Algebraic Property Testing: A Survey Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing: Personal Perspective Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing: Personal Perspective Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Distance: Definition: Notes: ± ( f ; g ) = P r x 2 D [ 6 ] F m i n ¼ ² · . F i s ( k ; ² ± ) - l o c a y t e b f 9 q u r T . 2 p w ¸ 1 ¡ j k - l o c a y t e s b i m p 9 ² ; ± > = O ( 1 ) n d r : A f 2 F w . April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Brief History [Blum,Luby,Rubinfeld – S’90] Linearity + application to program testing [Babai,Fortnow,Lund – F’90] Multilinearity + application to PCPs (MIP). [Rubinfeld+S.] Low-degree testing + Formal Definition [Goldreich,Goldwasser,Ron] Graph property testing. Since then … many developments Graph properties Statistical properties More algebraic properties April 1, 2009 Algebraic Property Testing @ DIMACS

Specific Directions in Algebraic P.T. More Properties Low-degree (d < q) functions [RS] Moderate-degree (q < d < n) functions q=2: [AKKLR] General q: [KR, JPRZ] Long code/Dictator/Junta testing [PRS] BCH codes (Trace of low-deg. poly.) [KL] All nicely “invariant” properties [KS] Better Parameters (motivated by PCPs). #queries, high-error, amortized query complexity, reduced randomness. April 1, 2009 Algebraic Property Testing @ DIMACS

Contrast w. Combinatorial P.T. U n i v e r s f : D ! R g ( A l s o u a y ) R i ¯ e d F P r p t = L n b c . M u s t a c e p O k o r j w . h F F Algebraic Property = Code! (usually) April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Goal of this talk Implications of linearity Constraints, Characterizations, LDPC structure One-sided error, Non-adaptive tests [BHR] Redundancy of Constraints Tensor Product Codes Symmetries of Code Testing affine-invariant codes Yields basic tests for all known algebraic codes (over small fields). April 1, 2009 Algebraic Property Testing @ DIMACS

Basic Implications of Linearity [BHR] Generic adaptive test = decision tree. f(i) 1 f(j) f(k) ² P i c k p a t h f o l w e d b y r n m g 2 F . 1 ² Q u e r y f a c o d i n g t p h . ² A c e p t i ® f o n a h s w m 2 F . ² Y i e l d s n o - a p t v r f F . April 1, 2009 Algebraic Property Testing @ DIMACS

Basic Implications of Linearity [BHR] Generic adaptive test = decision tree. f(i) 1 f(j) f(k) ² P i c k p a t h f o l w e d b y r n m g 2 F . 1 ² Q u e r y f a c o d i n g t p h . ² A c e p t i ® f o n a h s w m 2 F . ² Y i e l d s n o - a p t v r f F . April 1, 2009 Algebraic Property Testing @ DIMACS

Constraints, Characterizations ² S a y t e s q u r i 1 ; : k c p h f ( ) 2 V 6 = F ² ( i 1 ; : k V ) = C o n s t r a E v e y f 2 F ¯ . 1 D i1 i2 ik in V? 2 ² I f e v r y 6 2 F j c t d w . p o s i b h n a z Like LDPC Codes! April 1, 2009 Algebraic Property Testing @ DIMACS

Constraints, Characterizations ² S a y t e s q u r i 1 ; : k c p h f ( ) 2 V 6 = F ² ( i 1 ; : k V ) = C o n s t r a E v e y f 2 F ¯ . 1 D i1 i2 ik in V? 2 ² I f e v r y 6 2 F j c t d w . p o s i b h n a z Like LDPC Codes! April 1, 2009 Algebraic Property Testing @ DIMACS

Example: Linearity Testing [BLR] Constraints: Characterization: C x ; y = ( + V ) j 2 F n w h e r f a b g x in V? f i s l n e a r ® 8 x ; y C t ¯ d y x+y April 1, 2009 Algebraic Property Testing @ DIMACS

Insufficiency of local characterizations [Ben-Sasson, Harsha, Raskhodnikova] There exist families characterized by k-local constraints that are not o(|D|)-locally testable. Proof idea: Pick LDPC graph at random … (and analyze resulting property) F April 1, 2009 Algebraic Property Testing @ DIMACS

Why are characterizations insufficient? Constraints too minimal. Not redundant enough! Proved formally in [Ben-Sasson, Guruswami, Kaufman, S., Viderman] Constraints too asymmetric. Property must show some symmetry to be testable. Not a formal assertion … just intuitive. April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Redundancy? E.g. Linearity Test: Standard LDPC analysis: What natural operations create redundant local constraints? Tensor Products! ¡ ­ ( D 2 ) c o n s t r a i d m ¡ D i m e n s o ( F ) ¼ f r c t a . R q u # < l w h d y ! April 1, 2009 Algebraic Property Testing @ DIMACS

Tensor Products of Codes! Redundancy? F £ G = f M a t r i c e s u h v y o w n F d l m G g S u p o s e F , G y t m a i c F i r s t ` e n f d m b y h . Free F d e t r m i n G d e t r m i n d e t r m i n w c , b y F a G ! April 1, 2009 Algebraic Property Testing @ DIMACS

Testability of tensor product codes? Natural test: Given Matrix M Test if random row in F Test if random column in G Claim: If F, G codes of constant (relative) distance; then if test accepts w.h.p. then M is close to codeword of F x G Yields O(√n) local test for codes of length n. Can we do better? Exploit local testability of F, G? April 1, 2009 Algebraic Property Testing @ DIMACS

Robust testability of tensors? Natural test (if F,G locally testable): Given Matrix M Run Local Test for F on random row Run Local Test for G on random column Suppose M close on most rows/columns to F, G. Does this imply M is close to F x G? Generalizes test for bivariate polynomials. True for F, G = class of low-degree polynomials. [BFLS, Arora+Safra, Polishchuk+Spielman]. General question raised by [Ben-Sasson+S.] [P. Valiant] Not true for every F, G ! [Dinur, S., Wigderson] True if F (or G) locally testable. Test that random row close to F Test that random column close to G April 1, 2009 Algebraic Property Testing @ DIMACS

Tensor Products and Local Testability Robust testability allows easy induction (essentially from [BFL, BFLS]; see also [Ben-Sasson+S.]) ² L e t F n = - f o l d s r . ² G i v e n f : D ! F N a t u r l s P c k d o m x - p y j 2 April 1, 2009 Algebraic Property Testing @ DIMACS

Robust testability of tensors (contd.) Unnatural test (for F x F x F): Given 3-d matrix M: Pick random 2-d submatrix. Verify it is close to F x F Theorem [BenSasson+S., based on Raz+Safra]: Distance to F x F x F proportional to average distance of random 2-d submatrix to F x F. [Meir]: “Linear-algebraic” construction of Locally Testable Codes (matching best known parameters) using this (and many other ingredients). April 1, 2009 Algebraic Property Testing @ DIMACS

Redundant Characterizations (contd.) Redundant constraints necessary for testing [BGKSV] How to get redundancy? Tensor Products Sufficient to get some local testability Invariances (Symmetries) Sufficient? Counting (See Tali’s talk)  April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Testing by symmetries April 1, 2009 Algebraic Property Testing @ DIMACS

Invariance & Property testing Invariances (Automorphism groups): Hope: If Automorphism group is “large” (“nice”), then property is testable. F o r p e m u t a i n ¼ : D ! , s - v f 2 l ± . A u t ( F ) = f ¼ j i s - n v a r g o m p d e c . April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Examples Majority: Graph Properties: Algebraic Properties: What symmetries do they have? ¡ A u t g r o p = S D ( f l ) . ¡ E a s y F c t : I f A’ u ( ) = S D h e n i p o l R ; 1 ² - b . ¡ A u t . g r o p i v e n b y a m f c s [ F N S , B l ] w h April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Properties & Invariances Automorphism groups? Question: Are Linear/Affine-Inv., Locally Characterized Props. Testable? ([Kaufman + S.]) D = F n , R ( L i e a r t y o w - d g M u l ) O r D = K ¶ F , R ( u a l - B C H ) ( K ; F ¯ n i t e l d s ) L i n e a r t s f o m d . ¼ ( x ) = A w h 2 F £ (Linear-Invariant) A ± n e t r a s f o m i d . ¼ ( x ) = + b w h 2 F £ ; (Affine-Inv.) April 1, 2009 Algebraic Property Testing @ DIMACS

Linear-Invariance & Testability Unifies previous studies on Alg. Prop. Testing. (And captures some new properties) Nice family of 2-transitive group of symmetries. Conjecture [Alon, Kaufman, Krivelevich, Litsyn, Ron] : Linear code with k-local constraint and 2-transitive group of symmetries must be testable. April 1, 2009 Algebraic Property Testing @ DIMACS

Some Results [Kaufman + S.] Theorem 1: Theorem 2: F µ f K n ! g l i e a r , - v t k o c y h z d m p s ( ; ) b . F µ f K n ! g l i e a r , ± - v t h s k o c m p ( ; ) y b . April 1, 2009 Algebraic Property Testing @ DIMACS

Examples of Linear-Invariant Families ¡ L i n e a r f u c t o s m F . ¡ P o l y n m i a s F [ x 1 ; : ] f d e g r t ¡ T r a c e s o f P l y i n K [ x 1 ; : ] d g t m ¡ ( T r a c e s o f ) H m g n u p l y i d ¡ F 1 + 2 , w h e r a l i n - v t . P o y m s u p d b g ; 3 5 7 April 1, 2009 Algebraic Property Testing @ DIMACS

What Dictates Locality of Characterizations? ¡ P r e c i s l o a t y n u d : D p - f g . E x m F b + j h v k w ¡ F o r a ± n e - i v t f m l y d c ( s ) b h g ¡ F o r s m e l i n a - v t f , c b u h g d . E x a m p l e : K = F 7 ; 1 + 2 o y f d g r t s 6 u n i 3 . D ( ) ­ L c · 4 9 April 1, 2009 Algebraic Property Testing @ DIMACS

Property Testing from Invariances April 1, 2009 Algebraic Property Testing @ DIMACS

Key Notion: Formal Characterization ¡ F h a s i n g l e - o r b t c z f 9 C = ( x 1 ; : k V ) u ± ¼ 2 A . T h e o r m : I f F a s i n g l - b t c z y k ( w ) . Rest of talk: Analysis (extending BLR) April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS BLR Analysis: Outline ² H a v e f s . t P r x ; y [ ( ) + 6 = ] ± < 1 2 W n o h w c l m g F ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² I f c l o s e t F h n g w i b a d . ² B u t i f n o c l s e ? g m a y v b q d ¯ ! ² S t e p s : ¡ S t e p : P r o v f c l s g ¡ S t e p 1 : P r o v m s l i k y w h n g a j . ¡ S t e p 2 : P r o v h a g i s n F . April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS BLR Analysis: Step 0 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . C l a i m : P r x [ f ( ) 6 = g ] · 2 ± ¡ L e t B = f x j P r y [ ( ) 6 + ] ¸ 1 2 g ¡ P r x ; y [ l i n e a t s j c 2 B ] ¸ 1 ) P r x [ 2 B ] · ± ¡ I f x 6 2 B t h e n ( ) = g April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS V o t e x ( y ) BLR Analysis: Step 1 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² S u p o s e f r m x , 9 t w q a l y i k v . P b n d h c ? ² I f w e i s h t o g l n a r , d u c . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS V o t e x ( y ) BLR Analysis: Step 1 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² S u p o s e f r m x , 9 t w q a l y i k v . P b n d h c ? ² I f w e i s h t o g l n a r , d u c . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS BLR Analysis: Step 1 V o t e x ( y ) ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± f ( y ) ¡ f ( x + y ) ? f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) P r o b . R w / c l u m n s - z e · ± April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS BLR Analysis: Step 1 V o t e x ( y ) ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± f ( y ) ¡ f ( x + y ) ? f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) P r o b . R w / c l u m n s - z e · ± April 1, 2009 Algebraic Property Testing @ DIMACS

BLR Analysis: Step 2 (Similar) f ± < 1 2 , t h n 8 x ; y g ( ) + = P r o b . R w / c l u m n s - z e · 4 ± g ( x ) g ( y ) ¡ g ( x + y ) f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) April 1, 2009 Algebraic Property Testing @ DIMACS

Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e 1 ) ; : k i 2 V ] = ± ¿ ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] ² S t e p s : ¡ S t e p : P r o v f c l s g Step 1: Prove “most likely” is overwhelming majority. ¡ ¡ S t e p 2 : P r o v h a g i s n F . April 1, 2009 Algebraic Property Testing @ DIMACS

Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e 1 ) ; : k i 2 V ] = ± ¿ ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] S a m e s b f o r ² S t e p s : ¡ S t e p : P r o v f c l s g Step 1: Prove “most likely” is overwhelming majority. ¡ ¡ S t e p 2 : P r o v h a g i s n F . April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS V o t e x ( L ) Matrix Magic? ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] L e m a : 8 x , P r ; K [ V o t ( ) 6 = ] · 2 k ¡ 1 ± L ( x 2 ) L ( x k ) ¢ x K ( x 2 ) ? . K ( x k ) April 1, 2009 Algebraic Property Testing @ DIMACS

? Matrix Magic? L ( x ) ¢ L ( x ) x K ( x ) . K ( x ) ² W a n t m r k 2 ) ¢ L ( x k ) x K ( x 2 ) ? . K ( x k ) ² W a n t m r k e d o w s b c i . ² S u p o s e x 1 ; : ` l i n a r y d t h m .

Matrix Magic? ` L ( x ) ¢ L ( x ) x K ( x ) ` . K ( x ) ² S u p o s e Fill with random entries Matrix Magic? Fill so as to form constraints Tensor magic implies final columns are also constraints. ` L ( x 2 ) ¢ L ( x k ) x K ( x 2 ) ` . K ( x k ) ² S u p o s e x 1 ; : ` l i n a r y d t h m . April 1, 2009

Matrix Magic? ` L ( x ) ¢ L ( x ) x K ( x ) ` . K ( x ) ² S u p o s e Fill with random entries Matrix Magic? Fill so as to form constraints Tensor magic implies final columns are also constraints! ` L ( x 2 ) ¢ L ( x k ) x K ( x 2 ) ` . K ( x k ) ² S u p o s e x 1 ; : ` l i n a r y d t h m . April 1, 2009

Algebraic Property Testing @ DIMACS Summarizing Affine invariance + single-orbit characterizations imply testing. Unifies analysis of linearity test, basic low-degree tests, moderate-degree test (all A.P.T. except dual-BCH?) April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Concluding thoughts - 1 Didn’t get to talk about PCPs, LTCs (though we did implicitly) Optimizing parameters Parameters In general Broad reasons why property testing works worth examining. Tensoring explains a few algebraic examples. Invariance explains many other algebraic ones. (More about invariances in [Grigorescu,Kaufman,S. ’08], [GKS’09]) April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Concluding thoughts - 2 Invariance: Seems to be a nice lens to view all property testing results (combinatorial, statistical, algebraic). Many open questions: What groups of symmetries aid testing? What additional properties needed? Local constraints? Linearity? Does sufficient symmetry imply testability? Give an example of a non-testable property with a k-single orbit characterization. April 1, 2009 Algebraic Property Testing @ DIMACS

Algebraic Property Testing @ DIMACS Thank You! April 1, 2009 Algebraic Property Testing @ DIMACS