Satisfiability over Cross Product is NP

Transcription

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

2 Reminder: Complexity Theory P := { L {,1}* decidable in polynomial time } NP := { L verifiable in polynomial time } PSPACE := { L decidable in polyn. space } Def: Call L verifiable in polynomial time if L = { x {,1} n n N, y {,1} q(n) : x,y V } discrete "witness" Examples: for some V P and q N[N]. 3SAT 32 = { : Boolean formula in 2-CNF 3-CNF admits a satisfying assignment } P 3COL = { G : graph G admits a 3-coloring} HC = { G : G has a Hamiltonian cycle} EC = { G : G has a Eulerian cycle } P NP NP NP NP Martin Ziegler 2

3 Reminder: NP-completeness P := { L {,1}* decidable in polynomial time } NP := { L verifiable in polynomial time } Def: Polynom. reduction from A to B {,1}* is a f:{,1}* {,1}* computab. in polytime such that x A f(x) B. Write A ¹ B. NPc (e.g. 3SAT) NP P p A ¹p B, B ¹p C A ¹p C A ¹p B, B P A P For any L NP, L ¹p SAT (S. Cook / L. Levin 7ies) Martin Ziegler 3

4 Turing vs. BSS Machine Discrete: Turing Machine / Random-Access Machine (TM/RAM) Input/output: finite sequence of bits {,1}* or integers Z* Each memory cell holds one element of R={,1} / R=Z `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 º NP R º EXP R º (Tarski Quantifier Elimination) strict? NP R º -complete: Does a given real int. polynom.system have a real root? H R* real Halting problem Undecidable, too: Mandelbrot Set, Newton starting points R º? Martin Ziegler 4

5 Turing vs. BSS Complexity NP R -complete: Does a given multivariate integer polynomial have a real root? Theorem [Canny'88, Grigoriev'88, Heintz&Roy& &Solerno'9, Renegar'92]: NP R PSPACE ("efficient real quantifier elimination") No 'better' (e.g. in PH) algorithm known to-date! Allender, Bürgisser, Kjeldgaard-Pedersen, Miltersen 6: P R CH) Similarly with integer root: undecidable (Matiyasevich 7) Similarly with rational root: unknown (e.g. Poonen'9) Simil. with complex root: corp NP mod GRH (Koiran'96) Martin Ziegler 5

6 NP ⁰ Completeness R ⁰ does it have a satisfying assignment over subspaces of R³/C³? QSAT R : Given a term t(x 1,..X n ) over,,, FEAS R ⁰: Given a system of n-variate integer polynomial in-/equalities, does it have a real solution? CONV R ⁰:, is the solution set convex? DIM R ⁰: of dimension n? C.Herrmann& M.Z. 211 P. Koiran'99 Today: The following problem is NP R -complete: QUAD R : Given p Z[X 1,,X n ] of total N.E.Mnëv (8ies), J. Richter-Gebert'99 ⁰ degree Given 4, does a term it have t(x a real root? 1, X n ) over only, Is a given oriented matroid realizable? does the equation t(x Is a given arrangement of pseudolines, 1, X n ) =X stretchable? 1 Certain geometric have a solution properties over of graphs R³\{}? Peter W. Shor'91 M. Schaefer 21 Martin Ziegler 6

7 Cross Product in R³ (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 b) a) a a b (((a b) a) a) (a b) = (a b) a b a a b a, a b = a b sin (a,b) anti-commutative, non-associative. Martin Ziegler 7

8 Decision Problems with Cross Product Theorem: a) to c) and a') to b') are all equivalent to Polynomial Identity Testing RP (randomized polytime with one-sided error, Schwartz-Zippel) (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 ) d) to f) are all NP R -complete (((a b) a) a) (a b) = d') Given to f') a terms are equivalent t(v 1, Vto n ) Hilbert's and s(v 1 1th,..V n ) Problem built from over only: Q In a) particular Is there an there assignment exists a vcross 1,,vproduct n R³ s.t. equation t(v 1,..v n )? t(v b) 1 Is,..v there n )=v 1 an assignment satisfiable over v j R³ R³ s.t. but t(v not 1,..v over n ) s(v Q³. 1,..v n )? c) Is there an assignment v j R³ s.t. t(v 1,..v n )=e z? d) Is there an assignment v j R³ s.t. t(v 1,..v n )=v 1? e) Is there an assignment v j R³ s.t. t(v 1,..) v 1? f) Is there an assignment v j R³ s.t. t(v 1,..v n ) s(v 1,..v n )? a') to f') similarly but for assignments Q³ Martin Ziegler 8

9 Proof (Sketch, hardness) QUAD R (Does given p Z[X 1,..X n ] have a real root?) ¹ p e) e) Is there an assignment v j F³ s.t. t(v 1,..v n ) v 1? For the standard any right-handed orthonormal gonal basis e 1,e 2,e 3 of F³ and for r,s F, the following are easily verified: F(e 1 -r s e 2 ) = Fe 3 [ F(e 3 -r e 2 ) F(e 1 -s e 3 ) ] Encode s F F(e 1 -s e 3 ) = Fe 2 [ F(e 2 -e 3 ) F(e 1 -s e 2 ) ] as as projective affine F(e 3 -s e 2 ) = Fe 1 [ F(e 1 -e 3 ) F(e 1 -r e 2 ) ] point line F(e 1 -s 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 ) = Fe 2 [ F(e 1 -e 2 ) F(e 2 -e 3 ) ] Can thus express the arithmetic operations and - using the cross product and Fe 1 and Fe 2 and F(e 1 -e 2 ) and F(e 2 -e 3 ). Martin Ziegler 9

10 Proof (Sketch, hardness) QUAD R (Does given p Z[X 1,..X n ] have a real root?) ¹ p e) e) Is there an assignment v j F³ s.t. t(v 1,..v n ) v 1? For the standard any right-handed orthogonal basis e 1,e 2,e 3 of F³, can express and using cross product and Fe 1 and Fe 2 and F(e 1 -e 2 ) and F(e 2 -e 3 ). Encode s F as as projective affine terms V 1 (A,B,C), V 2 (A,B,C), V 12 (A,B,C), point line F(e e 1 -s e 1 -s e 2 2 ) V 23 (A,B,C) that for any assignment A,B,C P²F, either coincide with Fe 1 =A and Fe 2 and F(e 1 -e 2 ) and F(e 2 -e 3 ) for some right-handed orthogonal basis e i or evaluate to. 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 )= t p (F(e 1 -s 1 e 2 ),,F(e 1 -s n e 2 );A,B,C)=A Martin Ziegler 1

11 Conclusion Identified a new problem complete for NP R 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 currently developping into similarly rich structural theory [Baartse&Meer'13] PCP Theorem for NP over the Reals Question: Graph Coloring being NP-complete, how about Quantum Graph Coloring? [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 )= t p (F(e 1 -s 1 e 2 ),,F(e 1 -s n e 2 );A,B,C)=A Martin Ziegler 11