Ben Lee Volk. Joint with. Michael A. Forbes Amir Shpilka

Transcription

1 Ben Lee Volk Joint with Michael A. Forbes Amir Shpilka

2 Ben Lee Volk Joint with Michael A. Forbes Amir Shpilka

3 (One) Answer: natural proofs barrier [Razborov-Rudich]:

4 (One) Answer: natural proofs barrier [Razborov-Rudich]: A computationally-bounded observer cannot distinguish between the truth table of a random function with small circuit and that of a truly random function (assuming some crypto). So every lower bound proof attempt which yields such an algorithm cannot work.

5 (One) Answer: natural proofs barrier [Razborov-Rudich]: A computationally-bounded observer cannot distinguish between the truth table of a random function with small circuit and that of a truly random function (assuming some crypto). So every lower bound proof attempt which yields such an algorithm cannot work.

6 Def: A property S of boolean functions is natural if it is:

7 Def: A property S of boolean functions is natural if it is: 1. Useful: if f has S then f doesn t have a small ckt.

8 Def: A property S of boolean functions is natural if it is: 1. Useful: if f has S then f doesn t have a small ckt. 2. Large: random functions have S with large probability.

9 Def: A property S of boolean functions is natural if it is: 1. Useful: if f has S then f doesn t have a small ckt. 2. Large: random functions have S with large probability. 3. Constructive: Given truth table of f of size N = 2 n, there is an algorithm for deciding whether f S with running time poly(n) = 2 O(n).

10 Def: A property S of boolean functions is natural if it is: 1. Useful: if f has S then f doesn t have a small ckt. 2. Large: random functions have S with large probability. 3. Constructive: Given truth table of f of size N = 2 n, there is an algorithm for deciding whether f S with running time poly(n) = 2 O(n). natural proof: a lower bound proof which exhibits a natural property.

11 Def: A property S of boolean functions is natural if it is: 1. Useful: if f has S then f doesn t have a small ckt. 2. Large: random functions have S with large probability. 3. Constructive: Given truth table of f of size N = 2 n, there is an algorithm for deciding whether f S with running time poly(n) = 2 O(n). natural proof: a lower bound proof which exhibits a natural property. [Razborov-Rudich]: Most known lower bounds are natural and if there s a pseudorandom function in then no natural lower bound against.

12 f (x 1, x 2, x 3 ) [x 1, x 2, x 3 ] x 1 x 2 x 3 1

13 ...are also pretty hard.

14 ...are also pretty hard. We can prove lower bounds for restricted models, but all lower bounds eventually also apply to polynomials we think of as easy.

15 ...are also pretty hard. We can prove lower bounds for restricted models, but all lower bounds eventually also apply to polynomials we think of as easy. Most lower bounds seem natural, but unclear whether there are pseudorandom functions computed by low-degree algebraic circuits.

16 ...are also pretty hard. We can prove lower bounds for restricted models, but all lower bounds eventually also apply to polynomials we think of as easy. Most lower bounds seem natural, but unclear whether there are pseudorandom functions computed by low-degree algebraic circuits. (and even if not, maybe there are such functions that are only secure against algebraic circuits?)

17 ...are also pretty hard. We can prove lower bounds for restricted models, but all lower bounds eventually also apply to polynomials we think of as easy. Most lower bounds seem natural, but unclear whether there are pseudorandom functions computed by low-degree algebraic circuits. (and even if not, maybe there are such functions that are only secure against algebraic circuits?) Can we identify formal barriers?

18 ...are also pretty hard. We can prove lower bounds for restricted models, but all lower bounds eventually also apply to polynomials we think of as easy. Most lower bounds seem natural, but unclear whether there are pseudorandom functions computed by low-degree algebraic circuits. (and even if not, maybe there are such functions that are only secure against algebraic circuits?) Can we identify formal barriers? (also asked by [Aaronson-Drucker] and [Grochow])

19 Many lower bounds for restricted models of algebraic circuits have this form:

20 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f.

21 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f. 2. argue that if f is computed by a small ckt, rank(m f ) = small.

22 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f. 2. argue that if f is computed by a small ckt, rank(m f ) = small. 3. show some explicit f 0 with rank(m f0 ) = large.

23 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f. 2. argue that if f is computed by a small ckt, rank(m f ) = small. 3. show some explicit f 0 with rank(m f0 ) = large. (examples: evaluation dimension, partial derivatives, shifted partial derivatives,...)

24 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f. 2. argue that if f is computed by a small ckt, rank(m f ) = small. 3. show some explicit f 0 with rank(m f0 ) = large. (examples: evaluation dimension, partial derivatives, shifted partial derivatives,...) equivalently: for some r r submatrix, det(m ) 0, while f 0 det(m ) = 0 for all simple f. f

25 Many lower bounds for restricted models of algebraic circuits have this form: 1. given f, construct some matrix M f whose entries are coefficients of f. 2. argue that if f is computed by a small ckt, rank(m f ) = small. 3. show some explicit f 0 with rank(m f0 ) = large. (examples: evaluation dimension, partial derivatives, shifted partial derivatives,...) equivalently: for some r r submatrix, det(m ) 0, while f 0 det(m ) = 0 for all simple f. f Thus, the property {g : det(m g ) 0} is useful, constructive (determinant is efficiently computable) and large.

26 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if

27 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f

28 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f 2. (Constructiveness) D has a small algebraic circuit

29 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f 2. (Constructiveness) D has a small algebraic circuit (largeness comes for free)

30 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f 2. (Constructiveness) D has a small algebraic circuit (largeness comes for free) Important: D is an N-variate polynomial for N = n+d d (if we deal with n-variate degree d polynomials) or N = 2 n (if we deal with multilinear polynomials)

31 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f 2. (Constructiveness) D has a small algebraic circuit (largeness comes for free) Important: D is an N-variate polynomial for N = n+d d (if we deal with n-variate degree d polynomials) or N = 2 n (if we deal with multilinear polynomials) Toy example: is the class of perfect squares among polynomials of the form ax 2 + bx + c, and D(a, b, c) = b 2 4ac.

32 Def: A (distinguisher) polynomial D 0 is an algebraic natural proof against a class if 1. (Usefulness) D(coeff(f )) = 0 for all f 2. (Constructiveness) D has a small algebraic circuit (largeness comes for free) Important: D is an N-variate polynomial for N = n+d d (if we deal with n-variate degree d polynomials) or N = 2 n (if we deal with multilinear polynomials) Toy example: is the class of perfect squares among polynomials of the form ax 2 + bx + c, and D(a, b, c) = b 2 4ac. [Grochow]: almost all known lower bounds can be cast in this form.

33 [Mulmuley-Sohoni]

34 [Mulmuley-Sohoni] VP per VP det

35 [Mulmuley-Sohoni] VP per VP det VP is a zero set of a set of polynomials. What is the complexity of?

36 Given an algebraic circuit C, decide deterministically whether C computes the zero polynomial.

37 Given an algebraic circuit C, decide deterministically whether C computes the zero polynomial. Black-box: compute a hitting set, i.e., a set such that for every C there is α such that C(α) 0.

38 Given an algebraic circuit C, decide deterministically whether C computes the zero polynomial. Black-box: compute a hitting set, i.e., a set such that for every C there is α such that C(α) 0. Suppose D is a distinguisher for a class. Then D 0, and yet D(coeff(f )) = 0 for all f.

39 Given an algebraic circuit C, decide deterministically whether C computes the zero polynomial. Black-box: compute a hitting set, i.e., a set such that for every C there is α such that C(α) 0. Suppose D is a distinguisher for a class. Then D 0, and yet D(coeff(f )) = 0 for all f. = {coeff(f ) : f } is not a hitting set for D.

40 Given an algebraic circuit C, decide deterministically whether C computes the zero polynomial. Black-box: compute a hitting set, i.e., a set such that for every C there is α such that C(α) 0. Suppose D is a distinguisher for a class. Then D 0, and yet D(coeff(f )) = 0 for all f. = {coeff(f ) : f } is not a hitting set for D. In other words: if {coeff(f ) : f } is a hitting set for a class, then no natural proof for with the distinguisher coming from.

41 Def: Let [x 1,..., x n ] be a class of degree d polynomials, and [X 1,..., X N ] for N = n+d d. is a succinct hitting set for if := {coeff(f ) : f } is a hitting set for.

42 Def: Let [x 1,..., x n ] be a class of degree d polynomials, and [X 1,..., X N ] for N = n+d d. is a succinct hitting set for if := {coeff(f ) : f } is a hitting set for. Thm: If is a succinct hitting set for, no algebraically natural proof against with the distinguisher coming from.

43 Def: Let [x 1,..., x n ] be a class of degree d polynomials, and [X 1,..., X N ] for N = n+d d. is a succinct hitting set for if := {coeff(f ) : f } is a hitting set for. Thm: If is a succinct hitting set for, no algebraically natural proof against with the distinguisher coming from. Proof: If D is non-zero then D does not vanish on.

44 Def: Let [x 1,..., x n ] be a class of degree d polynomials, and [X 1,..., X N ] for N = n+d d. is a succinct hitting set for if := {coeff(f ) : f } is a hitting set for. Thm: If is a succinct hitting set for, no algebraically natural proof against with the distinguisher coming from. Proof: If D is non-zero then D does not vanish on. (also observed independently by [Grochow-Kumar-Saraf-Saks])

45 Def: Let [x 1,..., x n ] be a class of degree d polynomials, and [X 1,..., X N ] for N = n+d d. is a succinct hitting set for if := {coeff(f ) : f } is a hitting set for. Thm: If is a succinct hitting set for, no algebraically natural proof against with the distinguisher coming from. Proof: If D is non-zero then D does not vanish on. (also observed independently by [Grochow-Kumar-Saraf-Saks]) Question: are poly(n) size and poly(n) degree circuits a hitting sets for poly(n) size and poly(n) degree circuits? ( does VP hit VP? )

46 Let = {coeff(f ) : f VP(n)}. If is a hitting set for VP(N) then there are no VP-algebraic natural proofs against VP.

47 Let = {coeff(f ) : f VP(n)}. If is a hitting set for VP(N) then there are no VP-algebraic natural proofs against VP.

48 Let = {coeff(f ) : f VP(n)}. If is a hitting set for VP(N) then there are no VP-algebraic natural proofs against VP. (note: may a-priori be infinite but we ll soon see that this actually implies there exists some small )

49 Def: A polynomial map : l N is a generator for a class if for every non-zero F, F( (y)) 0.

50 Def: A polynomial map : l N is a generator for a class if for every non-zero F, F( (y)) 0. (want: deg poly(n), l as small as possible)

51 Def: A polynomial map : l N is a generator for a class if for every non-zero F, F( (y)) 0. (want: deg poly(n), l as small as possible) generators hitting sets

52 Def: A polynomial map : l N is a generator for a class if for every non-zero F, F( (y)) 0. (want: deg poly(n), l as small as possible) generators hitting sets ( = : evaluate. = : interpolate.)

53 Def: A polynomial map : l N is a generator for a class if for every non-zero F, F( (y)) 0. (want: deg poly(n), l as small as possible) generators hitting sets ( = : evaluate. = : interpolate.) succinct hitting sets =?

54 Def: A polynomial G(x, y) is a -succinct generator for if:

55 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α).

56 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α). 2. The polynomial map = coeff x (G(x, y)) is a generator for.

57 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α). 2. The polynomial map = coeff x (G(x, y)) is a generator for. G(x, y) = 1 (y 1 + y 2 ) + x 1 (y 1 y 3 2 ) + x 2 (y y 2) + x 1 x 2 1 (y) = (y 1 + y 2, y 1 y 3 2, y2 1 + y 2, 1)

58 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α). 2. The polynomial map = coeff x (G(x, y)) is a generator for. G(x, y) = 1 (y 1 + y 2 ) + x 1 (y 1 y 3 2 ) + x 2 (y y 2) + x 1 x 2 1 (y) = (y 1 + y 2, y 1 y 3 2, y2 1 + y 2, 1) {G(x, α) : α l } is a -succinct hitting set against. So succinct generator = succinct hitting set (and even a uniform one).

59 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α). 2. The polynomial map = coeff x (G(x, y)) is a generator for. Aside: why not just require G(x, y)?

60 Def: A polynomial G(x, y) is a -succinct generator for if: 1. For every α, G(x, α). 2. The polynomial map = coeff x (G(x, y)) is a generator for. Aside: why not just require G(x, y)? We can, but 1. unnecessary for succinct hitting sets which imply barriers 2. G(x, α) might be in even smaller class (e.g., if y has high deg)

61 Recall: succinct generator = succinct hitting sets.

62 Recall: succinct generator = succinct hitting sets. Other direction?

63 Recall: succinct generator = succinct hitting sets. Other direction? Interpolating has complexity poly( ) which is not succinct.

64 Recall: succinct generator = succinct hitting sets. Other direction? Interpolating has complexity poly( ) which is not succinct. But still true because of the existence of universal circuits.

65 Recall: succinct generator = succinct hitting sets. Other direction? Interpolating has complexity poly( ) which is not succinct. But still true because of the existence of universal circuits. In particular, if := {coeff(f ) : f VP(n)} is an (infinite) hitting set for VP(N), there is a N poly log(n) size hitting set ( is in the image of a universal circuit).

66 Conjecture: VP hits VP.

67 Conjecture: VP hits VP. How to obtain evidence to support this conjecture?

68 Conjecture: VP hits VP. How to obtain evidence to support this conjecture? Can we construct algebraic pseudorandom functions?

69 Conjecture: VP hits VP. How to obtain evidence to support this conjecture? Can we construct algebraic pseudorandom functions? [Aaronson-Drucker] s candidate: det l i, j (x) where l i, j s are random linear functions.

70 Conjecture: VP hits VP. How to obtain evidence to support this conjecture? Can we construct algebraic pseudorandom functions? [Aaronson-Drucker] s candidate: det l i, j (x) where l i, j s are random linear functions. Conjecture: this is pseudorandom (maybe only against alg ckts?)

71 Conjecture: VP hits VP. How to obtain evidence to support this conjecture? Can we construct algebraic pseudorandom functions? [Aaronson-Drucker] s candidate: det l i, j (x) where l i, j s are random linear functions. Conjecture: this is pseudorandom (maybe only against alg ckts?) Challange: establish this under some crypto hardness assumption.

72 This work: nearly all the hitting sets we know for restricted models can be made succinct.

73 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s:

74 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas,

75 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree,

76 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree, sparse polynomials,

77 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree, sparse polynomials, Σm ΣΠ O(1) formulas,

78 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree, sparse polynomials, Σm ΣΠ O(1) formulas, commutative roabps,

79 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree, sparse polynomials, Σm ΣΠ O(1) formulas, commutative roabps, depth-o(1) Occur-O(1) formulas

80 This work: nearly all the hitting sets we know for restricted models can be made succinct. Thm: coefficient vectors of poly(log s, n) size multilinear formulas are hitting sets for N = 2 n variate size s: Σ k ΠΣ formulas, ΣΠΣ formulas of constant transcendence degree, sparse polynomials, Σm ΣΠ O(1) formulas, commutative roabps, depth-o(1) Occur-O(1) formulas circuits composed with sparse polynomials of transcendence degree O(1).

81 Not as succinct but still worth mentioning:

82 Not as succinct but still worth mentioning: width w 2 length-n roabps are hitting set for width-w length-n roabps (in certain variable orders).

83 Not as succinct but still worth mentioning: width w 2 length-n roabps are hitting set for width-w length-n roabps (in certain variable orders). (open: make the w 2 closer to poly log(w) and remove the restriction on ordering)

84 Not as succinct but still worth mentioning: width w 2 length-n roabps are hitting set for width-w length-n roabps (in certain variable orders). (open: make the w 2 closer to poly log(w) and remove the restriction on ordering) We can t make all known hitting sets succinct but we have some excuses (more on that later).

85 Toy example: [X 1,..., X N ] is the class of polynomials with monomials of support poly log(n).

86 Toy example: [X 1,..., X N ] is the class of polynomials with monomials of support poly log(n). Hitting set for : {v : supp(v) poly log(n) = poly(n)}.

87 Toy example: [X 1,..., X N ] is the class of polynomials with monomials of support poly log(n). Hitting set for : {v : supp(v) poly log(n) = poly(n)}. ( guess vars in small support monomials, brute-force over them)

88 Toy example: [X 1,..., X N ] is the class of polynomials with monomials of support poly log(n). Hitting set for : {v : supp(v) poly log(n) = poly(n)}. ( guess vars in small support monomials, brute-force over them) Q: Is this succinct?

89 Toy example: [X 1,..., X N ] is the class of polynomials with monomials of support poly log(n). Hitting set for : {v : supp(v) poly log(n) = poly(n)}. ( guess vars in small support monomials, brute-force over them) Q: Is this succinct? A: Yes. Each v is a coefficient vector of a poly(n) ΣΠ circuit in x 1,..., x n (only poly(n) monomials with non-zero coefficient).

90 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s.

91 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials.

92 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials. Exercise: if F(X) has sparsity s, F(X + 1) has non-zero monomial of support log s. [Forbes15, Gurjar-Korwar-Saxena-Theirauf16]

93 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials. Exercise: if F(X) has sparsity s, F(X + 1) has non-zero monomial of support log s. [Forbes15, Gurjar-Korwar-Saxena-Theirauf16] Cor: {v + 1 : supp(v) poly log(s)} hitting set for.

94 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials. Exercise: if F(X) has sparsity s, F(X + 1) has non-zero monomial of support log s. [Forbes15, Gurjar-Korwar-Saxena-Theirauf16] Cor: {v + 1 : supp(v) poly log(s)} hitting set for. Q: Is this succinct?

95 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials. Exercise: if F(X) has sparsity s, F(X + 1) has non-zero monomial of support log s. [Forbes15, Gurjar-Korwar-Saxena-Theirauf16] Cor: {v + 1 : supp(v) poly log(s)} hitting set for. Q: Is this succinct? A: Yes. 1 is coeff vector of n i=1 (x i + 1). Now take the sum of this and the circuit from previous slide.

96 Less-of-a-toy example: [X 1,..., X N ] is the class of polynomials of sparsity at most s. F doesn t necessarily contain small support monomials. Exercise: if F(X) has sparsity s, F(X + 1) has non-zero monomial of support log s. [Forbes15, Gurjar-Korwar-Saxena-Theirauf16] Cor: {v + 1 : supp(v) poly log(s)} hitting set for. Q: Is this succinct? A: Yes. 1 is coeff vector of n i=1 (x i + 1). Now take the sum of this and the circuit from previous slide. Cor: poly(log s, n)-σπσ succinct hitting set for sparse polynomials.

97 In the paper: many other succinct generators, most of them follow from various combinations of basic constructs such as Shpilka-Volkovich generator and Gabizon-Raz s rank condenser, which we make succinct.

98 In the paper: many other succinct generators, most of them follow from various combinations of basic constructs such as Shpilka-Volkovich generator and Gabizon-Raz s rank condenser, which we make succinct. Builds on a lot of previous work: [Dvir-Shpilka, Karnin-Shpilka, Kayal-Saraf, Saxena-Seshadhri, Shpilka-Volkovich, Forbes-Shpilka, Forbes-Shpilka-Saptharishi, Beecken- Mittmann-Saxena, Agrawal-Saha-Saxena-Saptharishi,...].

99 In the paper: many other succinct generators, most of them follow from various combinations of basic constructs such as Shpilka-Volkovich generator and Gabizon-Raz s rank condenser, which we make succinct. Builds on a lot of previous work: [Dvir-Shpilka, Karnin-Shpilka, Kayal-Saraf, Saxena-Seshadhri, Shpilka-Volkovich, Forbes-Shpilka, Forbes-Shpilka-Saptharishi, Beecken- Mittmann-Saxena, Agrawal-Saha-Saxena-Saptharishi,...]. Cor: Super-polynomial lower bounds on defining equations of VP in many models.

100 conspicuously missing: Klivans-Spielman (Kronecker) generator.

101 conspicuously missing: Klivans-Spielman (Kronecker) generator. X i y ki mod p y new var, k integer and p prime chosen from sufficiently large set.

102 conspicuously missing: Klivans-Spielman (Kronecker) generator. X i y ki mod p y new var, k integer and p prime chosen from sufficiently large set. Hits sparse polynomials and very useful in other constructions.

103 conspicuously missing: Klivans-Spielman (Kronecker) generator. X i y ki mod p y new var, k integer and p prime chosen from sufficiently large set. Hits sparse polynomials and very useful in other constructions. Challenge: make it succinct, i.e., find a small circuit in {x 1,..., x n, y} such that the coefficient of x S is y kbin(s) mod p, bin(s) = integer whose binary expansion is the characteristic vector of S.

104 The generator X i y ki mod p 1 1 y ki mod p m m for m = O(log n) hits roabps (in any order) and read-once determinants (polys of the form det(m) where each entry in M contains a var or a constant and each var appears at most once) [Agrawal-Gurjar-Korwar-Saxena, Fenner-Gurjar-Thierauf, Gurjar-Thierauf]

105 The generator X i y ki mod p 1 1 y ki mod p m m for m = O(log n) hits roabps (in any order) and read-once determinants (polys of the form det(m) where each entry in M contains a var or a constant and each var appears at most once) [Agrawal-Gurjar-Korwar-Saxena, Fenner-Gurjar-Thierauf, Gurjar-Thierauf] [Raz] and [Raz-Yehudayoff] showed: small multilinear formulas are not a hitting set for read-once determinants.

106 The generator X i y ki mod p 1 1 y ki mod p m m for m = O(log n) hits roabps (in any order) and read-once determinants (polys of the form det(m) where each entry in M contains a var or a constant and each var appears at most once) [Agrawal-Gurjar-Korwar-Saxena, Fenner-Gurjar-Thierauf, Gurjar-Thierauf] [Raz] and [Raz-Yehudayoff] showed: small multilinear formulas are not a hitting set for read-once determinants. Our constructions are all small multilinear formulas, so we might need new ideas.

107 More models for which we know of hitting sets but not succinct ones:

108 More models for which we know of hitting sets but not succinct ones: roabps in any order read-k oblivious ABPs bounded-depth multilinear formulas

109 More models for which we know of hitting sets but not succinct ones: roabps in any order read-k oblivious ABPs bounded-depth multilinear formulas Also: pseudorandom polynomials? Is [Aaronson-Drucker] s construction pseudorandom?

110 More models for which we know of hitting sets but not succinct ones: roabps in any order read-k oblivious ABPs bounded-depth multilinear formulas Also: pseudorandom polynomials? Is [Aaronson-Drucker] s construction pseudorandom?