Một số tính chất về sự tính toán và độ phức tạp tính toán trên trường số thực theo mô hình máy blum shub smale và mô hình tổng quát trên cấu trúc đại số

Tên đề tài: Một số tính chất về sự tính toán và độ phức tạp tính toán trên trường số thực theo mô hình máy Blum-Shub-Smale, và mô hình tổng quát trên cấu trúc đại sỗ 2.. Mục tiêu và nội

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC Tự NHIÊN

BÁO CÁO TÓM TẮT

1 Tên đề tài: Một số tính chất về sự tính toán và độ phức tạp tính toán trên trường số thực theo mô hình máy Blum-Shub-Smale, và mô hình tổng quát trên cấu trúc đại sỗ

2 Mã số: QT 01 - 029

PGS.TS Vũ Ngọc Loãn

5 Mục tiêu và nội dung nghiên cứu

a) Mục tiêu nghiên cứu

- Mô hình tính toán trên số thực được đưa ra bởi ba nhà khoa học Bluin, Shub, Smale từ năm 1989 và thường được gọi là máy BSS.

- Một trong những mục tiêu chính theo cách tiếp cận của Blum-Shub-Smale

là phải xây dựng lý thuyết độ phức tạp tính toán một cách đồng đều nhằm giải quyết các vấn đề trên cơ sở nền tảng cơ bản là tính giải tích và tính tô-pô, và muốn chỉ ra một sô vấn đề thực sự khó ngay cả khi ở đây các số thực bất kỳ được xử lý như là những thực thể cơ bản.

- Nhiều khái niệm và kết quả cơ bản của ]ý thuyết độ phức tạp tính toán cổ điển được chuyển saníí mô hình BSS trên số thực tương tự là các vấn đề tính được bằng máy BSS-đơn định (ký hiệu là PR) trong thời gian đa thức và máy BSS-không đơn định (ký hiệu là NPR ) trong thời gian đa thức.

- Một mô hình tổng quát hon trên cấu trúc đại số được nghiên cứu bởi Hemmerling từ năm 1995 (University Greifswald, Germany) về lý thuyết độ phức tạp tính toán và tương ứng là một số kết quả tổng quát về các bài toán NP- đầy đủ.

b) Nội dunu nghiên cứu

- Nghiên cứu các tính chất cùa độ phức tạp tính toán theo mó hình máy Blum-Shub-Smale xử lý trên số thực

Title of project:

Some properties of the computability and the complexity over the reals numbers with the computational model of Blum-Shub-Smale, and over other algebaic structures with more general model Hemmerling

2 Code of project: Q T 01 - 029

3 Head of research group:

Prof.Dr Tran Tho Chau

4 Participants: Prof Dr Dang Huy Ruan

Prof Dr Vu Ngoc Loan

5 Aims and contents of project:

- Studying some properties of the computation a n d complexity o v e r th e

reals of the model Blum-Shub-Smale

- Studying some properties of the computation and complexity over the algebraic structures of the general model Hemmerlinc

- Nghiên cứu các tính chất của độ phức tạp tính toán theo mô hình mở rộng của Hemmerling trên cấu trúc đại số.

6 Các kết quả đạt được

- Tổng quan một số kết quả quan trọng theo hai mô hình: mô hình BSS trên sô thực và mô hình tổng quát trên cấu trúc đại số

- Về nghiên cứu cơ bản:

Đưa ra hai kết quả về sự tính toán trên cấu trúc đại số và một hàm tính được theo mô hình mở rộng của Hemmerling

7 Tình hình sử dụng kinh phí

b) Sử dụng:

+ Thuê khoán chuyên môn: 6.500.000đ

+ Hội nghị khoa học, Xêmina: l.OOO.OOOđ

+ In ấn và các việc khác: 500.000đ

Hà nội, ngày 31 thánạ 12 năm 2002

XÁC NHẬN CỦA TRƯỜNG

Introduction to th e Theory of C om putation and

>mplexity over the Real N um bers and other Algebraic

Introduction

ce the achievement of a formal definition of the concept of algorithm, the m athem atical eory of C om putation has developed into a broad and rich discipline T he notion of complexity

an algorithm yields an im p o rta n t area of research, known as complexity theory, t h a t can be

proached from several points of view of the structural approach is outlined there.

The theory can be defined in a m athem atically rigorous way, it will be neccessary to ln- xiuce formal counterparts for many of the informal notions, such as ’’problem s” and ” algo- hms” Indeed, one of the main goals is to make explicit the connection between the formal rminology and the more intuitive, informal shorthand th a t is commonly used in its place, nee we have this connection well in hand, it will be possible for us to pursue the discussions imarily at the informal level, reverting to the formal level only when neccessarv for clarity

id rigor

A problem will be a general question to be answered, usually possessing several param eters,

r free variables, whose values are left unspecified A problem is described by giving [9]:(1) a general description of all its parameters,

(2) a statem en t of w hat properties the answer, or solution, is required to satisfy

An instance of a problem is obtained by specifying particular values for all the problem larameters For example, we consider the classical problem ’'Traveling Salesman P roblem ” , flic param eters of this problem consist of a finite set c = { c i,c 2 , c m } of "cities" and for

■aril pair of cities d, Cj in c , the ’’distance” d(Cị,Cj) between them A solution is an ordering

c7t(i)- ■ ■ ■ ■ cw{,n)) of the given cities th a t minimizes

This expression gives the length of the ’’tour" th a t starts at c*(1) visits each city ill sequence, and then returns directly to C-(1) from the last city c-(m ■

Algorithms are general, step-b) -step procedures for solving problems For concreteness,

we can think of them simplv as being com puter programs, w ritten in some precise.' com puter

1

1 Title of project:

Some properties of the computability and the complexity over the reals numbers with the computational model of Blum-Shub-Smale, and over other algebaic structures with more general model Hemmerling

2 Code of project: Q T 01 - 029

3 Head of research group:

Prof.Dr Tran Tho Chau

4 Participants: Prof Dr Dang Huy Ruan

Prof Dr Vu Ngoc Loan

5 Aims and contents of project:

reals of the model Blum-Shub-Smale

- Studying some properties of the computation and complexit) over the algebraic structures of the General model Hemmerlinu

6 Main obtained results:

- Surveying some crucial results from two above models:

the computation and complexity over the reals of the model Bliim-Shub- Smale and the computation and complexity over the other algebraic structures

- Obtaining two results about the computability over the algebraic structures

7 Finance

a) Receiving (From Nat Uni Ha noi):

b) Spendings:

(i) For research works:

(ii) For scientific conferences and seminars:

(iii)Other works:

8.000.000 đ

6.500.000đ

1 OOO.OOOđ 500.000đ

Ha noi, December 31- 2002

Prof.Dr Tran Tho Chau

MỤC LỤC

Trang

2.3 Quan hệ với 1Ĩ1Ô hình máy BSS-yếu và máy tuyến tính cộng 11

3.2 Khả năng đoán nhận và các khái niệm liên quan 20 3.3 Chương trình vạn năng và một số kết quả 21 3.4 Các mối quan hệ giữa các lớp độ phức tạp đa thức 23

Ernst-Moritz-Amdt-Universitat Greifswald

Preprint-Reihe Mathematik

Introduction to the Theory of Computation and Complexity over the Real Numbers and

other Algebraic Structures

Tran Tho Chau

Nr 20/2002

In tr o d u c tio n to th e T h eo ry o f C o m p u ta tio n and

C o m p le x ity over th e R ea l N u m b ers and o th e r A lgeb raic

S tru ctu res

T ran T h o C h a u , N a tio n a l U n iv e r sity H a n o i

In this paper, the BSS model of com putation over the reals and other rings as well as a more general model of com putation over arbitrary algebraic structures are introduced and discussed Some crucial results concerning computability and com putational complexity within both frameworks are given and explained

Since th e ach iev em en t of a form al definition of th e concept of algorithm, th e m a th e m a tic a l

T h e o ry of C o m p u t a t i o n has developed into a b road and rich discipline T h e n o tio n of complexity

o f an algorit hm yields an i m p o r t a n t a r e a of research, known as c o m p lex ity theory, t h a t can be

ap p ro a c h e d from several p o in ts of view of th e structural a p p ro ac h is o u tlin e d there.

T h e th e o ry can be defined in a m a th e m a tic a lly rigorous way, it will be neccessary to in­tro d u c e form al c o u n t e r p a r ts for m a n y of th e inform al notions, such as ’’p ro b le m s ” and ’’algo­

r i t h m s ” Indeed, one of th e m a in goals is to m ake explicit th e co n n ec tio n betw een th e formal

te rm in o lo g y a n d th e m ore in tu itiv e, inform al s h o r th a n d t h a t is c o m m o n ly used in its place

O nce we have th is c o n n ec tio n well in h a n d , it will be possible for us to p u rs u e th e discussions

p rim a rily a t th e inform al level, rev ertin g to th e form al level only when neccessary for clarity

a n d rigor

A p ro b le m will be a g eneral q u e stio n to be answ ered, usually possessing several p a ra m e te rs ,

or free variables, w hose values are left unspecified A p roblem is describ e d by giving [9]:

(1) a general d e s c rip tio n of all its p a ra m e te rs ,

(2) a s t a t e m e n t of w h a t p ro p e rtie s th e answer, or solution, is required to satisfy

An in s ta n c e of a p ro b le m is o b ta in e d by specifying p a r tic u la r values for all th e p ro b le m

p a r a m e te rs For ex am p le, we consider th e classical p roblem ’■ I raveling S a le sm an P ro b le m ''

T h e p a r a m e te r s of th is p ro b le m consist of a finite set c — { c i c 2 , c m } of "cities" a n d for

each p a ir of cities Cj,Cj in c , t h e ’’d i s t a n c e ” d(cj, cj) bet ween t h e m A s ol uti on is an or deri ng (tV(i)< £;r(2), ■ • • , C7T(7h)) of th e given cities t h a t m inim izes

T h is expression gives th e le n g th of th e ’’t o u r ’" th a t s ta r t s at C.-Iij, visits each city in sequence,

a n d th e n r e tu r n s d ire c tly to C~(1) from th e last city c-(m).

A lg o rith m s are general, s t e p - b y - s t e p p rocedures for solving p roblem s For concreteness,

we can th i n k of th e m s im p ly as b ein g c o m p u te r p ro gram s, w ritte n ill som e precise conip u to r

A b s t r a c t

1

language An algorithm is said to solve a problem n if that algorithm can be applied to any

instance I of n and is garanted always to produce a solution for that instance I.

T h e tim e re q u ire m e n t of an alg o rith m are conveniently expressed in te rm s of a single vari­able, th e ’’size” of a pro b lem in stance, which is intended to reflect th e a m o u n t of input d a t a needed no describ e th e instance O ften th e size of a problem in sta n ce is m e a su re d in an informal wav

We observe t h a t th e descrip tio n of a problem in stance t h a t we provide as in p u t to th e com ­

p u t e r c an be viewed as a single finite s trin g of sym bols chosen from a (finite) in p u t ’’a l p h a b e t ”

A lth o u g h th e re are m a n y different ways in which instances of a given pro b lem m ight be de­scribed, one p a r ti c u la r way has been chosen in advance a n d each p ro b le m has associated w ith

a fixed e n c o d in g scheme, which m a p s p roblem instances into th e s trin g s describing them T h e

in p u t le n g th for an in sta n ce / of a p roblem n is defined to be th e n u m b e r of symbols in the

d e s c rip tio n of / o b ta in e d form th e encoding scheme for n It is th is n u m b e r, th e in p u t length,

t h a t is used as th e form al m e asu re of in sta n ce size

A lgebraic com p lex ity th e o ry considers alg o rith m s t h a t include exact ra tio n a l o p e ra tio n s on real n u m b e rs a n d e x a c t c o m p ariso n s of real num bers, b u t requires t h a t th e d im ensionality of

th e d a t a p resen te d to any a lg o rith m is fixed Thus, in s tu d y in g a pro b lem such as m a trix

m u ltip lic a tio n , a s e p a ra te alg o rith m is required for each co m b in a tio n of m a tr ix dim ensions,

even th o u g h w h a t we really w ant is a u n i f o r m alg o rith m t h a t will accept m a trice s of any size

Moreover, th e tools of algebraic com plexity theory are best suited to pro b lem s whose com plexity

is clearlv p o ly n o m ia l b o u n d e d , a n d c a n n o t address questions a b o u t p o l y n o m i a l - t i m r solvability

T h e field of in f o r m a tio n - b a s e d co m plexity [31] also allows exact ra tio n a l o p e ra tio n s on reals,

b u t m a in ly considers p ro b lem s such as in teg ratio n , in which th e in p u t is a function ra th e r th a n

a finite a rr a y of real num b e rs

T h e tim e complES itv fu n ctio n for an alg o rith m expresses its tim e re q u ire m e n t by giving, for each possible in p u t length, th e largest a m o u n t of tim e needed by th e alg o rith m to solve a

p roblem in s ta n c e of t h a t size T h is function is not w ell-defined until one fixes th e encoding schem e to be used for d e t e rm in in g in p u t length and the c o m p u te r or c o m p u te r model to be used for d e te r m in i n g ex ec u tio n tim e

In 1989, L B lum , M S hub a n d s Sm ale [4] in tro d u ced a m odel for c o m p u ta tio n s over

th e real n u m b e rs (and o th e r rings as well) which is now usually called a BSS m achine One

m o tiv a tio n for this com es from scientific c o m p u ta tio n In th e use of th e c o m p u te r, a reasonable ide aliz ation m e asu re s th e cost of o p e ra tio n s and tests in d e p e n d e n t of th e size of th e num ber

1 liis c o n tra s ts to th e usual th e o re tic a l c o m p u te r science p ic tu re which takes into account the

n u m b e r of bitas of th e o p eran d s A n o th e r m o tiv a tio n is to bring th e th e o ry of c o m p u ta tio n into

th e d o m a in of analysis, g e o m e try an d topology T h e m a th e m a tic s of these su b je c ts can then

be used in th e s y s te m a tic analysis of algorithm s A novelty of th e ap p ro a c h 01 B lum Shub and

S m alo is t h a t th e ir m odel is uniform (for all in p u t- le n g th s ) w hereas th e n otions explored in

a lg eb raic c o m p le x ity ( s tr a ig h t- lin e , p rogram s, a rith m e tic circuits, decision trees) arc typically iro n -u n ifo rm O n e of th e m a in p u rp o ses of the BSS approach was to cre a te a uniform c o m p le x it

th e o ry d e a lin g w ith p ro b lem s h aying an analy tic al and topological b a 'k g r o u n d and to show

t h a t c e r ta in p ro b le m s re m a in h a rd even if a r b itr a r y reals arc t r e a te d as l>asic entities

M an y basic c o n ce p ts a n d fu n d a m e n ta l results of classical c o m p u ta b il ity ailf 1 c-oriipkxity

theory reappear in the BSS model: the e) stence of universal machines, the classes PiR and

NPtr (real analogues of p and NP) and the existence of /VPiR-complete problems Of course

these n o tio n s a p p e a r in a different form, w ith a stro n g an a ly tic flavour: typical exam ples of

u n d ecidable , recursive en u m e ra b le sets are com p lem e n ts of c e rta in J u lia sets, a n d th e first

p ro b le m t h a t was shown to be /VPiR-com plete is th e q uestion \vhether a given m u ltiv a ria te

p o ly n o m ial of degree four has a real ro o t [4], In th e Boolean p a r ts all pro b lem s in th e class TVPjR are decid ab le w ith in single ex p o n e n tia l tim e (b u t this is n o t as triv ial as in th e classical

case), th e P w versus N P w q uestion is solved in K o ir a n ’s m odel [17] a n d th e PiR versus N

q u e stio n is one of th e m a jo r o pen pro b lem s [11, 3]

Based on th e c o m p u ta ti o n m odel in tro d u c e d in [12] for s trin g functio n s over single sorted,

t o t a l algebraic s tru c tu re s , A H em m erlin g [12] studies some basic featu res of a general theory

of c o m p u ta b ility T h e conce p t generalizes th e B l u m - S h u b - S m a le s e ttin g of c o m p u ta b ility over the reals a n d o th e r rings T h e conce p t of 5 - c o m p u t a b i l i t y of strin g fu nctions over th e universe

of th e s tr u c t u r e s is defined a n d shown to be general enough to include classical recursion

theory M oreover, n o n d e te rm in is tic c o m p u ta tio n s of two kinds are considered, n am ely bv non-

d e te r m in is tic b ra n c h in g w ith in p ro g ra m a n d by guessing of elem ents of th e universe [13] In

th is g en era liza tio n is gives corresp o n d in g ly general results of TV P -com pleteness including b oth

C o o k ’s basic th e o re m on th e ./V P-com pleteness of SAT a n d M eggido’s genera liza tio n of the com pleteness resu lts by B lu m - S h u b - S m a le

T h e prin cip al te chnique used for d e m o n s tr a tin g t h a t two p roblem s are rela ted is t h a t of

’’re d u c in g ” one to th e o th e r, by giving a constru ctiv e tra n s fo rm a tio n t h a t m a p s any instance of

th e first p ro b le m into an in s ta n c e of th e second one Such a tr a n s fo rm a tio n provides the m eans for co nverting any a lg o rith m t h a t solves th e second problem into th e co rre s p o n d in g alg o rith m for solving th e first problem s

T h e f o u n d a tio n s for th e th e o ry of N P -c o m p le te n e s s were laid in a p a p e r of S tep h e n Cook,

p resen te d in 1971, e n title d ’’T h e C o m p lex ity of T h eo re m P roving P ro c e d u r e s ” [$] In this brief

b u t elegant p a p e r Cook did several i m p o r ta n t things:

• Signifiance of ” p o ly n o m ia l tim e re d u cib ility ” , t h a t is, red u ctio n s for which th e required t r a n s ­

fo rm a tio n can be ex ecuted by a p o ly n o m ial tim e algorithm

• O n th e class N P th e decision p ro b le m can be solved in p o lynom ial tim e by a n o n d e te rm in istic

co m p u te r

• O ne p a r ti c u la r p ro b le m in -VP, called th e ’’satisfiability problem , has th e p ro p e rty t h a t

every o th e r p ro b le m in N P can be polyn o m ially reduced to it If th e satisfiability problem can be solved w ith a p o ly n o m ia l tim e a lg o rith m , th e n so can even- p ro b le m in N P , a n d if any

p roblem in N P is in tr a c ta b le , th e n th e satisfiability p roblem also m ust be in tra c ta b le T hus, ill a sense, th e satisfiab ility pro b lem is th e " h a rd e s t" p roblem in X P

T h is survey is organized as follows: in section 2 th e BSS m odel over th e real n u m b e rs is in tro ­duced to g e th e r w ith th e basic definitions, notions, and some results c g n ce rn in g rh e com plexity

t h e o ry over IR and v aria tio n s of th e BSS m odel (additive, linear, weak m achines) Section 3 generalizes th e BSS m odel over th e reals to the model over a r b i t r a r y st.rur-1 uros and it gives

co rre s p o n d in g ly general resu lts c o n c e a lin g c o m p u ta b ility th(33rv a n d re la tio n s h ip between t h ' “

co m p le x ity classes

3

D e fin itio n 2.1 Let Y c IR00 A BSS-machine M over R with admissible input set Y is given

by a finite s e t I of in s tr u c tio n s la belle d by 0 , 1 , , N A configuration of M is a q u a d r u p le (n , i , j , x ) E / X IN X IN X IR00 H ere n d e n o te s th e c u rre n tly e x e c u te d in s tr u c tio n , i a n d j are used as a d d re s s e s (c o p y -re g iste rs) a n d X is th e a c tu a l c o n te n t of th e re g isters of M T h e initial

c o n f ig u ra tio n of M ’s c o m p u t a t i o n on i n p u t y 6 Y is ( 1 ,1 , 1 , l e n g t h ( y ) , y ).

If n = N a n d th e a c t u a l co n fig u ra tio n is (N , i , j , x ), th e c o m p u t a t i o n s to p s w ith o u t p u t X

T h e i n s tr u c tio n s M ca n p e rfo r m are of th e following types:

• C o m p u t a t i o n :

- D a t a c o m p u ta ti o n s : n : x s <— x k on Xi, w here on e { + , —, * , / } or TL : x s <— a for som e

c o n s t a n t a € IR T h e re g is te r x s will g et th e value Xk On X 1 or a resp All o th e r register -entries

re m a in u n c h a n g e d T h e n e x t in s tr u c tio n will be n + 1.

- In d e x c o m p u t a t i o n s : I <— I + 1 or i i— 1; j i— J + 1 or j <— 1.

• Branch: n : if x0 > 0 g o to P ( n ) else g o to n + 1 A cco rd in g to th e answ er of th e te st th e next

in s tr u c tio n is d e t e r m in e d (here p ( n ) e I ) All o th e r registers are n o t changed.

• Copy: n : 2; <— Xj, i e th e c o n te n t of th e ’’r e a d ” - r e g is te r is copied in to th e V\v rite” -re g is te r

T h e n e x t i n s t r u c t i o n is 71 + 1 All o th e r registers re m a in u n c h a n g e d All Q a p p e a r in g a m o n g

th e c o m p u t a t i o n - i n s t r u c t i o n s b u ilt up th e (finite) set of m a c h in e c o n s ta n t s of M

R e m a r k 2 1 - T h e k ind of o p e r a t io n s allowed d e p e n d s on t h e u n d e rly in g s tr u c t u r e A b ra n c h

X > 0? for e x a m p le do es o nly m a k e sense in o rd ered set T h e c o p v - r e g is te r s a n d - i n s t r u c t i o n

a rc n e cc essarv in o rd e r to deal w ith a r b i t r a r y long in p u ts from IR00 T h e way of c h an g in g th e

e n trie s in t h e c o p v - r e g i s t e r th e (’’a d d re s s in g " I seems to be r a t h e r re s tric tiv e a p a r t from th e fact t h a t t h e r e is no in d ire c t a d d ressin g However it is g en era l e n o u g h for o u r p u rp o ses, see

r e m a r k 2.2

- In t h e in itia l c o n fig u ra tio n on i n p u t y, th e le n g th is in c lu d ed , since it c a n n o t be seen from

t h e re g is te r c o n t e n ts if y t e r m i n a t e s w ith som e c o m p o n e n ts equal to 0.

Vow to a n y BSS- m a c h in e M over } ’ th e re co rre s p o n d s in a n a t u r a l way th e functio n

c o m p u t e d by M It is a p a r ti a l fu n ctio n from y to IR ^ a n d is given as t h ' “ result of V s

c o m p u t a t i o n 011 an in p u t y Ễ y

D e f i n i t i o n 2 2 Let 4 c B c IR0® a n d M be a B S S - m a c h i n e OYPr B

4

ì) T h e o u t p u t - s e t of M is th e set <ĩ>Aí(i?) T h e hal t in g- set of M is th e set of all in p u ts y for

vhich is defined.

o) A is called recurcively enumerable over B iff A is th e o u t p u t - s e t recursive of a B S S -m a c h in e

jver B (If B = IR00, A is sim p ly called recursively enumerable.)

3) A p a ir (B , A ) is called a decision problem It is Sc id decidable iff th e re exists a B S S -m achine

M w ith adm issible in p u t set B such t h a t is th e ch ara cteristic function of A in B In t h a t

case M decides (B , A ).

As can be seen easily (B , A ) is d ecidable iff A an d B \ A are b o th h a ltin g sets over B

D e f i n i t i o n 2 3 a ) For X 6 IR00 such t h a t X — ( x i , , Xjfc, 0, 0, • ■ •) it is

s i z e ( x ) := k.

b ) Let M be a B S S - m a c h in e over Y 6 R ° ° , y G Y T h e r un n in g t i m e of M on y is defined by

m , ^ Í n u m b e r of o p e ra tio n s executed by M on in p u t y, if is defined,

T h e first im p o r t a n t difference to classical com plexity th e o ry of th is above definition:

• s ta te s t h a t any real n u m b e r - in d e p e n d e n tly if its m a g n itu d e - is considered as entity

• defines th e cost of any basic o p e ra tio n to be 1-no m a t t e r a b o u t th e o p eran d s

b ) { B , A ) belongs to class N P jr ( n o n d e te r rn im s ti c poly no mi al t i m e ) iff th e re exists a B S S

-m a ch in e M w ith adm issible i n p u t - s e t B X IR30 and c o n s ta n ts k e IN c e IR such t h a t the

following c o n d itio n s hold:

(1 ) <!>,,(!/ ĩ ) e {0,1}

(3) \fy G 4 3 ; e IR30 (<Ĩ>Af ( y z) = 1 and T \ , ( y z) < c ■ s i z e ( y ) k ^

Herein- (he in p u t p a rt 2 can be considered as a "guess", and this will be done in some inform al

d e s c rip tio n s of A'P|R alg o rith m s

c ) ( B A ) belongs to class CO — A PiR iff ( B B \ .4) G

XPui-R e m a r k 2 2 • T h e class YPift would not be changed if a more g e n e ra l way of ad d ressin g ÍỊ used in th e definition of B S S -in a c lú n e

• In a sim ila r way as above one defines f u rth e r com plexity classes, for e x a m p le s E X P lu mid

X E X P\ị{ (here th e ru n n in g tim e is b o u n d e d to be s in g le -e x p o n e n tia l).

e x a m p l e s

-) T h e c o m p u ta ti o n of th e g re a te s t integer in X , for X > 0 in IR (see F ig u re 1.)

I n p u t X 6 IR as th e second c o o rd in a te of a p o in t in R 2 w ith first c o o rd in a te 0

Replace (k , x ) by (k+ 1 , 2 — 1)

O u t p u t k, th e first co o rd in a te

Input X e R as the

F ig u r e 1

2) Let 5 c z +, the positive integers We co n s tru c t a m a chine M s over IR t h a t ’’decides” 5

T h a t is, for each in p u t n £ z +, M s o u t p u t s 1 (yes) if n € s an d 0 (no) if n ị s M s has a

built-in c o n s ta n t s e IR defined by its b in a ry expansion

s = •s1s2 s n , w here s n = ị ,

[ 0, otherw ise

Figure 2

M's w ith its bu ilt-in c o n s ta n t s, plays a role analogous to an ’’o ra c le ’’ for a T u rin g m achine

t h a t answ ers queries ” Is n G s?" a t a cost of n lo g n (see F igure 2.)

D e f i n i t i o n 2 5 Let n € !>■ a n d 5 c IR" T h e n the set 5 is s emialgebraic if s is th e set of

e lem en ts in IR" t h a t satisfy a finite system of p o lynom ial eq u alities and ine q u alities over IR or

6

equivalently if s is finite union of su b se ts of th e form:

{ y e IRn : f ( y) = 0 A gi ( y ) > 0 , ,gr{y) > 0},

w here / , g \ , gr £ IR [x i, x 2) • , x n] are polynom ials.

E x a m p l e 2 1 C o n sid er th e following functio n / : R 3 — » R

i c this set 4 c o n ta in s all p o in ts w ith n o n -n e g a tiv e c o o rd in a te s on th e face of s p h e re w ith

ra d iu s R = \ / 2 (only ị of sp here) T h e n A is decidable (see F ig u re 3.).

T h e class PiR can be considered as a th e o re tic a l fo rm a liz a tio n of th e p r o b le m s bein g effi­ciently solvable T h e r u n n in g tim e increases only p o ly n o m ia lly w ith th e p r o b le m size T h e

n o n d e te rm in is m in b) of th e definition refers to th e vector 2 T h e .V P iR -m ach in e is n o t a l­

lowed to answ er "yes" if th e in p u t does not belong to A a n d for each y e 4 th e re m u s t be a

" g u e s s ” ; t h a t proves th is fact in p o ly n o m ia l time It is evident t h a t p ữi c -VP[R T h e q uestion

P m Ỷ NPịR can be co nsidered as th e m a in unsolved pro b lem in real c o m p le x ity th e o ry a n d th e

a n a lo g u e to th e classical p versus N P in th e T u rin g theory T h e difference b e tw ee n p m and

.YPir is th e difference of fast p r o o f-fin d in g versus fast p ro o f-c h e c k in g [23]

E x a m p le 2.2 T h e T raveling S a le sm an decision problem over IR is in X p

Here is th e flow c h a r t for th e N P m a ch in e as follows (figure 4.):

D e fin itio n 2.6 Let (B?, A2) be decision problems.

a ) (£?2, 4 2) is reducible in p o ly n o m i a l t i me to iff th e re exists a B S S - m a c h i n e M over

B 2 such t h a t c B ị , $Aí(y) £ A i < = > y € A 2 a n d M works in p o ly n o m ia l tim e.

N o t a t i o n : (#2,-42) <nt (J3i, A i )

any P ro b le m in N P f c is reducible to it in p olynom ial tim e).

c) { B \ , Ax) G CO — NPfil is CO — /V PiR-com plete iff it is u n iversal w r t < R in CO — N P R

R e m a r k 2.3 Complete problems are essential for complexity classes because they represent

th e h a rd e s t p ro b le m s in it As in classical theory, th e re la tio n <IR is b o t h tr a n s itiv e a n d

reflective T h is im plies PjR — NP]R iff it exists a /V ffo -c o m p le te p ro b le m in class Pfo.

E x a m p l e 2 3 For k E IN consider th e sets

F k '■= { / I / p o ly n o m ia l in n unknow ns w ith real coefficients , d e g ( f ) < k n e IN}

^zcro := { / £ F k \ f has a rea-l zer0 } ’ a n d

F; ero+ := { / £ F k \ f has a real zero w ith all c o m p o n e n ts b ein g n o n n e g a tiv e },

w here a polynomial is rep resen ted as an elem ent of in th e following way:

T h e p o ly n o m ia l / : IRn — » IR of degree < 4 is powerfreely r epr es ent ed in IR00 as (4 n)

f o l l o w e d b y a s e q u e n c e o f ( a , a a ) w h e r e a = ( q j , q 2, Q3 Q 4); a , € [ 0 , , n ] , Q, < a i+1 a n d

fla € ]R T h e p a ir ( a , a Q) s t a n d s for th e m on o m ial aa x ữ ì x a2x a3x a<, w ith x 0 = 1 to allow for

te r m s of degree less t h a n 4 T h e s e ( a , a a ) are su pposed o rd e re d by th e le x ic o g ra p h ic o rd e r on

th e a T h u s f ( x ) = £ a a x 0 ĩ x Qĩx ữ3x 0 ^ N ote t h a t / can be consid ere d as a p o ly n o m ia l on IR ^

8

which does n o t d e p e n d on x l for i > n For each degree d € z + , it is clear how to generalize

th is d e s c rip tio n to g et th e powerfree representati on in IR°° of p o ly n o m ia ls / : IRn — > IR of

degree < d.

T h e b o t h decision p ro b lem s ( F k , F ^ ero) a n d ( F k , F ^ ero+) belong to NP-íỉi for all k 6 IN by guessing a (n o n n eg ativ e) zero X , p luging it into / and e v alu atin g f ( x )

R e m a r k 2.4 - The semi-algebraic subsets of IR are finite unions of intervals - bounded or

u n b o u n d e d , o pen, closed (including single p o in t sets), or half open [4]

- T h e decision p ro b le m s (IR, Q), ( R , Z ), (IR, IN) and ( Q ,Z ) do n o t b elo n g to th e class P ịr

(see [2 1])

- T h e p ro b le m (IR, <Q) ist n o t decid ab le (see [21]); ( R , Z ), (IR, IN), (Q, Z ) a re decidable

2.2 B a sic c o m p le x ity r e su lts

We coưic back to C o o k ’s fu n d a m e n ta l theorem : the honor of being th e ’’first” N P - c o m p le t e

p ro b le m goes to a decision p ro b le m from Boolean logic, which is u s u a lly referred to as th e

S A T I S F I A B I L I T Y pro b lem (SAT for sh o rt) T h e te rm s are defined as follows:

L et u = { lii, 1Í2, • • • u m } be a set of Boolean variables A truth a s s i g n m e n t for u is a

fu n ctio n t : u — > { T r u e , F a l s e } If u is a variable in Ư, th e n u a n d Ũ are literals over u

A clause over u is a set of literals over u , for exam ple {líi, Ũ3, lig} It represen ts th e

d is ju n c tio n of th o se literals a n d is satisfied by a t r u t h assig n m e n t if a n d onlv if a t least one

of its m e m b e r s is tr u e u n d e r t h a t assignm ent T h e clause above will be satisfied bv t unless

t ( u i ) — F , t ( u s ) = T , t.(ug) = F A collection c of danses over u is satisfiable if a n d only

if th e re exists som e t r u t h a s s ig n m e n t for Ư t h a t sim ultaneously satisfies all th e clauses in c Such a t r u t h a s s ig n m e n t is called a satisfying t r u t h assignm ent for c

T h e S A T - p r o b le m ist specified as follows:

I n s t a n c e : A set u of v ariables a n d a collection c of clauses over u ■

Q u e s t i o n : Is th e re a satisfy in g t r u t h a s sig n m e n t for c ?

F o r e x a m p l e : u — { u ^ u 2} a n d c = {{líi, ũ 2}, {ũ\ u 2}} provide th e answ er of SAT ” ves;! ( t r u t h assig n m e n t: t ( u i) = t { u 2) — T ) O n th e o th e r h an d , rak in g C' = { { u i u2}, { u 1; ũ 2},

I 'í71}} yield th e answ er '’no ” , i e C ' is not satisfiable.

T h e o r e m 2 1 (C o o k ’s T h e o re m [9]) S A T I S F I A B I L I T Y IS N P - c o m p l e t e

T h e e x p erien ce can still narrow th e choises down to a core of basic p ro b le m s t h a t have been useful in th e p a s t Fven th o u g h ill th e o ry any known v P - c o m p l e te p ro b le m can serve j u s t as

well as any o th e r for prov in g ail new p roblem N P - com plete, in p ra c tic e c e r ta in p ro b lem s do

seem to be m uch h o tt e r s u ite d for th is task T h e following six p r o b le m s are a m o n g those t h a t have been used m ost frequently a n d these six can serve as a “ basic core" of know n V P -com plete

p ro b le m s for t h a b e g in n e r (see [9]):

9

1) 3—Satisfiability (3SAT) 2) 3-Dimensional Matching (3DM) 3) Vertex cover (VC)

T he following diagram shows the sequence of transformations used to prove th a t the six basic problems are /V P-com plete in classical complexity theory

SATSFIABILITY

ị3SAT

Now we want return to continue the work on problems like Fir versus N P ỵị { Obviously if

problems in NP]R would not be decidable then it would not make sense to speak a b o u t their

complexity Moreover, it is im p o rtan t to know whether iVF]R-complete problems exist and how they look like We know t h a t proving completeness results for decision problems in principe is possible by reducing known complete problems to those in question Nevertheless it remains the task to find a "first’’ complete problem This is one of the m ajor results in [4],

T h e o r e m 2.2 (B him -S hub-S m a le [4])

a) For any k > 4 the problem ( F k,F^ero) is N P]R-complete.

b) All problems in class N P-ỊỊị are decidable in single exponential time.

P a rt a) is proved by an adaption of Cook's famous -V P-completeness result for the 3- Satisfiabilitv problem in Turing theory to the BSS-model The decidability of problems in -YPir is much harder to show than in discrete complexity theory The problem is closely connected with so called quantifier-elimination over real closed fields: The problem whether a

f G F k lias a zero can be formulated via the first-order formula

3'X-i 3xnf ( x 1, ,x„) = 0.

R e m a r k 2.5 If P]K 7^ -VPn? is assumed then k = 4 is a sharp bound in Theorem 2.2 This is

a result bv Triesch [31], who proved ( F k F ; eru) to belong to P|R for k = 1.2.3.

• { Fk,F*ero+) ( N P ^ - c o m p l e te for k > 4 [20])

• Is a sem i-algebraic set, given by polynomials of degree at most d, convex (CO — jV.Fr-complete for d > 2 [7])

Especially with respect to classical complexity theory, { F2,F^ero+) is interesting This

bounds the complexity of many im p o rtan t combinatorial problems if considered over the reals, for examples, 3SAT, HC, Travelling Salesman are all reducible to it in polynomial time Starting with the first m aster problem according to theorem 2.2 a) one can hopefully reach many new yvp-com pleteness problems those are used most frequently too

2.3 R elation s w ith w ea k -B S S -m o d e ls and linear additive m achines

The ’’weak B S S -m odel” introduced by Koiran [17] as well as linear variants of BSS-machines are im p o rta n t (later one has many other variations of BSS-model) T he weak BSS-model was inspired by the idea to bring the BSS-theory nearer to classical complexity theory by using a different cost-m easure for the operations For example the following program:

input X e IR for i = 1 to n

do X < r— X2

od

o u t p u t X

Here with n steps the value X2" is computed, i e a polynomial of degree 2n can be computed

in n steps Koiran introduced a different cost-m easure for the allowed operations The idea is

to consider the real constants of a machine as separate inputs in order to speak about the bit size of numbers appearing during a computation

Let M be a BSS-m achine with real constants X \ X S For any input-size n M realizes

an algebraic co m putation tree If any node V is passed by M during this com putation, the

value computed by M up to this node is of the form /i,(asj., , Qs , X\ , , xn) where /„ G

Ọ ( f ti, ., ữ s, Xi, ., x n ) is a rational function with rational coefficients only And now the

weak cost of the according operation is fixed as maximum of d e g ị f ư) and the m axim um height

of all coefficients of /„ (here the height of a rational 2 is given by |_log(|p| + 1) + log(|ợ|)J)

D e f i n i t i o n 2 7 ([17]) The weak B SS - m o d e l is given as the BSS-model together with the weak cost measure, i G the weak running time of a BSS-machine M on input X £ IR00 is the sum

of the weak costs related to all operations M performs until $ v ( x ) is computed.

Weak determ inistic and lion-determ inistic polynomial time as well as weak polynomial time

reducibility are defined in a straightforward manner (and denoted by P\\ XP\v etc.)

D e f i n i t i o n 2.8 a ) ([17] [3]) Let c be a complexity class over the reals, the boolean part B P (C )

o f c d e n o t e s

{L n {0 1}*

11