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ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC Tự NHIÊN TÊN ĐỂ TÀI MỘT s ố TÍNH CHẤT VỂ s ự TÍNH TỐN VÀ ĐỘ PHỨC TẠP TÍNH TỐN TRÊN TRƯỜNG s ố THựC THEO MƠ HÌNH MÁY BLUM-SHUB-SMALE, VÀ MƠ HÌNH TỔNG QT TRÊN CẤU TRÚC ĐAI s ố MÃ SỐ: QT 01 - 029 CHỦ TRÌ ĐỂ TÀI : PGS.TS TRAN H A N Ơ I - 2002 thọ châu BÁO CÁO TÓM TẮT Tên đề tài: Một số tính chất tính tốn độ phức tạp tính tốn trường số thực theo mơ hình máy Blum-Shub-Smale, mơ hình tổng quát cấu trúc đại sỗ Mã số: QT 01 - 029 Chủ trì đề tài: PGS.TS.Trần Thọ Châu Cán tham gia: PGS.TS Đặng Huy Ruận PGS.TS Vũ Ngọc Loãn Mục tiêu nội dung nghiên cứu a) Mục tiêu nghiên cứu - Mơ hình tính tốn số thực đưa ba nhà khoa học Bluin, Shub, Smale từ năm 1989 thường gọi máy BSS - Một mục tiêu theo cách tiếp cận Blum-Shub-Smale phải xây dựng lý thuyết độ phức tạp tính tốn cách đồng nhằm giải vấn đề sở tảng tính giải tích tính tơ-pơ, muốn sơ vấn đề thực khó số thực xử lý thực thể - Nhiều khái niệm kết ]ý thuyết độ phức tạp tính tốn cổ điển chuyển saníí mơ hình BSS số thực tương tự vấn đề tính máy BSS-đơn định (ký hiệu PR) thời gian đa thức máy BSS-không đơn định (ký hiệu NPR ) thời gian đa thức - Một mơ hình tổng qt hon cấu trúc đại số nghiên cứu Hemmerling từ năm 1995 (University Greifswald, Germany) lý thuyết độ phức tạp tính tốn tương ứng số kết tổng quát toán NPđầy đủ b) Nội dunu nghiên cứu - Nghiên cứu tính chất cùa độ phức tạp tính tốn theo mó hình máy Blum-Shub-Smale xử lý số thực SUMMARY 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 Code of project: Q T 01 - 029 Head of research group: Prof.Dr Tran Tho Chau Participants: Prof Dr Dang Huy Ruan Prof Dr Vu Ngoc Loan Aims and contents of project: a) Re arch aims' In 1989, L.Blum, M.Shub and S.Smale introduced a model for computation over the reals numbers, and this model is now usually called a BSSmachine One of the main purposes of the BSS approach was to create a uniform complexity theory dealing with problems having an analytical and topological background, and to show that certains problems remain hard even if arbitrary reals are treated as basic entities Many basic concepts and fundamental results of classical computability and complexity theory reappear in the BSS-model: the classes Pp> and NPR b) Main contents: - Studying some properties of the computation a n d complexity o v e r the 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 tính chất độ phức tạp tính tốn theo mơ hình mở rộng Hemmerling cấu trúc đại số Các kết đạt - Tổng quan số kết quan trọng theo hai mơ hình: mơ hình BSS sơ thực mơ hình tổng qt cấu trúc đại số - Về nghiên cứu bản: Đưa hai kết tính tốn cấu trúc đại số hàm tính theo mơ hình mở rộng Hemmerling Tình hình sử dụng kinh phí a) Được cấp: 8.000.000 đồng b) Sử dụng: + Th khốn chun mơn: 6.500.000đ + Hội nghị khoa học, Xêmina: l.OOO.OOOđ + In ấn việc khác: 500.000đ Hà nội, ngày 31 thánạ 12 năm 2002 Ý kiến Ban chủ nhiệm khoa Chủ trì đề tài XÁC NHẬN CỦA TRƯỜNG Introduction to the Theory of C om putation and >mplexity over the Real Num bers and other Algebraic Structures Tran Tho Chau, N ational U niversity Hanoi A bstract In this paper, the BSS model of computation over the reals and other rings as well as a more general model of computation over arbitrary algebraic structures are introduced and discussed Some crucial results concerning computability and computational complexity within both frameworks are given and explained 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 The notion of complexity an algorithm yields an im po rtan t area of research, known as complexity theory, th 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 athematically rigorous way, it will be neccessary to lnxiuce formal counterparts for many of the informal notions, such as ’’problems” and ” algohms” 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 parameters, r free variables, whose values are left unspecified A problem is described by giving [9]: ( ) a general description of all its parameters, ( ) a statem ent of what 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 Problem ” , flic param eters of this problem consist of a finite set = { ci,c2 , 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 c 7t(i)■■■■cw{,n)) of the given cities th a t minimizes c This expression gives the length of the ’’tour" that 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 computer programs, w ritten in some precise.' com puter SUMMARY 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 Code of project: Q T 01 - 029 Head of research group: Prof.Dr Tran Tho Chau Participants: Prof Dr Dang Huy Ruan Prof Dr Vu Ngoc Loan Aims and contents of project: a) Rearch aim s: In 1989, L.BIum, M.Shub and S.Smalc introduced a model for computation over the reals numbers, and this model is now usually called a BSSmachine One of the main purposes of the BSS approach was to create a uniform complexity theory dealing with problems having an analytical and topological backeround, and to show that certains problems remain hard even if arbitrary reals are treated as basic entities Many basic concepts and fundamental results of classical computability and complexity theory reappear in the BSS-model: the classes PR and NPR b) Main contents: - Studying some properties of the computation and complexity o v e r the reals of the model Blum-Shub-Smale - Studying some properties of the computation and complexit) over the algebraic structures of the General model Hemmerlinu Main obtained results: - Surveying some crucial results from two above models: the computation and complexity over the reals of the model Bliim-ShubSmale and the computation and complexity over the other algebraic structures - Obtaining two results about the computability over the algebraic structures Finance a) Receiving (From Nat Uni Ha noi): 8.000.000 đ b) Spendings: (i) For research works: 6.500.000đ (ii) For scientific conferences and seminars: OOO.OOOđ (iii)Other works: 500.000đ Ha noi, December 31- 2002 Prof.Dr Tran Tho Chau MỤC LỤC Trang Mở đầu Mơ hình máy Blum-Shub-Smale 2.1 Các khái niệm định nghĩa 2.2 Các kết 2.3 Quan hệ với 1Ĩ1Ơ hình máy BSS-yếu máy tuyến tính cộng 11 Mơ hình tính toán cấu trúc đại 13 3.1 Các khái niệm định nghĩa 13 3.2 Khả đoán nhận khái niệm liên quan 20 3.3 Chương trình vạn số kết 21 3.4 Các mối quan hệ lớp độ phức tạp đa thức 23 Tài liệu tham khảo 25 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 tro d u ctio n to th e T h eory o f C o m p u ta tio n and C o m p le x ity over th e R eal N u m b ers and o th er A lgeb raic S tru ctu res Tran T h o C h au , N a tio n a l U n iv ersity H an oi A bstract In this paper, the BSS model of computation 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 computational complexity within both frameworks are given and explained In tro d u ctio n Since th e achievem ent of a form al definition of the concept of algorithm, th e m a th e m a tic a l T h eo ry of C o m p u ta tio n has developed into a broad and rich discipline T h e no tion of complexity of an algorithm yields an i m p o r ta n t a re a of research, known as com plexity theory, th a t can be ap p ro ac h ed from several p o in ts of view of th e structural approach is ou tlin ed 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 in tro d u c e form al c o u n te r p a rts for m a n y of th e inform al notions, such as ’’p ro b le m s” and ’’algo r ith m s ” Indeed, one of th e m ain goals is to make explicit the connection betw een the formal term inology an d th e m ore intuitive, inform al s h o rth a n d t h a t is com m only used in its place O nce we have th is connection well in h an d , it will be possible for us to pu rsu e th e discussions p rim arily a t th e inform al level, reverting to th e formal level only when neccessary for clarity an d rigor A p rob lem will be a general q uestion to be answered, usually possessing several param eters, or free variables, whose values are left unspecified A problem is described by giving [9]: ( ) a general d escrip tio n of all its p aram eters, ( ) a s ta 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 instan ce of a p rob lem is o b ta in e d by specifying p a rtic u la r values for all the pro blem p a ram eters For exam ple, we consider the classical problem ’■ I raveling S alesm an P ro b lem '' T h e p a m e te rs of this p ro blem consist of a finite set — { c i c , c m } of "cities" an d for each pair of cities Cj,Cj in c , th e ’’d ist an ce” d(cj,cj) between them A solution is an ordering c (tV(i)< £;r(2 ), ■• • , C7T(7h)) of th e given cities t h a t m inimizes T h is expression gives th e leng th of th e ’’t o u r ’" th a t s ta r ts at C.-Iij, visits each city in sequence, a n d th e n re tu r n s d irec tly to C~(1) from the last city c-(m) A lg o rith m s are general, s te p - b y - s t e p procedures for solving problem s For concreteness, we can th in k of th e m sim ply as being c o m p u te r program s, w ritten ill some precise coniputor all pointers Pt Pi p2 P3 Pa r - move(p2); r - move(p3); r - move(p3); r - move(p4); r - move(p4); r - app(p4); Pi t : = Pi t *Pi t; P2 t : = P2 t *P2 t; P3 T:= P3 t *P3 T; Pi t := Pi t +P'2 T; Pi i := Pa t +P3 Í; Pa t : = P4 t - ; P Í:= P t-l; Ml : if p4 T / then goto Ml M : halt { infinite cycle } Here we have the halting set Q {{x, y, ) € 1R3 :f ( x y z) = 0} To obtain a decision, we only have to replace the instructions after the labels Mị resp M by the followingsubprograms: Ml : p4 t : = 1; {O u u t } del (p ,); del {pi); del(pi): halt M : Pa | : = 0; {O u t p u t } d e l (/)]); d e l (/>]): dcK/Jj): halt M a n y a u t h o r s , like B SS a n d F r i e d m a n - M a n s f i i ’M allow i hc u s f of a r b i t r a r y d e m e n t s of thr' universe as constants in their programs This sspms to i)f> f | i i i t P ronmirm with 18 to HAM model over the integers Rem ark th a t any constant i can be considered as term ” + + • ■■+ 1” (i t i m e s 1) i f i > 0, a n d ” ( - ) + ( - ) H -f ( - ) ” (i t i m e s - ) if I < We shall strictly distinguish between s - programs, where only the finitely many base constants of the stru ctu re are allowed to occur as direct operands, and the (S —quasiprograms which are analogously defined but allowing arbitrary elements of the universe as direct operands Those will be denoted as quasiconstants Now we shall see th a t the com putability of relations by quasiprograms can be characterized by the com putability by means of programs L e m m a 3.1 A relation ip : s + = > s + (subset of s + X s +) is computable by a ( D - , N\ — or N 2~ ) s -quasiprogram iff there are a relation V? : s + = > s + and a string CJ0 e S ’ such that "0 = ^(wo) and V computable by an S-program of the same type N o t a t i o n : A ” Q ” in the prefix denotes the concepts ’’Quasiprogram” corresponding the D Q computability, A^iQ-reconizability, etc L e m m a 3.2 For every X G {D, Ni, N , D Q , N iQ , N Q}, the class of all X -recognizable sets of string is closed under (finite) union and intersection D e f in it io n An element s G s is said to be (S-) str uct ib le if the total constant functions '2 Then e v e r y p a r t i a l r e c u r s i v e f u n c t i o n y : -4+ — » + IS d e t e r m i s t i c a l l y S —c o m pu t ab l e D e f in it io n 3.5 A s tru ctu re s is said to be bipotent if it contains at least two constructible e l e m e n t s r , /'!• Examples of bipotent structures are the structures with the number domains specifying in section 2, and the nonbipotent structures are groups Q or vector spaces V in section R e m a r k 3.2 Over bipotcnt structure, one can use auxiliary tracks within string processing in a r a t h e r n a t u r a l way e x a m p l e : s t r i n g Ò) = r n £] r ,2s2 ■■■1Ir,s n (i] £ {0- 1} ) i n s t e a d of t i l t c u r r e n t string LO — S1S2 ■■■sn B ipotent structures also allow a rather simple painruj of string: p a i r ( s I in- -Qi s 'm) = +, w = dom(ip) And the synonymous denotation halting set is used - An output set w is the range of a D —computable string function cp, w = ran(ip) - A set w c s + is said to be (S) — enumerable if it is empty or there are an (5) —constructible string w and a D —com putable (partial) string function Ip such th a t w = {^'!(u;o) : i G N} where ỉpl d e n o t e s t h e z-th i t e r a t i o n o f Ip This representation of w means th a t it can be exhausted by an 5-effective counting process starting with the constructible string LO0 ĩp is called a successor function enumerating w P r o p o s itio n 3.2 ([12]) A s t r u c t u r e o f f i n i t e s i g na t ur e IS c o n s t r u c t i v e iff i ts u ni v e r s e IS e n u mer abl e T h e o r e m 3.2 ([12]) F o r a n a r b i t r a r y s t r u c t u r e (cS), the f o l l o w i n g c o n d i t i o n ure equivalent: a) The universe of (S) IS enumerable b) Every halting set over (s ) is enumerable c ) E v e r y o u t p u t s e t o v e r ( S ) IS e n um er a b le D e f i n i t i o n 3.6 Let w c s + Its projection is defined by n ( I F ) = df { w i : t he r e T h e o r e m 3.3 is a U’2 e + such t h a t p a i r ( wI , U'2) e w } ([12]) O v e r a r b i t r a r y s t r u c t u r e s , the f o l l o w i n g c o n d i t i o n s are equivalent: a ) T h e c l a s s e s o f all l t i n g s e t s a n d o f all o u ut s e t s respect ivel y, coinci de b) The c l a ss o f all h a l t i n g s e t s IS d o s e d u n d e r p ro j ec t io n c) The c las s o f all h a l t i n g s e t s IS d o s e d u n d e r i m a g e s o f D - c o m p u t a b l e functions R e m a r k 3.3 There exists the structure owning an output set th a t is not a halting set, for example: - Let M = (]N:0 1: =: *) where denotes the multiplication: and are the only constructible elements of the univerw of M Hence IX is a halting set which is not V -enum erable The set of square numbers { k : k e IN} is an output set but not a hairing WT Iiidood for inputs of length 1: U' — r c EV by the possible promises of conditional expressions with t Ilfo n l y v a r i a b l e X t h e e l e m e n t s f r o m IN \ {0 1} c a n n o t b e Pi' parat ed - Over the (unordered) field of real numbers, 1Z = II-Í-H 1: r > } = { r : r ± } is an o u t p u t set, but not a h a l t i n g set 20 m r h from t l i f ot h( r t hr* SH IIU = {r : 3.3 U niversal program s and som e results Here we suppose th a t the considered structures (S) have finite signature Using two constructible elements r and r u every ( S )-q u asip ro g ram (V) can be encoded in a straightforward m anner by a string denoted by code(V) The keywords of the programming language, the technical symbols, base constants, base relations and base functions be encoded by strings from {r0, r i } + , and indices of pointer variables and the goal labels be binarilv encoded over {Vo,ri} The quasi-constants of quasiprograms are encoded by themselves It is convenient to use only the track of even-numbered placcs in strings for these encodings where as the odd-nu m bered places are filled by r or T1 such th a t the starting places of codes of the syntactic units can uniquely be identified More precisely, instead of the direct encoding ” s 1s s „ ” o f s o m e s y n t a c t i c u n i t u, we use t h e p a d d e d s tr i ng code(n) = r i s i r0s ■■■r 0s n Then for quasiprograms V = U]U2 u m, where ut are the syntactic units (i = 1, , m), let codeựp) = code(ui) ■code(u2) ■■■code(um) Now, the parts of code(V) which represent the codes of the syntactic units can be identified by a deterministic S —program T h e o r e m 3.4 ([12]) There is a D —program u such that, for all D —quasiprograms V and ail strings w E s + : (fu(pair(codeịv), w)) = ự>p{w) For i — 1,2 there IS an N t —program Ui such that, for all Nị —quasiprograms V : p-p = {(to, w')\(pair(code{V), w), lo') € Pu, }• This is proved by applying standard techniques of simulation and programming, as they are well-known from classical com putation theory based on the concept of Turing machine Notice th a t the concept of S —program corresponds to the notion of multihead Turing machine Thus, the simulated program V may use arbitrarily many pointer variables, whereas the uni versal program u and Ut respectively, are equipped with some fixed number of pointers only Thus, in each step of the simulation, the positions of the ^ - p o i n t e r s Pj must be marked by t he s i m u l a t i n g p r o g r a m U[7] b y m e a n s o f e n c o d i n g s o f Pj at t he c o r r e s p o n d i n g plac es o f the (encoding of the) current ^ -c o n fig u tio n To compute the values of base functions or base relations, the universal programs can use (finitely many) suitable subroutines The simulation of nondeterministic steps can analogously be performed by subroutines The encoding of pairs of strings is used to define inductively the encoding of K tuples of strings, for k G u f Let tu p l e \{u') = w tiupie2 (u ',w ) = putriw LL') t upLek+](u' » Uk) = p a i r i i i Q t u p l f We s i m p l y w r i t e [?(■-[, u-k i) for k > .It'k) for k > I i n s t e a d of D e f i n i t i o n A k - a r y partial function ỹ : {S+)k — t s ~ is said to be (dctcnnin isticallyj computable over iff the unary string function is (](tenninist ir allv 5-fompuLfcWf where-: ( [ t r , u-k}) = r- ( w i II-k) for //•:
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