satisfies the PCPverifier from Algorithm 1 with probability at least 1 − b ε for b ε = 1 − ε − ε ′ − (6 · √ ε + ε ′ = 1 − O ( √ ε + ε ′ ). In particular, if t ′ > K ′ log ( N ) for a sufficiently large constant K ′ , then ¯ k ≥ ¯ C log( N ).25herefore, if ε > L with probability greater than 39 /
40, and by Theorem 4.1 we conclude that C ( x ) = 1.This completes the proof of Theorem 8.1. In this paper we establish a conditional result on the existence of a PCP system that is soundagainst non-signaling proofs with constant locality. There are several natural research directionsleft open for future work.
Resolving the hypothesis.
The implications of Hypothesis 2 motivates the study of geometryof non-signaling proofs. In particular, as a natural intermediate step toward settling Hypothesis 2,one can study the validity of a weaker version of hypothesis, requiring that the rounded proof isclose to the given almost non-signaling proof on all subsets of size at most 2 (instead of 4) assumingthat F is linear (instead of almost linear), i.e., requiring that ∆ ( F , F ′ ) ≤ ε ′ . We remark thatalthough Hypothesis 2 requires that ∆ ( F , F ′ ) is small, in fact, it suffices to show that ∆ ( F , F ′ )is small, i.e., prove the hypothesis for subsets of size at most 3. Reducing the alphabet.
While we answer Question 1.2 affirmatively up to Hypothesis 2, we mayrequire the proof to be of smaller alphabet. In the classical PCPs literature, the standard techniquefor alphabet reduction is known as proof composition , where the given “outer” proof over largealphabet is composed with a collection of “inner proofs of proximity” over small alphabet [BGH + Extending our approach to polynomial size nsPCPs.
Our PCP construction is basedon exponential-length
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As mentioned in the introduction, we can think of k -non-signaling functions as points in the polytope P k ⊆ R d , for d = P ni =0 (cid:0) ni (cid:1) i , which corresponds the solutions of the k ’th level relaxation of theSherali-Adams hierarchy. Analogously, we can think of ( ε, k )-non-signaling functions as pointsin the polytope P εk ⊆ R d , which corresponds the solutions of the noisy version of the k ’th levelrelaxation of Sherali-Adams hierarchy, where for any two sets S, T ⊆ [ n ] the marginal distributionsinduced by P S on S ∩ T is ε -close in total variation distance to the marginal distributions inducedby P T on S ∩ T . Then, Hypothesis 2 can be rephrased as follows: for any p ∈ P εk there exists p ′ ∈ P k ′ such that ∆ ( p, p ′ ) ≤ ε ′ .We remark that sensitivity analysis of linear programs has been studied in the past (see, e.g.,[Sch86] Section 10). However, the parameters obtained by these results seem to be too weak forour application. Nonetheless, it is possible that this approach could still work for our setting,since we are looking for an approximate solution with respect to the ∆ distance, which is rathernon-standard.In [CMS20], the following lemma, in the same spirit as the hypothesis, was proved. Lemma A.1 ((see [CMS20, Lemma C.3])) . For every ( ε, k ) -non-signalling function F : D → { , } there exists k -non-signalling function F ′ such that ∆ k ( F , F ′ ) ≤ O (4 k · ε )While the guarantee of O (4 k · ε ) on the distance in the lemma is too large for our applications,Hypothesis 2 is somewhat more specific, and it is plausible that proving it is easier than improvingLemma A.1. We discuss Hypothesis 2 below.1. Note that unlike Lemma A.1, Hypothesis 2 assumes that F is almost linear. We do not knowwhether this is essential, however, it is reasonable to believe that being almost linear addsconstraints on the structure of F , thus making it easier to prove Hypothesis 2.2. In Hypothesis 2 the requirement on the distance between the given almost non-signaling function,and the rounded function is only on sets of size at most 4. In fact, it is not difficult to see thatproving that ∆ ( F , F ′ ) ≤ ε ′ also suffices for the applications. This seems to be a significantrelaxation compared to ∆ k proved in Lemma A.1.3. In fact, our proof of soundness would go through even with a weaker version of the hypothesis,where we replaced the “worst-case” notion of ∆ with the “average-case”. Specifically, given an( ε, k )-almost non-signaling proof F that satisfies every constraint of the linear ALMSS verifierwith high probability, we want the rounded proof to satisfy the linear ALMSS verifier with highprobability with respect to the distribution induced by the verifier on the 4-query sets.Furthermore, since our almost non-signaling proof F is obtained by flattening the repeated proof d F ( t ) , we may assume that F satisfies every constraints of the Ω( k )-sequential repetition of the30inearity test, i.e., for some ℓ = Ω( k ) it holds that ∀ x , y , . . . , x ℓ , y ℓ ∈ { , } n Pr [ F ( x i ) + F ( y i ) = F ( x i + y i ) ∀ i ∈ [ ℓ ]] ≥ − ε , and, similarly, F satisfies every constraint of the Ω( t )-sequential repetition of the linear ALMSSverifier with high probability, and the goal is to get a rounded proof to satisfy the linear ALMSSverifier with high probability with respect to the distribution induced by the verifier on the4-query sets.4. An alternative way to prove our main theorem is to prove that Theorem 4.1 holds for almostnon-signaling proofs. This question seems to be well motivated by the application to delegationof computation . Indeed, Kalai et. al [KRR14] constructed PCP systems (of polynomial size) thatare sound against ( ε, polylog ( N ))-non-signaling proofs, for some negligible ε >
0. However, theirproof seems to break for constant ε >
0. Our work motivates studying the power of almostnon-signaling proofs for constant ε >ε >