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• 1. Learning Read-Constant Polynomials of Constant Degree Modulo Composites 1 2 Arkadev Chattopadhyay ` Ricard Gavalda 3 4 Kristoffer Arnsfelt Hansen ´ Denis Therien 1 University of Toronto 2 ` Universitat Politecnica de Catalunya 3 Aarhus University 4 McGill University CSR 2011 — June 14, 2011Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 1/27
• 2. This talk Boolean functions represented by polynomials over Zm , with m composite. Angluin’s exact learning model. Programs over groups/monoids.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 2/27
• 3. Modular polynomials P(x1 , . . . , xn ) is a polynomial over Zm . A ⊆ Zm is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ ALearning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 3/27
• 4. Modular polynomials P(x1 , . . . , xn ) is a polynomial over Zm . A ⊆ Zm is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ A We also say that P is a “generalized” representation of f .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 3/27
• 5. Modular polynomials P(x1 , . . . , xn ) is a polynomial over Zm . A ⊆ Zm is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ A We also say that P is a “generalized” representation of f . Most important parameter: The degree deg(P) of P.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 3/27
• 6. State of affairs We don’t really understand the computational power of modular polynomials.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 4/27
• 7. State of affairs We don’t really understand the computational power of modular polynomials. 1/r “Surprising” upper bounds of degree O(n ) for some fundamental functions.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 4/27
• 8. State of affairs We don’t really understand the computational power of modular polynomials. 1/r “Surprising” upper bounds of degree O(n ) for some fundamental functions. Best degree lower bound is Ω(log n).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 4/27
• 9. State of affairs We don’t really understand the computational power of modular polynomials. 1/r “Surprising” upper bounds of degree O(n ) for some fundamental functions. Best degree lower bound is Ω(log n). This talk: Polynomials of constant degree.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 4/27
• 10. Angluin’s exact learning model Let C be a concept class of Boolean functions, together with a scheme for representing the functions. Teacher holds Boolean function f ∈ C (the target function). Learner, having no knowledge of f , wishes to learn f (i.e ﬁnd a representation of f ).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 5/27
• 11. Angluin’s exact learning model Let C be a concept class of Boolean functions, together with a scheme for representing the functions. Teacher holds Boolean function f ∈ C (the target function). Learner, having no knowledge of f , wishes to learn f (i.e ﬁnd a representation of f ). Two types of queries:Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 5/27
• 12. Angluin’s exact learning model Let C be a concept class of Boolean functions, together with a scheme for representing the functions. Teacher holds Boolean function f ∈ C (the target function). Learner, having no knowledge of f , wishes to learn f (i.e ﬁnd a representation of f ). Two types of queries: n Membership: Learner presents x ∈ {0, 1} . Teacher responds with f (x).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 5/27
• 13. Angluin’s exact learning model Let C be a concept class of Boolean functions, together with a scheme for representing the functions. Teacher holds Boolean function f ∈ C (the target function). Learner, having no knowledge of f , wishes to learn f (i.e ﬁnd a representation of f ). Two types of queries: n Membership: Learner presents x ∈ {0, 1} . Teacher responds with f (x). Equivalence: Learner presents (representation of) Boolean function g. Teacher responds with EQUAL if f = g, n or presents x ∈ {0, 1} such that g(x) = f (x).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 5/27
• 14. A slight generalization of modular polynomials P(x1 , . . . , xn ) is a polynomial over ﬁnite commutative ring with unity R. A ⊆ R is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ ALearning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 6/27
• 15. A slight generalization of modular polynomials P(x1 , . . . , xn ) is a polynomial over ﬁnite commutative ring with unity R. A ⊆ R is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ A Remarks: l Without loss of generality: R = Zm for some m and l.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 6/27
• 16. A slight generalization of modular polynomials P(x1 , . . . , xn ) is a polynomial over ﬁnite commutative ring with unity R. A ⊆ R is an accepting set. The pair (P, A) computes the Boolean function n f : {0, 1} → {0, 1} given by f (x) = 1 if and only if P(x) ∈ A Remarks: l Without loss of generality: R = Zm for some m and l. Equivalently: A constant sized Boolean combination of Boolean functions computed by polynomials over Zm .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 6/27
• 17. Our goal with this research Theorem (?) Let R be ﬁxed. The class of Boolean functions computed by polynomials over R of constant degree can be exactly learned in deterministic polynomial from membership queries.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 7/27
• 18. Our goal with this research Theorem (?) Let R be ﬁxed. The class of Boolean functions computed by polynomials over R of constant degree can be exactly learned in deterministic polynomial from membership queries. Still an open problem, even when R = Zm for composite m.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 7/27
• 19. Our result Theorem Let R be ﬁxed. The class of Boolean functions computed by read-constant polynomials over R of constant degree can be exactly learned in deterministic polynomial from membership queries.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 8/27
• 20. Our result Theorem Let R be ﬁxed. The class of Boolean functions computed by read-constant polynomials over R of constant degree can be exactly learned in deterministic polynomial from membership queries. Read-constant: Every variable occurs in only constantly many monomials.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 8/27
• 21. Programs over groups/monoids Let M be a ﬁnite monoid. Instruction Triple < i, g, h >, i ∈ [n], g, h ∈ M. Program List L = ( 1 , . . . , m) of instructions, and accepting set A ⊆ M.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 9/27
• 22. Programs over groups/monoids Let M be a ﬁnite monoid. Instruction Triple < i, g, h >, i ∈ [n], g, h ∈ M. Program List L = ( 1 , . . . , m) of instructions, and accepting set A ⊆ M. The pair (L, A) computes the Boolean function function f as follows. g if xi = 1 < i, g, h >(x) = h if xi = 0Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 9/27
• 23. Programs over groups/monoids Let M be a ﬁnite monoid. Instruction Triple < i, g, h >, i ∈ [n], g, h ∈ M. Program List L = ( 1 , . . . , m) of instructions, and accepting set A ⊆ M. The pair (L, A) computes the Boolean function function f as follows. g if xi = 1 < i, g, h >(x) = h if xi = 0 m f (x) = 1 if and only if ∏ i (x) ∈A i=1Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 9/27
• 24. Programs vs. circuit classes Algebraic structure Circuit class 1 Any monoid NC 0 Solvable monoid ACC 0 Aperiodic monoid AC . . .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 10/27
• 25. Programs vs. circuit classes Algebraic structure Circuit class 1 Any monoid NC 0 Solvable monoid ACC 0 Aperiodic monoid AC . . . 0 Solvable group CC 0 0 Nilpotent group NC ◦ MOD ◦ NC 0 Abelian group NC ◦ MOD 0 p-group MODp ◦ NC . . .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 10/27
• 26. Programs vs. circuit classes Algebraic structure Circuit class 1 Any monoid NC 0 Solvable monoid ACC 0 Aperiodic monoid AC . . . 0 Solvable group CC 0 0 Nilpotent group NC ◦ MOD ◦ NC 0 Abelian group NC ◦ MOD 0 p-group MODp ◦ NC . . . The class of Boolean functions computed by constant degree polynomials over a ﬁnite commutative ring with unity R is a very natural concept class!Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 10/27
• 27. Notation M ⊆ [n] represent the monomial ∏i∈M xi . n χM ∈ {0, 1} is characteristic vector of M. cM is coefﬁcient of monomial M. n If w ∈ {0, 1} , Iw ⊆ [n] is the indices i where wi = 1.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 11/27
• 28. Structural properties of polynomials ´ ´ Theorem (Peladeau and Therien) Let R be a ﬁnite commutative ring with unity and d any number. Then there exist a constant c = c(R, d) such that: For any polynoimial P over R of degree at most d and for n any r ∈ range(P) there exist w ∈ {0, 1} with |Iw | ≤ c such that P(w) = r .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 12/27
• 29. Structural properties of polynomials ´ ´ Theorem (Peladeau and Therien) Let R be a ﬁnite commutative ring with unity and d any number. Then there exist a constant c = c(R, d) such that: For any polynoimial P over R of degree at most d and for n any r ∈ range(P) there exist w ∈ {0, 1} with |Iw | ≤ c such that P(w) = r . Remarks: Proved using an inductive Ramsey-theoretic argument. Strong relation to the degree for representing the AND function over R.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 12/27
• 30. Boolean function determined by low-weight inputs Corollary There exist a constant c = c (R, d) such that: Let P and Q be polynomials of degree at most d with accepting sets A and B. If the Boolean functions computed by the pairs (P, A) and n (Q, B) agree on all inputs w ∈ {0, 1} with |Iw | ≤ c , then the two Boolean functions are identical.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 13/27
• 31. Boolean function determined by low-weight inputs Corollary There exist a constant c = c (R, d) such that: Let P and Q be polynomials of degree at most d with accepting sets A and B. If the Boolean functions computed by the pairs (P, A) and n (Q, B) agree on all inputs w ∈ {0, 1} with |Iw | ≤ c , then the two Boolean functions are identical. Proof. Let c = c(R × R, d). Consider the polynomial P × Q with P and Q as separate coordinates. Any value of the range is assumed on input of weight at most c .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 13/27
• 32. A consistency test Input: Polynomial Q with accepting set A. Membership query access to Boolean function f . Task: Decide if the pair (Q, A) computes the function f .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 14/27
• 33. A consistency test Input: Polynomial Q with accepting set A. Membership query access to Boolean function f . Task: Decide if the pair (Q, A) computes the function f . n Query f on all w ∈ {0, 1} with |Iw | ≤ c (R, d) Return true if and only if for each queried w, f (w) = 1 if and only if Q(w) ∈ ALearning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 14/27
• 34. A consistency test Input: Polynomial Q with accepting set A. Membership query access to Boolean function f . Task: Decide if the pair (Q, A) computes the function f . n Query f on all w ∈ {0, 1} with |Iw | ≤ c (R, d) Return true if and only if for each queried w, f (w) = 1 if and only if Q(w) ∈ A This also gives equivalence queries for free.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 14/27
• 35. A template for learning Deﬁne a search space: Let P be guaranteed to include the target (P presumably depends on answers to membership queries).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 15/27
• 36. A template for learning Deﬁne a search space: Let P be guaranteed to include the target (P presumably depends on answers to membership queries). Exhaustive search: for each P ∈ P perform consistency test on P, return ﬁrst P that passes.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 15/27
• 37. Difﬁculties Different polynomials can represent the same Boolean function.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 16/27
• 38. Difﬁculties Different polynomials can represent the same Boolean function. A query χM with |M| = 1 only reveals whether cM ∈ A.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 16/27
• 39. Difﬁculties Different polynomials can represent the same Boolean function. A query χM with |M| = 1 only reveals whether cM ∈ A. A query χM in general only reveals whether ∑I⊆M cI ∈ A.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 16/27
• 40. Difﬁculties Different polynomials can represent the same Boolean function. A query χM with |M| = 1 only reveals whether cM ∈ A. A query χM in general only reveals whether ∑I⊆M cI ∈ A. Our approach: Deﬁne an equivalence relation on the monomials, such that equivalent monomials can be assumed to have same coefﬁcient.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 16/27
• 41. An attempt Let M and M be of degree d. Say M ≡d M if: ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose we know the equivalence classes of ≡d ’s and let P be the set of polynomials respecting these.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 17/27
• 42. An attempt Let M and M be of degree d. Say M ≡d M if: ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose we know the equivalence classes of ≡d ’s and let P be the set of polynomials respecting these. Good: Small search space (in fact constant size).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 17/27
• 43. An attempt Let M and M be of degree d. Say M ≡d M if: ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose we know the equivalence classes of ≡d ’s and let P be the set of polynomials respecting these. Good: Small search space (in fact constant size). Good: Deﬁnition looks a little like membership queries...Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 17/27
• 44. An attempt Let M and M be of degree d. Say M ≡d M if: ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose we know the equivalence classes of ≡d ’s and let P be the set of polynomials respecting these. Good: Small search space (in fact constant size). Good: Deﬁnition looks a little like membership queries... Bad: but not exactly, so assumption is probably unrealistic!Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 17/27
• 45. An attempt Let M and M be of degree d. Say M ≡d M if: ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose we know the equivalence classes of ≡d ’s and let P be the set of polynomials respecting these. Good: Small search space (in fact constant size). Good: Deﬁnition looks a little like membership queries... Bad: but not exactly, so assumption is probably unrealistic! Importantly: Does it include the target function?Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 17/27
• 46. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 47. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ].Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 48. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ]. n Let x ∈ {0, 1} be arbitrary. We need only consider x such that M ⊆ Ix .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 49. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ]. n Let x ∈ {0, 1} be arbitrary. We need only consider x such that M ⊆ Ix . Deﬁne r = P(x) − P(χM ).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 50. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ]. n Let x ∈ {0, 1} be arbitrary. We need only consider x such that M ⊆ Ix . Deﬁne r = P(x) − P(χM ). r + P(χM ) = P(x) − P(χM ) + P(χM ) = P(x).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 51. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ]. n Let x ∈ {0, 1} be arbitrary. We need only consider x such that M ⊆ Ix . Deﬁne r = P(x) − P(χM ). r + P(χM ) = P(x) − P(χM ) + P(χM ) = P(x). r + P(χM ) = P(x) − P(χM ) + P(χM ) = P(x) − P(χM ) + P (χM ) = P (x).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 52. Changing coefﬁcients Suppose M ≡d M : ∀r ∈ R : r + P(χM ) ∈ A if and only if r + P(χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1 Mi . Suppose (by induction?) that cMi = cM for all i, and deﬁne i P = P[cM ← cM ]. n Let x ∈ {0, 1} be arbitrary. We need only consider x such that M ⊆ Ix . Deﬁne r = P(x) − P(χM ). r + P(χM ) = P(x) − P(χM ) + P(χM ) = P(x). r + P(χM ) = P(x) − P(χM ) + P(χM ) = P(x) − P(χM ) + P (χM ) = P (x). Conclusion: P and P compute the same function.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 18/27
• 53. A problem The equivalence relation deﬁned is highly representation dependent.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 19/27
• 54. A problem The equivalence relation deﬁned is highly representation dependent. Changing coefﬁcients may possibly change the equivalence classes as well!Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 19/27
• 55. A problem The equivalence relation deﬁned is highly representation dependent. Changing coefﬁcients may possibly change the equivalence classes as well! Our real equivalence relation is deﬁned only in terms of the function represented together with a property of the polynomial that is preserved by altering coefﬁcients (in the similar way)!Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 19/27
• 56. Recall: Structural properties of polynomials ´ ´ Theorem (Peladeau and Therien) Let R be a ﬁnite commutative ring with unity and d any number. Then there exist a constant c = c(R, d) such that: For any polynoimial P over R of degree at most d and for n any r ∈ range(P) there exist w ∈ {0, 1} with |Iw | ≤ c such that P(w) = r .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 20/27
• 57. Magic sets for polynomials Corollary There exist a constant s = s(R, d), such that for every multilinear polynomial P over R of degree at most d, there exist a set J ⊂ {1, . . . , n} with the following properties: |J| ≤ s. n For every r ∈ range(P) there exist w ∈ {0, 1} with Iw ⊆ J such that P(w) = r . We call J a “magic set” for P.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 21/27
• 58. Using a magic set Assume P has magic set J.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 22/27
• 59. Using a magic set Assume P has magic set J. Let K be variables sharing monomial with variable in J. Let N = [n] (J ∪ K ). (We say N is at distance ≥ 2 from J).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 22/27
• 60. Using a magic set Assume P has magic set J. Let K be variables sharing monomial with variable in J. Let N = [n] (J ∪ K ). (We say N is at distance ≥ 2 from J). n Let w, x ∈ {0, 1} with Iw ⊆ J, Ix ⊆ N.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 22/27
• 61. Using a magic set Assume P has magic set J. Let K be variables sharing monomial with variable in J. Let N = [n] (J ∪ K ). (We say N is at distance ≥ 2 from J). n Let w, x ∈ {0, 1} with Iw ⊆ J, Ix ⊆ N. Then P(w ∨ x) = P(w) + P(x).Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 22/27
• 62. Using a magic set Assume P has magic set J. Let K be variables sharing monomial with variable in J. Let N = [n] (J ∪ K ). (We say N is at distance ≥ 2 from J). n Let w, x ∈ {0, 1} with Iw ⊆ J, Ix ⊆ N. Then P(w ∨ x) = P(w) + P(x). Note: When P is read-constant also K is of constant size.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 22/27
• 63. Properties of polynomials with magic sets Lemma Let P(x) = ∑I⊆[n],|I|≤d cI ∏i∈I xi be any polynomial over R, with a magic set J. Let N be the set of indices at distance ≥ 2 from J. Then r + ∑I⊆N,|I|≤d λI cI ∈ range(P), for all r ∈ range(P) and all λI ∈ {0, . . . , |R| − 1}.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 23/27
• 64. Properties of polynomials with magic sets Lemma Let P(x) = ∑I⊆[n],|I|≤d cI ∏i∈I xi be any polynomial over R, with a magic set J. Let N be the set of indices at distance ≥ 2 from J. Then r + ∑I⊆N,|I|≤d λI cI ∈ range(P), for all r ∈ range(P) and all λI ∈ {0, . . . , |R| − 1}. Proof. By induction, repeatedly moving supports of inputs to J.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 23/27
• 65. The real attempt Assume J is magic set for P, and let M and M be of degree d. Say M ≡d,J M if: ∀w, Iw ⊆ J : P(w ∨ χM ) ∈ A if and only if P(w ∨ χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1,J Mi .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 24/27
• 66. The real attempt Assume J is magic set for P, and let M and M be of degree d. Say M ≡d,J M if: ∀w, Iw ⊆ J : P(w ∨ χM ) ∈ A if and only if P(w ∨ χM ) ∈ A. For all immediate sub-monomials M1 , . . . , Md and M1 , . . . , Md we have Mi ≡d−1,J Mi . If IM , IM ⊆ N, we can change coefﬁcients as before, and the induction now works.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 24/27
• 67. The search space Let P be the set of polynomials satisfying: There is a magic set J, such that for all monomials M, M of degree d with IM , IM ⊆ N, if M ≡d,J M then cM = cM .Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 25/27
• 68. The search space Let P be the set of polynomials satisfying: There is a magic set J, such that for all monomials M, M of degree d with IM , IM ⊆ N, if M ≡d,J M then cM = cM . Note: Given J and K , equivalence classes of ≡d,J can be computed using membership queries. Remains to specify: Coefﬁcients of monomials containing variables from J ∪ K . Coefﬁcients of equivalence classes.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 25/27
• 69. The search space Let P be the set of polynomials satisfying: There is a magic set J, such that for all monomials M, M of degree d with IM , IM ⊆ N, if M ≡d,J M then cM = cM . Note: Given J and K , equivalence classes of ≡d,J can be computed using membership queries. Remains to specify: Coefﬁcients of monomials containing variables from J ∪ K . Coefﬁcients of equivalence classes. Conclusion: P is of polynomial size, and can be exhaustively searched using membership queries!Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 25/27
• 70. Extensions to higher degree Using a result of Tardos and Barrington on the MODm degree of AND, we can show r −1 l l l d c(Zm , d), c (Zm , d), s(Zm , d) ≤ γ where r is the number of distinct prime divisors of m, and γ depends on m and l.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 26/27
• 71. Extensions to higher degree Using a result of Tardos and Barrington on the MODm degree of AND, we can show r −1 l l l d c(Zm , d), c (Zm , d), s(Zm , d) ≤ γ where r is the number of distinct prime divisors of m, and γ depends on m and l. Result: Sub-exponential learning algorithm for slowly growing d and k for learning degree d read-k polynomials.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 26/27
• 72. Conclusion New Result: A Polynomial time learning algorithm for read-constant polynomials of constant degree over ﬁnite rings.Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 27/27
• 73. Conclusion New Result: A Polynomial time learning algorithm for read-constant polynomials of constant degree over ﬁnite rings. Questions: Can we avoid the read-constant assumption? Other uses of “magic sets”?Learning Read-Constant Polynomials of Constant Degree Modulo Composites - CSR 2011 Page 27/27