Understanding and Implementing F5

Transcription

1 Understanding and Implementing F5 John Perry University of Southern Mississippi Understanding and Implementing F5 p.1

2 Overview Understanding F5 Description Criteria Proofs Implementing F5 Maple, Singular Relation to Buchberger s Criteria Some optimizations Understanding and Implementing F5 p.2

3 Motivation challenge! compare with other criteria Buchberger criteria Pivoting, Extended First Criteria open source / something to work with Understanding and Implementing F5 p.3

4 Part 1 Understanding F5 Understanding and Implementing F5 p.4

5 Introduction BA (1965) GB algorithm GM (1988) fast GB algorithm selection strategy: normal near-optimal use of Buchberger s Criteria Understanding and Implementing F5 p.5

6 Introduction F4 (1999) F5 (2002) FAST GB algorithm selection strategy: increasing total degree sparse linear algebra FAAAAAAAST GB algorithm (sometimes*) incremental selection strategy ( f 1 ),( f 1, f 2 ),..., f 1, f 2,..., f m new criteria Buchberger criteria disregarded regular sequence= no reduction to zero * sometimes : If close to GB, and most real-world systems in engineering are close to GB. (Faugère) Understanding and Implementing F5 p.5

7 Introduction F4 (1999) F5 (2002) FAST GB algorithm well understood many implementations, widely available FAAAAAAAST GB algorithm (sometimes) not well understood few implementations (5)* fewer work *Stegers count, (mid-2005) Understanding and Implementing F5 p.5

8 Introduction 2002 paper limited length algorithm vs. proofs? full description of algorithm sketch of proofs problems solved w/f5 no new description of algorithm no proofs 2007 HDR? not in wide circulation Understanding and Implementing F5 p.6

9 Idea Linear system: f 1 = 2x+ 3y 8 f 2 = 4x+ 5y 12 Understanding and Implementing F5 p.7

10 Idea Linear system: Understanding and Implementing F5 p.7

11 Idea Linear system: ( f 3 = 2 f 1 f 2 ) Understanding and Implementing F5 p.7

12 Idea Linear system: Moral: f 3 = 2 f 1 f 2 discard f 1 or f 2 from matrix linear dependence f 2 is rewritable by f 3 R criterion Understanding and Implementing F5 p.7

13 Linear to non-linear Compute GB Triangularize Sylvester submatrix (Lazard, 1983) S-poly two rows needing cancellation p G 0 iff linear dependence in Syl(G) Understanding and Implementing F5 p.8

14 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f x f y f f f 2 Understanding and Implementing F5 p.9

15 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f x f 2 f y f f f 2 f 3 = y f 1 x f 2 Understanding and Implementing F5 p.9

16 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f y f f x f f y f f 3 Understanding and Implementing F5 p.9

17 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f y f f x f f y f 3 f f 3 f 4 = f 2 y f 3 Understanding and Implementing F5 p.9

18 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f y f f x f f f f 3 Understanding and Implementing F5 p.9

19 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f y f f f f f 3 x f 3 f 5 f 5 = f 1 x f 3 *Note f 5 = f 2 no new info f 1 x f 3 useless Understanding and Implementing F5 p.9

20 Linear to non-linear Example: f 1 = x 2 1, f 2 = xy 1 x 3 x 2 y xy 2 y 3 x 2 xy y 2 x y x f y f y f f f f f 3 Higher order cancellations also useless. Basis: G={ f 1, f 2, f 3, f 4 } Reduced basis: R={ f 3, f 4 } Understanding and Implementing F5 p.9

21 Idea Signatures: multiplier and index f 3 = y f 1 x f 2 f 4 = f 1 x f 3 = f 1 x(y f 1 x f 2 ) =( xy+ 1) f 1 + x 2 f 2 f 5 = f 2 y f 3 = f 2 y(y f 1 x f 2 ) = y 2 f 1 +(xy+ 1) f 2 Understanding and Implementing F5 p.10

22 Idea Signature leading term of representation rewritable linear dependence remainders ( f 3, f 4, f 5 ) not rewritable respect the signatures!!! consider all cancellations in Syl avoid cancellations that might affect signature top-reduction: new polynomial? Understanding and Implementing F5 p.11

23 Summary of idea compute GB triangularize Syl R criterion: older rows newer rows signatures: track linear dependencies Understanding and Implementing F5 p.12

24 Basic description of algorithm compute GB of( f 1, f 2 ) G 2 = g 1, g 2,..., g m2 compute GB of( f 1, f 2, f 3 ) same as GB of g 1, g 2,..., g m2, f 3 G 3 = g 1, g 2,..., g m3 iterate: after computing G i, compute GB of g1, g 2,..., g mi, f i+1 1< i< m Use signatures to avoid linear dependencies / useless cancellations Understanding and Implementing F5 p.13

25 New criterion Question: Could we detect f 1 x f 3 F 0? Answer: Principal Syzygies (PS) f 1 f 2 f 2 f 1 = 0 Sums of monomial multiples of f 1, f 2 linear dependence in Sylvester matrix Understanding and Implementing F5 p.14

26 New criterion Example: f 1 x f 3 =( xy+ 1) f 1 + x 2 f 2. But f 1 f 2 f 2 f 1 = 0= x 2 1 f 2 (xy 1) f 1 = 0 = x 2 f 2 =(xy 1) f 1 + f 2. Thus f 1 x f 3 =( xy+ 1) f 1 +[(xy 1) f 1 + f 2 ]=1 f 2. Lowered signature: already considered any cancellations. Understanding and Implementing F5 p.15

27 Theorem Integrate R Criterion with PS criterion: Theorem. For G prev G, suppose G prev is a Gröbner basis of f 1, f 2,..., f m 1, and for all 1 i<j m such that S g i, g j = u gi v g and g i f 1,..., f m \ f1,..., f m 1 we have (A) or (B) or (C) where (A) u g i or v g j has index m and satisfies PS wrt G prev ; (B) u g i or v g j satisfies R; (C) S g i, g j G 0. Then G is a Gröbner basis of f 1, f 2,..., f m. Understanding and Implementing F5 p.16

28 Why? sketch S := S g i, g j = p F h p p (A) = syzygy σ s.t. S = S σ has lower signature (B) = syzygyσ s.t. S= S σ and = rewritable S-polys have lower signature new rep new cancellations? new S-polys new S-polys satisfy (A), (B), or (C), repeat (A), (B), (C) never increase signature (A), (B) decrease signature of rewritable (C) lowers lcms finitely many S-polys= finitely many signatures decrease signature until no rewritable S-polys all must satisfy (C)= GB Understanding and Implementing F5 p.17

29 Why? sketch (Eder) S (1) g i, g i = h f h (1) f m m (A), (B), (C) S g i, g j =µ1 S (2) g i1, g j1 + h g h (2) g m m µ 1. S g i, g j = µλ S g iλ, g jλ Understanding and Implementing F5 p.18

30 Why? sketch (Perry) S g i, g i = h (1) 1 f h (1) m f m (A), (B), (C) S g i, g i = h (2) 1 g h (2) m g m. i= 1,2,..., m lt S f i, f j = h (r) 1 g h (r) m g m h (r) g i i lt S f i, f j i= 1,2,..., m Understanding and Implementing F5 p.18

31 Part 2 Implementing F5 Understanding and Implementing F5 p.19

32 Challenge understanding? pseudocode difficult to follow unfamiliar notation fatal & misleading errors complex: subalgorithms, different cases some systems work with a broken implementation false sense of confidence... stages interdependent (Stegers) Understanding and Implementing F5 p.20

33 Maple implementation 2007 Modification of traditional Buchberger algorithm Maple programming language (Faugère: C) excruciatingly slow Faugère: very easy Stegers (MAGMA): considerable effort Perry: months pseudocode & commentary: Understanding and Implementing F5 p.21

34 Maple implementation 2008 Slight modification to use matrix not really F4-ish slower!!! gave up Understanding and Implementing F5 p.22

35 Singular implementation 2008 basic modification of traditional Buchberger algorithm Singular programming language much faster than Maple Cyclic-6: < 3 minutes vs. > 15 minutes two days work (correct pseudocode helps) to-do move to kernel sparse linear algebra? compiled C++ vs. interpreted Singular Understanding and Implementing F5 p.23

36 Buchberger s Criteria? Sometimes BC= (PS or R) but not always! Question: Can we improve F5 using BC? Understanding and Implementing F5 p.24

37 Buchberger s Criteria? Sometimes BC= (PS or R) but not always! Question: Can we improve F5 using BC? NO Understanding and Implementing F5 p.24

38 Buchberger s Criteria Example: ( 29 ) f 1 = 3x 4 y+ 18xy 4 + 4x 3 y z+ 20xy 3 z+ 3x 2 z 3 f 2 = 3x 3 y 2 + 7x 2 y y 2 z 3 f 3 = 12xy x 4 z+ 27y 4 z+ 11x 3 z 2 S( f 1, f 4 ) caught by Buchberger s 2nd Criterion (via f 6 ) BUT! skipping S( f 1, f 4 ) does not give a Gröbner basis 28 polynomials in result (vs. 30 if no skip) 66 S-polynomials do not reduce to zero S( f 1, f 4 ) is not one of these Understanding and Implementing F5 p.25

39 Buchberger s Criteria Why fail? certainly S( f 1, f 4 ) should reduce to zero S( f 1, f 6 ), S( f 4, f 6 ) reduce to zero distinct lcms= no three-way trap triangularization: missed step! Buchberger s Criteria ignore signature signatures x 2 z f 1 and y 3 f 2 neither signature is considered if we skip top-reduction: some reductions forbidden unmodified F5: top-reduction of S( f 1, f 4 ) stops with polynomial whose signature is x 2 z f 1 Note: in 11, does not appear to terminate! Understanding and Implementing F5 p.26

40 Other optimizations Stegers, pg. 41 compute reduced GB R prev after each increment reduce, top-reduce wrt R prev, not wrt G prev safe: smaller signatures in R prev do not replace G prev w/r prev signature corruption Understanding and Implementing F5 p.27

41 Other optimizations Stegers, pg. 41 compute reduced GB R prev after each increment reduce, top-reduce wrt R prev, not wrt G prev safe: smaller signatures in R prev do not replace G prev w/r prev signature corruption Yes, but... Understanding and Implementing F5 p.27

42 Other optimizations Can replace G prev with R prev =(g 1, g 3,..., g M ) create appropriate signatures compute basis for g 1, g 2, g 3,..., g M, f m signatures 1g 1, 1g 2,..., 1g M+1 (g M+1 = f m ) also basis for f 1, f 2,..., f m! no more signature corruption! create appropriate rules S g i, g j Rprev 0 signatures for zero polynomial Understanding and Implementing F5 p.28

43 Other optimizations Can replace G prev with R prev =(g 1, g 3,..., g M ) create appropriate signatures compute basis for g 1, g 2, g 3,..., g M, f m signatures 1g 1, 1g 2,..., 1g M+1 (g M+1 = f m ) also basis for f 1, f 2,..., f m! no more signature corruption! create appropriate rules S g i, g j Rprev 0 signatures for zero polynomial much more efficient than published F5 Understanding and Implementing F5 p.28

44 Other optimizations Can replace G prev with R prev =(g 1, g 3,..., g M ) create appropriate signatures compute basis for g 1, g 2, g 3,..., g M, f m signatures 1g 1, 1g 2,..., 1g M+1 (g M+1 = f m ) also basis for f 1, f 2,..., f m! no more signature corruption! create appropriate rules S g i, g j Rprev 0 signatures for zero polynomial A new, more efficient algorithm for computing Gröbner bases with reduction to zero? Understanding and Implementing F5 p.28

45 Some example systems system basic reduce w/r prev compute w/r prev f Katsura Katsura Cyclic (time in seconds, using Singular timer) Interpreted Gebauer-Möller algorithm typically slower. Understanding and Implementing F5 p.29

46 Other optimizations 2 : f 2 = f (Bardet, Faugère, Salvy 2003) general: syzygies that lower signature? Understanding and Implementing F5 p.30

47 Conclusion algorithm better understood (we hope) termination? unresolved new, open-source implementations in interpreted languages slow, okay, yes compilation, linear algebra? significant speedup optimization: possibilities Understanding and Implementing F5 p.31

48 Thanks Christian Eder (joint work) Jean-Charles Faugère Aleksandr Sem nov Till Stegers Alekseī Zobnin Technische Universität Kaiserslautern Understanding and Implementing F5 p.32