Algebraic Property Testing: A Survey Madhu Sudan MIT 1 1 April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS
Algebraic Property Testing: Personal Perspective Madhu Sudan MIT 2 2 April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS
Algebraic Property Testing: Personal Perspective Madhu Sudan MIT 3 3 April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS
Property Testing Property Testing � Distance: Distance: δ ( f, g ) = Pr x ∈ D [ f ( x ) 6 = g ( x )] � δ ( f, F ) = min g ∈ F { δ ( f, g ) } f ≈ ² g if δ ( f, g ) ≤ ² . � Definition: Definition: � F is ( k, ², δ )-locally testable if ∃ a k -query tester T s.t. T f accepts w.p. ≥ 1 − ² f ∈ F ⇒ T f rejects w.p. ≥ ² . δ ( f, F ) ≥ δ ⇒ � Notes: Notes: k -locally testable implies ∃ ², δ > 0 � locally testable implies ∃ k = O (1) One-sided error: Accept f ∈ F w.p. 1 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 4 4 April 1, 2009 April 1, 2009
Brief History Brief History � [ [ Blum,Luby,Rubinfeld Blum,Luby,Rubinfeld – – S S’ ’90] 90] � � Linearity + application to program testing Linearity + application to program testing � � [ [ Babai,Fortnow,Lund Babai,Fortnow,Lund – – F F’ ’90] 90] � � Multilinearity Multilinearity + application to PCPs (MIP). + application to PCPs (MIP). � � [ [ Rubinfeld+ S Rubinfeld+ S. .] ] � � Low Low- - degree testing + degree testing + Formal Definition Formal Definition � � [ [ Goldreich,Goldwasser,Ron Goldreich,Goldwasser,Ron] ] � Graph property testing. Graph property testing. � � � Since then Since then … … many developments many developments � � Graph properties Graph properties � � Statistical properties Statistical properties � � More algebraic properties More algebraic properties � Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 5 5 April 1, 2009 April 1, 2009
Specific Directions in Algebraic P.T. Specific Directions in Algebraic P.T. � More Properties More Properties � � Low Low- - degree (d < q) functions [ degree (d < q) functions [ RS RS] ] � � Moderate Moderate- - degree (q < d < n) functions degree (q < d < n) functions � � q= 2: [ q= 2: [ AKKLR AKKLR] ] � � General q: [ General q: [ KR, JPRZ KR, JPRZ] ] � � Long code/ Dictator/ Junta testing [ Long code/ Dictator/ Junta testing [ PRS PRS] ] � � BCH codes (Trace of low BCH codes (Trace of low- - deg. poly.) [ deg. poly.) [ KL KL] ] � � All nicely All nicely “ “ invariant invariant ” ” properties [ properties [ KS KS] ] � � Better Parameters (motivated by PCPs). Better Parameters (motivated by PCPs). � � # queries, high # queries, high- - error, amortized query error, amortized query � complexity, reduced randomness. complexity, reduced randomness. Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 6 6 April 1, 2009 April 1, 2009
Contrast w . Com binatorial P.T. Contrast w . Com binatorial P.T. Universe (Also usually) R is a fi eld F { f : D → R } Property = Linear subspace. F Must accept F Ok to accept Must reject w.h.p. Algebraic Property = Code! (usually) Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 7 7 April 1, 2009 April 1, 2009
Goal of this talk Goal of this talk � Implications of linearity Implications of linearity � � Constraints, Characterizations, LDPC structure Constraints, Characterizations, LDPC structure � � One One- - sided error, Non sided error, Non- - adaptive tests [ BHR] adaptive tests [ BHR] � � Redundancy of Constraints Redundancy of Constraints � � Tensor Product Codes Tensor Product Codes � � Symmetries of Code Symmetries of Code � � Testing affine Testing affine- - invariant codes invariant codes � � Yields basic tests for all known algebraic Yields basic tests for all known algebraic � codes (over small fields). codes (over small fields). Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 8 8 April 1, 2009 April 1, 2009
Basic I m plications of Linearity [ BHR] Basic I m plications of Linearity [ BHR] � Generic adaptive test = decision tree. Generic adaptive test = decision tree. � f(i) 1 0 f(k) f(j) • Pick path followed by random g ∈ F . 0 1 • Query f according to path. • Accept i ff f on path consistent with some h ∈ F . • Yields non-adaptive one-sided error test for linear F . Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 9 9 April 1, 2009 April 1, 2009
Basic I m plications of Linearity [ BHR] Basic I m plications of Linearity [ BHR] � Generic adaptive test = decision tree. Generic adaptive test = decision tree. � f(i) 1 0 f(k) f(j) • Pick path followed by random g ∈ F . 0 1 • Query f according to path. • Accept i ff f on path consistent with some h ∈ F . • Yields non-adaptive one-sided error test for linear F . Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 10 10 April 1, 2009 April 1, 2009
Constraints, Characterizations Constraints, Characterizations • Say test queries i 1 , . . . , i k accepts h f ( i 1 ) , . . . , f ( i k ) i ∈ V 6 = F k • ( i 1 , . . . , i k ; V ) = Constraint 1 i 1 Every f ∈ F satis fi es it. 2 in V? i 2 • If every f 6 ∈ F rejected i k w. positive prob. then F characterized by constraints. D Like LDPC Codes! • Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 11 11 April 1, 2009 April 1, 2009
Constraints, Characterizations Constraints, Characterizations • Say test queries i 1 , . . . , i k accepts h f ( i 1 ) , . . . , f ( i k ) i ∈ V 6 = F k • ( i 1 , . . . , i k ; V ) = Constraint 1 i 1 Every f ∈ F satis fi es it. 2 in V? i 2 • If every f 6 ∈ F rejected i k w. positive prob. then F characterized by constraints. D Like LDPC Codes! • Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 12 12 April 1, 2009 April 1, 2009
Exam ple: Linearity Testing [ BLR] Exam ple: Linearity Testing [ BLR] • Constraints: C x,y = ( x, y, x + y ; V ) | x, y ∈ F n where V = { ( a, b, a + b ) | a, b ∈ F } x • Characterization: in V? f is linear i ff y ∀ x, y, C x,y satis fi ed x+ y Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 13 13 April 1, 2009 April 1, 2009
I nsufficiency of local characterizations I nsufficiency of local characterizations � [ Ben [ Ben- - Sasson Sasson, , Harsha Harsha, , Raskhodnikova Raskhodnikova] ] � F � There exist families There exist families characterized characterized by by k k- - local local � constraints that are not that are not o(| D o(| D| ) | ) - - locally testable locally testable. . constraints � Proof idea: Pick LDPC graph at random Proof idea: Pick LDPC graph at random … … � (and analyze resulting property) (and analyze resulting property) Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 14 14 April 1, 2009 April 1, 2009
W hy are characterizations insufficient? W hy are characterizations insufficient? � Constraints too minimal. Constraints too minimal. � � Not redundant enough! Not redundant enough! � � Proved formally in [ Ben Proved formally in [ Ben- - Sasson Sasson, , � Guruswami, Kaufman, S., , Kaufman, S., Viderman Viderman] ] Guruswami � Constraints too asymmetric. Constraints too asymmetric. � � Property must show some symmetry to be Property must show some symmetry to be � testable. testable. � Not a formal assertion Not a formal assertion … … just intuitive. just intuitive. � Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 15 15 April 1, 2009 April 1, 2009
Redundancy? Redundancy? � E.g. Linearity Test: E.g. Linearity Test: � − Ω ( D 2 ) constraints on domain D � Standard LDPC analysis: Standard LDPC analysis: � − Dimension( F ) ≈ D − m for m constraints. − Requires #constraints < D . − Does not allow much redundancy! � What natural operations create redundant local What natural operations create redundant local � constraints? constraints? � Tensor Products! Tensor Products! � Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 16 16 April 1, 2009 April 1, 2009
Tensor Products of Codes! Tensor Products of Codes! F × G � Tensor Product: Tensor Product: � = { Matrices such every row in F and every column in G } � Redundancy? Redundancy? � Suppose F , G systematic First ` entries free rest determined by them. Free F determined G determined determined twice, by F and G ! Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 17 17 April 1, 2009 April 1, 2009
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