Satisfiability over Cross Product is NP NP R -complete Christian - PowerPoint PPT Presentation

Satisfiability over Cross Product is NP NP R -complete º Christian Herrmann, Johanna Sokoli, Martin Ziegler

Re Remi mind nder er: : Co Comp mplex exity ty Th Theo eory ry P := { L  { 0 , 1 }* decidable in polynomial time }  NP NP := { L verifiable in polynomial time }  PSP SPAC ACE := { L decidable in polyn. space } Def: Call L verifiable in polynomial time if L = { x  { 0 , 1 } n | n  N ,  y  { 0 , 1 } q ( n ) :  x , y  V } for some V  P and q  N [ N ] . discrete "witness" Examples: 2-CNF 3SAT = {  : Boolean formula  in 3-CNF in 3-CNF 3 2  P  NP NP admits a satisfying assignment }  NP NP 3COL = {  G  : graph G admits a 3-coloring}  NP NP HC = {  G  : G has a Hamiltonian cycle}  P  NP NP EC = {  G  : G has a Eulerian cycle } Martin Ziegler 2

Re Remi mind nder er: : NP NP -co comp mplet eteness eness P := { L  { 0 , 1 }* decidable in polynomial time }  NP := { L verifiable in polynomial time } Def: Polynom. reduction from A to B  { 0 , 1 }* is a f :{ 0 , 1 }*  { 0 , 1 }* computab. in polytime such that x  A  f ( x )  B . Write A ¹ p B . • A ¹ p B, B ¹ p C  A ¹ p C NP c (e.g. • A ¹ p B, B  P  A  P 3SAT) NP • For any L  NP , L ¹ p SAT (S. Cook / L. Levin 70ies) P • SAT ¹ p 3SAT, HC, 3COL… Martin Ziegler 3

Tu Turing ring vs vs. . BS BSS S Ma Mach chine ine Discrete: Turing Machine / Random-Access Machine ( TM/RAM ) Input/output: finite sequence of bits { 0 , 1 }* or integers  Z * Each memory cell holds one element of R ={0,1} / R = Z R ? º `Program' can store finitely many constants from R operates on R (for TM :  ,  ,  ; for RAM :  ,  ,  ,  ) Computation on algebras/structures [Tucker&Zucker], [Poizat] on R *:=  U k R k : Algebra ( R ,  ,  ,  ,  ,<) → real-RAM , BSS-machine [Blum&Shub&Smale'89],[Blum&Cucker&Shub&Smale'98] P R  NP NP R  EXP XP R (Tarski Quantifier Elimination) strict? º º º NP R -complete: Does a given NP int. real º polynom.system have a real root? H  R * real Halting problem Undecidable, too: Mandelbrot Set, Newton starting points Martin Ziegler 4

Tu Turing ring vs vs. . BS BSS S Co Comp mplexity lexity NP R -complete: Does a given multivariate ° NP integer polynomial have a real root? The heor orem em [Canny'88, Grigoriev'88, Heintz&Roy& ° &Solerno'90, Renegar'92]: NP NP R  PS PSPACE PACE ("efficient real quantifier elimination") No 'better' (e.g. in PH ) algorithm known to-date! ° (Allender, B ü rgisser, Kjeldgaard-Pedersen, Miltersen ‘ 06: P R  CH ) Similarly with integer root: undecidable (Matiyasevich ‘ 70) Similarly with rational root: unknown (e.g. Poonen'09) Simil. with complex root: coRP NP mod GRH (Koiran'96) Martin Ziegler 5

NP R ‒ Completeness Completeness NP ⁰ ⁰ QSAT R : Given a term t ( X 1 ,.. X n ) over  ,  ,  , does it have a satisfying assignment C.Herrmann& over subspaces of R ³/ C ³? M.Z. 2011 FEAS R : Given a system of n -variate ⁰ integer polynomial in-/equalities, does it have a real solution? ⁰ CONV R : … , is the solution set convex? P. Koiran'99 ⁰ Tod oday ay: DIM R : … of dimension n ? The following problem is NP R -complete: N.E.Mnëv (80ies), 0 QUAD R : Given p  Z [ X 1 ,…, X n ] of total ⁰ J. Richter-Gebert'99 degree 4, does it have a real root? Given a term t ( X 1 ,… X n ) over  only, • Is a given oriented matroid realizable? Peter W. Shor'91 does the equation t ( X 1 ,… X n ) = X 1 • Is a given arrangement of pseudolines, stretchable? have a solution over R ³\{0} ? • Certain geometric properties of graphs M. Schaefer 2010 Martin Ziegler 6

Cr Cross oss Pr Prod oduct uct i in n R ³ ( a x , a y , a z )  ( b x , b y , b z ) = ( a y ·b z -a z ·b y , a z ·b x -a x ·b z , a x ·b y -a y ·b z ) a  b ( parallel)  ( ( ( a  b )  a )  a )  ( a  b ) = 0  a  b = 0 ( ( a  b )  a )  a a  b ( a  b )  a b a | a  b | = | a |·| b |· sin  ( a , b ) a  b  a , anti -commutative, non -associative. Martin Ziegler 7

Decision cision Problem blems s with th Cross ss Product duct Theorem: a) to c) and a') to b') are all equivalent ( a x , a y , a z )  ( b x , b y , b z ) = ( a y ·b z -a z ·b y , a z ·b x -a x ·b z , a x ·b y -a y ·b z ) to Polynomial Identity Testing  RP RP ( randomized polytime with one-sided error, Schwartz-Zippel) ( ( ( a  b )  a )  a )  ( a  b ) = 0 d) to f) are all NP NP R -complete 0 d') to f') are equivalent to Hilbert's 10th Problem over Q built from  only: Given a term t ( V 1 ,… V n ) terms and s ( V 1 ,.. V n ) a) Is there an assignment v 1 ,…, v n  R ³ s.t. t ( v 1 ,.. v n )  0 ? In particular there exists a cross product equation t ( v 1 ,.. v n )= v 1  0 satisfiable over R ³ but not over Q ³. b) Is there an assignment v j  R ³ s.t. t ( v 1 ,.. v n )  s ( v 1 ,.. v n ) ? c) Is there an assignment v j  R ³ s.t. t ( v 1 ,.. v n )= e z ? d) Is there an assignment v j  R ³ s.t. t ( v 1 ,.. v n )= v 1  0 ? e) Is there an assignment v j  R ³ s.t. t ( v 1 ,..)  v 1  0 ? f) Is there an assignment v j  R ³ s.t. t ( v 1 ,.. v n )  s ( v 1 ,.. v n ) ? a') to f') similarly but for assignments  Q ³ Martin Ziegler 8

Pr Proof oof (S (Ske ketch, tch, har hardne dness ss) 0 QUAD AD R (Does given p  Z [ X 1 ,.. X n ] have a real root?) ¹ p e) e) Is there an assignment v j  F ³ s.t. t ( v 1 ,.. v n ) ≈ v 1 ≠ 0 ? any go nal For the standard right-handed ortho norm al basis e 1 , e 2 , e 3 of F ³ and for r , s  F , the following are easily verified: • F ( e 1 - r·s e 2 ) = F e 3  [ F ( e 3 - r · e 2 )  F ( e 1 - s · e 3 ) ] Encode s  F • F ( e 1 - s· e 3 ) = F e 2  [ F ( e 2 - e 3 )  F ( e 1 - s · e 2 ) ] as affine as projective line ( e 1 - s· e 2 ) point F ( e 1 - s· e 2 ) • F ( e 3 - s· e 2 ) = F e 1  [ F ( e 1 - e 3 )  F ( e 1 - r · e 2 ) ] • e 1 -( r - s ) · e 2 = e 3  [ ( [( e 2 - e 3 )  ( e 1 - r · e 2 )]  [ e 2  ( e 1 - s e 3 ) ] )  e 3 ] • F ( e 1 - e 3 ) = F e 2  [ F ( e 1 - e 2 )  F ( e 2 - e 3 ) ] Can thus express the arithmetic operations · and - using the cross product and F e 1 and F e 2 and F ( e 1 - e 2 ) and F ( e 2 - e 3 ) . Martin Ziegler 9

Pr Proof oof (S (Ske ketch, tch, har hardne dness ss) 0 QUAD AD R (Does given p  Z [ X 1 ,.. X n ] have a real root?) ¹ p e) e) Is there an assignment v j  F ³ s.t. t ( v 1 ,.. v n ) ≈ v 1 ≠ 0 ? any For the standard right-handed ortho go nal basis e 1 , e 2 , e 3 of F ³, can express  and · using cross product and F e 1 and F e 2 and F ( e 1 - e 2 ) and F ( e 2 - e 3 ) . Encode s  F as projective as affine point F ( e 1 - s· e 2 ) line e 1 - s· e 2 ↝ terms V 1 ( A , B , C ) , V 2 ( A , B,C ) , V 12 ( A , B , C ) , V 23 ( A , B,C ) that for any assignment A,B,C  P² F , either coincide with F e 1 = A and F e 2 and F ( e 1 - e 2 ) and F ( e 2 - e 3 ) for some right-handed orthogonal basis e i ‒ or evaluate to 0 . Using these terms, one can express (in polytime) any given p  Z [ X 1 ,…, X n ] as term t p ( Y 1 ,…, Y n ; A , B , C ) over  s.t. p ( s 1 ,…, s n )=0  t p ( F ( e 1 - s 1 · e 2 ),…, F ( e 1 - s n · e 2 ); A , B , C ) = A Martin Ziegler 10

Co Conc nclusion lusion • Identified a new problem complete for NP R 0 • defined over  only, i.e. conceptionally simplest • normal form for equations over  : t ( Z 1 ,…, Z n )= Z 1 NP R is an important Turing (!) complexity class as NP 0 currently developping into similarly rich structural theory [Baartse&Meer'13] PCP Theorem for NP over the Reals Question: Graph Colorin ing being NP -complete, how about Qu Quantum tum Gr Graph Co Coloring ng ? [LeGall'13] Using these terms, one can express (in polytime) any given p  Z [ X 1 ,…, X n ] as term t p ( Y 1 ,…, Y n ; A , B , C ) over  s.t. p ( s 1 ,…, s n )=0  t p ( F ( e 1 - s 1 · e 2 ),…, F ( e 1 - s n · e 2 ); A , B , C ) = A Martin Ziegler 11