Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) 405 Appled Mathematcs & Informaton Scences An Internatonal Journal http://dx.do.org/0.785/ams/080536 Contnuous Turng Machne: Real Functon Computablty and Iteraton Issues Xaolang Chen, Wen Song, Zexa Huang and Mngwe Tang School of Mathematcs & Computer Scence, Xhua Unversty, Chen du 60039, P. R. Chna Receved: 8 Sep. 03, Revsed: 6 Dec. 03, Accepted: 7 Dec. 03 Publshed onlne: Sep. 04 Abstract: Contemporary computer theory s governed by the dscretzaton of contnuous problems. Classcal Turng machnes (TMs) are orgnally bult to solve computaton and computablty problems, whch man feature s dscreteness. However, even some smple numercal calculatons problems, e.g., teratons nr n, generate dffcultes to be descrbed or solved by constructng a TM. Ths paper explores the computablty of contnuous problems by proposng a class of contnuous Turng machnes (CTMs) that are an extenson of TMs. CTMs can be appled to the standard for the precson of algorthms. Frst, computable real numbers are precsely defned by CTMs and ther computatons are regarded as the runnng of the CTMs. CTMs ntroduce the coded recursve descrptons, machne states, and operatons wth the characters of computer nstructons n essence compared wth usual computable contnuous models. Hence, they can precsely present contnuous computatons wth the form of processes. Second, the concepts of CTM computable and CTM handleable are proposed. Moreover, the basc concepts on approxmaton theory such as convergency, metrc space, and fxed-pont n R n are defned n a new space CTM R n. Fnally, an teratve algorthm s shown by constructng a CTM to solve lnear equatons. Keywords: Computatonal mathematcs, computer theory, contnuous Turng machne, real number computablty, teraton. Introducton The development of computablty theory n nformaton scence begns wth the generaton of recursve functons that depend on logcal theory. These recursve functons are consdered as the precse defntons of ntutve algorthms [. Turng descrbes computatons by a class of mathematcal machnes (theoretcal computers), usually called Turng machnes (TMs). The machnes precsely present the concept of computatons wth the form of processes by ntroducng machne states and the operatons wth respect to the characters of computer nstructons. TMs are equvalent to recursve functons. Hence, computablty problems are equvalent to Turng computablty [. A theory s sad to be an systematc approach f ts deducton and reasonng depend on a standard mathematcal model. Computatons are model-based processes n the soluton of a gven calculated problem. However, the exstng TM computablty theory cannot properly presents computable real functons snce TMs are dscrete n essence [3, 4. Thereafter, some mathematcal models and approaches are developed to analyze the computablty of real numbers. Mazur [5 defnes computable real functons by the proposed sequence computablty. Kretz and Wehrauch [6, 7 tae nto account the presentaton of real number computablty by ntroducng type- theory of effectveness (TTE), whch s based on the theory of representatons and s an approach of computable analyss. Edalat [8, 9, 0, studes computable real functons by doman theory. Many constructve analyss methods are also proposed. Moore [ proposes µ herarchy to nterpret recurson theory on reals and constructs flowcharts of contnuous tme to handle real number computablty and haltng problems. Doraszels and Satterthwate [3 defne computable real numbers by the establshed Marov arthmetc. Blum, Cucer et al. [4, 5 analyze the computable problems of real numbers by constructng real-ram models. However, the fact s that not all real number computablty that are descrbed above are equvalent. For example, Banach-Mazur computablty s not equvalent to Marov computablty for computable real numbers [6. On the other hand, the descrbed Correspondng author e-mal: xdxlchen@gmal.com c 04 NSP
406 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... theores and models are dffcult to compatble wth the classcal model TMs. Hence, ths paper attempts to construct a class of extenson TM to deal wth real number computablty. Numercal analyss nvolves the methods for real number calculatons. However, t does not consder computablty problems. Dfferent computatonal models can obtan nconsstent results on whether a real number problems s computable. For example, a serous dstorton or an entre wrong concluson may be obtaned f the real numbers computablty s consdered by TMs. Hence, dscrete machnes do not properly demonstrate real number computablty. On the other hand, although the theory of numercal analyss maes great achevements n the past, ts developments necessarly need a relable computable theory. Ths paper begns wth an extenson from dscrete TMs to contnuous TMs. Then, a class of autonomous contnuous Turng machnes (CTMs) s proposed n secton. The ratonalty by usng CTMs to explore the computablty of real numbers s consdered n secton 3. CTMs have manly two strengths compared wth the usual models. Frstly, the classc methods of TMs deal wth the computatons of natural numbers, the sets of natural number, and the arthmetcal functons. A CTM covers contnuous computatons and nclude dscrete computatons. Second, t s realstc and feasble snce the concept of computable s defned by constructng CTMs. An algorthm s sad to be computable f a CTM can be constructed for a certan nput to reach an output at fnte steps. CTMs have smple structures, basc operatons, and precse descrptons of computatons n the form of processes. A CTM seres wth respect to greater power can be constructed by an teratve or recursve constructon of CTMs. Iteratve technology based on CTMs s consdered n secton 4, whch demonstrates an approach to prove CTM-computable and to explan how to construct a complex machnes. Fnally, a typcal example s gven to llustrate real functon computablty, whch can be regarded as a methodology to solve a class of computable problems. In secton 5, we state the results of ths paper. Extenson: dscrete TM to contnuous TM The smplest way for a TM to compute four arthmetc operatons s that the representaton of numbers only uses 0 [3, where notaton 0 s a character n the tape of TMs, whch s dstngushed wth the numercal zero. However, the representaton method can lead to the ncrease of storages. Importantly, by consderng teratve computatons, a self-teratve TM can dffcultly be constructed snce computatons, e.g., teratons, cannot be easly represented by ntegers. Many researchers mae extensons from dscrete models to contnuous ones, where contnuous automatons, contnuous Petr nets, and Hybrd nets et al [7,8 are proposed. These models do not means that the number of new models s ncrease n the seres of computatonal models. Its purpose s to correctly and easly present, solve, and analyze a class of contnuous problems. A good approxmaton generally proves very valuably to solve a complex problem. Therefore, an approxmate method s consdered n TMs n ths paper. Generally, contnuous models are tme-related. However, CTMs are regarded as tme-ndependent. Ths secton expands TMs to CTMs by ntroducng an example for a non-output and two-type nondetermnstc TM M (Fg. (a)). Frst, M can be constructed by the followng algorthm. Algorthm for M constructon. TM M := On nput ω = ω,ω : // ω and ω represent ntal nputs of the two tapes, respectvely. DO { step: If there s a 0 on tape, then move t to tape or retan t n tape n a random manner. step: If there s a 0 n tape, then move t to tape or retan t n tape n a random manner. }whle.t. //Notaton.T. means that the logcal condton of the loop whle s always true. Accordng to the vew of machne computatons, a character 0 n M can be consdered as a certan mount of resources. Hence, a sngle resource s represented by a sngle 0 and several resources are represented by multple 0 (0 ) n M. TMs are theoretcal models of computers. In a real-world computer, the number of 0 can be represented as the amount of nformaton. For example, a sngle 0 can be nterpreted as G nformaton and 00000 as 5G nformaton. Second, a transformaton s consdered to dvde each 0 nto equal parts. Ths new TM s denoted by M and shown n Fg. (b). The world bloc s assumed as an unt of 0 n ntal confguratons. Each bloc s dvded nto. The new unt that s one -th of bloc s called pece. For example shown n Fg, the ntal confguraton of M (Fg (a)) leads to the confguraton of M (Fg (b)) n whch the resources are expressed n peces. Generally, the transton functons of multple nondetermnstc Turng machne (MNTM) have the form : Q Γ K P(Q Γ {L,R,S} ), where Q s the set of states, Γ s the tape alphabet, s the number of tapes, and P s power set. The expresson (q,a,,a ) P(q j,b,,b,l,...,r),a,b j Γ means that f the sate of a machne s q and read-wrte head through are readng symbols a through a, respectvely. The machne goes to one of possble states q j and wrtes symbols b through b. Correspondngly, transton functons drects each head to move left, rght, c 04 NSP
Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) / www.naturalspublshng.com/journals.asp 407 Controller M a Evolve confguratons for M 0 0 tape tape q 0 0 q Controller () M b q 0 0 q d 0 0 tape q c tape q3 0 0 q3 q m m c q3 c and addng a pece to tape. Hence, the evolved confguratons can be expressed n blocs (nteger) or n peces (ratonal number f s fnte). Let C be a confguraton that s expressed n peces n M and C = C / be the correspondng confguraton that s expressed n blocs n M. Obvously, the computatonal processes of M are ncluded n the processes of M. : = [ m c Expressed n blocs Evolve confguratons () for M Expressed n peces { q 0 0 q { q q 0 0 q q q c c q 0 q 0 q - q 0 0 q q q q q L L 0 0 e q q m m L q3 0 0 q3 q 0 0 q q3 q3 - q 0 0 q q3 q3 L f q3 q3 q3 0 0 q3 Fg. : Transformaton of a TM: (a) ordnary TM M, (b) transformed ordnary TM M, represented for =4, (c) state graph of M, (d) evolve confguratons for M, (e) mpled evolve confguratons n Fg(d), and (f) evolve confguratons for M. or to stay put. By consderng M as an example and ts state graph shown n Fg (c), ts evolutons contan three types of transton functons: :(q,0, )=(q +,,0,R,R) m :(q,α, )=(q,α,,l,l), α Γ c :(q,,0)=(q,0,,s,s) Symbol Γ denotes that there has not a resource at correspondng postons on the tape,.e., blan. Now, we consder M wth an strategy for the segmentaton of resources. Its transton functons are smlar to M. However, let the scale of ts evolutons be the unt of pece. For example,, m, and c n M are represented by the followng functons combnatons: : { (q,α, α)=(q,α, α+,s,s), α =,,, (q,, )=(q +,,,R,R) m :(q,{α, }, )=(q,{α, },,L,L), α =,,, c :(q, α,α)=(q, α+,α,s,s), α =,,, By consderng M and ts evolved confguratons shown n Fg. (d), the executon of transton functon conssts of removng a bloc from tape and addng a bloc to tape. Correspondngly, by consderng M and ts evolved confguratons shown n Fg. (f), the executon of conssts of removng a pece from tape tapes tape tape tape () M Controller / / / / / / Fg. : Structure of -type M K. tape tape tape The fact that transton functons of a -type TM execute smultaneously s denoted by ( j ), where, j,, and are transton functons from tape to, respectvely. The structure of M, whch can execute tmes smultaneously, s shown n Fg.. The executon of n M equals the executon of ( ) n M. We can change a way to descrbe transferred processes by ntroducng some new notatons. It facltates to dscuss the extenson from dscrete TMs to contnuous TMs. Let [ \ j \ \ be a class of orderly executve sequences of transton functons. Let [ j be a class of synchronzed executve sequences of transton functons. Let [ α = [( ) α be a class of specal executve sequences, whch performance means that TM mplements transton functon total α tmes smultaneously and removes or adds α pece resources n ts tapes to produce a new confguraton, where α s a non-negatve number. Fg. 3(a) shows a set of possble transtons of M that are concerned wth two bloc resources. In addton to sngle executon of or, multple transtons by the executon of[ \,[, and[ are also represented. The possble transtons of M for =4 are shown n Fg. 3(b). M contans many and fnte multple transtons, e.g., [ 3, [ and [ 6. We apostrophe read-wrte head and state alphabet q for smplfcaton. By observng the executon of [ 3, ts transton process can be expressed n peces as: [ [ tape 4 [ 3 > tape 3 4 c 04 NSP
408 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... Or t can be expressed n blocs as: [ tape 0.5 [ 0.75 > tape 0.75 [ 0.5 0.5 Furthermore, we can also descrbe n peces as: or express n blocs as: 8 4 tape [ 3 (5,3) [ 3 >(,6) (.5,0.75) [ 0.75 >(0.5,.5) 6 [ tape 0 [ 0 0 tape (bloc) 0 0 4 8(pece) (b) [ [ \ (a) [ tape α tape [ α α [ α (c) tape Fg. 3: From dscrete to contnuous turng machne: (a) graph of resource transton of M n Fgure (a), (b) graph of resource transton of M n Fgure (b) for =4, and (c) graph of resource transton of M, for. The number of possble multple transtons become nfnte f tends to nfnty. These transtons can be denoted by an segment of a lne between (,0) and (0,) shown n Fg. 3(c). For example the descrpton of the followng transton ( α,α) [ β >( α+ β,α β), mples that the confguratons ( α, α) and ( α + β,α β) are expressed n peces. Transton functon can be executed β tmes from the confguraton ( α,α) at a moment, where α s any real number n the range [0, and β s called a transferred quantty and satsfes the nequalty α β. Smlarly, f there s an executon of [ γ, then γ satsfes 0 γ α. Multple executons taen a form of [( ) β ( ) α are possble. TMs M and M dscussed above belong to MNTM n essence. The defned machnes are recognzers [3 of languages n ths paper f there are not specal remars. Lemma.Two-type nondetermnstc TM M equvalent fve-type determnstc TM M d. has an Proof. The deal s mae mutual smulatons between M and M d. The fact that M smulates M d s smple snce determnstc TM M s an specal case of nondetermnstc TM M d and we only needs to construct a nondetermnstc computatonal branch n M. On the other hand, f M d s constructed to smulate M, M d needs tryng all possble branches of nondetermnstc computatons of M. The machne M d can be establshed by constructng fve tapes. As shown n Fg. 4, we assume that every tape has a partcular functon. Tape and are smlar to the tapes n M. They contan constant strngs that copy from the ntal nputs of M. Tape 3 and 4 are smulaton tapes that mantan a copy from the tapes of M for a branch of ts nondetermnstc computatons. The data n tape 3 and 4 contan evolved confguratons at the branch. The functon of tape 5 s to generate the address strng ω address of nondetermnstc computatonal branches from the length one to length nfnte, constantly. Let Σ be an nfnte set of all address strngs, whch contans all possble branches of nondetermnstc computatons. Any address strng conssts of fnte nds of alphabets, whch are connected wth the number of states. By consderng M d, ts address strngs consst of three nds of alphabets,, and 3, whch come from the subscrpts of the three states of M (Fg. (c)). Σ s countable accordng to Cantor s theory snce the number of strngs n some certan length s fnte and the unon of denumerable countable sets s a countable set. A lst of Σ can be constructed by wrtng down all strngs of length zero, length one, length two and so on. The total number of address strngs can be expressed as + n=0 3n. We can easly mae a mappng from any strng to n N. Not all the address strngs are vald. For example, address strng s vald, whch represents that the current state s q whch confguraton s dsplayed n tape 3 and 4. The next state s q that s obtaned by executng m, where m s stored n controller of M d. Then, the fnally state s q that s obtaned by executng stored. The process can be denoted by [ m \ q > q. However, address strng 33 s nvald snce there not exsts a transton functon n the process from state q 3 to q 3 n Fg. (c). The exstence of nvald strngs s reasonable snce they can be consdered as null addresses. Proved process s just a constructve process, we construct M d as follow: c 04 NSP
Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) / www.naturalspublshng.com/journals.asp 409 Algorthm for M d constructon: TM M d := On nput ω = ω,ω,ω 3,ω 4,ω 5 : step: Intal tape and. Chec correctness of ω, ω. Tape 3 to 5 stay empty. step: Generate a strng ω address n tape 5 accordng to the rule of the ncrease of strng lengths. step3: Chec the valdty of ω address by checng transton functonal grd n controller. If ω address s vald, go to step4, else go to step. step4: Copy the data n tape and to tape 3 and 4, respectvely. Smulate the transton of states from the frst alphabet of ω address to the last one. Go to step. M Sd Controller B00 B B 0 B0 B## #3# sngle tape Fg. 5: Structure of sngle-type determnstc M sd wth a snapshot n Fg. 4. { nput tapes tape 0 0 tape M d Controller tape5 tape3 0 tape4 0 ##3###3## #33## #3# address tape Fg. 4: Structure of fve-type determnstc M d. } smulaton tapes Lemma 3.Two-type nondetermnstc TM M has an equvalent sngle-type determnstc TM M sd Proof. By Lemma and, we have the concluson. Lemma 4.Two-type nondetermnstc TM M has an equvalent sngle-type determnstc TM Msd. Lemma.Fve-type determnstc TM M d has an equvalent sngle-type determnstc TM M sd. Proof. The deal s mae mutual smulatons between M d and M sd. The fact that M d smulates M sd s smple. We only use any one of the tape n M d that can smulate the sngle-tape of M sd. The deal s to show how to smulate M d wth M sd. Fg. 5 llustrates that sngle tape can be used to represent fve tapes. M sd smulates the functons of M d by storng ther nformaton on ts sngle tape, where we use symbol B as a delmter to separate dfferent wor-spaces. Tape symbol wth a small above t s used to mar the poston of the head on the tape. These new symbols have been added to the tape alphabet. In other words, M sd contans vrtual wor-spaces and heads. we construct M sd as follow: Algorthm for M sd constructon: TM M sd := On nput ω = Bω Bω Bω 3 B ω 4 Bω 5 : step: M sd puts ts tape nto the format and the formatted tape contans Bˆ00 B ˆ B ˆ B ˆ B ˆ step: M sd scans ts tape from the left frst B n order to determne the symbols under the vrtual heads. Then M sd update ts tape accordng to M d s transton functons. step3: When M sd moves vrtual heads to the rght onto another B. It means that the correspondng head of M d has been moved onto the blan porton of the tape. Then, t contnuous the smulaton as before. Proof. Two-type TM M can be expressed by -type TM M wth an unt of peces, whch s denoted by Fg.. We assume that the unt of M sd s also expressed n peces. By consderng the -type structure of M shown n Fg., accordng to Lemma 3, the computatons n the two tapes of any level are equvalent to the computatons of M sd. Hence, the computatons of -type M equal the computatons of M sd machnes. Msd can be constructed by combnng the resources of the correspondng postons of M sd machnes, where resources are expressed n blocs. Hence, M s equvalent to M sd. Lemma 5.Sngle-type determnstc TM M sd has an equvalent sngle-type determnstc TM M sd. Proof. The fact that Msd smulates M sd can be acheved by settng the parameter =. Usng M sd to smulate Msd needs lttle changes about the nputs of M sd. A new wder range delmter s ncreased to separate pece wor-groups (a pece wor-group equals fve orgnal wor-spaces, whch s shown n Fg. 5). The pece wor-groups are combned by copyng the ntal data of M sd total tmes to ts sngle type n order of prorty. Then, smulaton can be acheved by changng the unt of blocs nto peces and the extended transton functons. Corollary.Two-type nondetermnstc TM M has an equvalent two-type nondetermnstc TM M Proof. By lemma 3, 4 and 5, we have the concluson. Theorem.Sngle-type determnstc TM Msd equvalent sngle-type determnstc TM M sd. has an c 04 NSP
40 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... Proof. Msd can be constructed by M sd f tends nfnte. We have ts equvalent machne M sd by Lemma 4, where the number of wor-spaces tends nfnte. Hence, the mutual smulatons between them are enabled. Of course, the length of the sngle tape of M sd s necessarly nfnte and M sd s a non-haltable TM n general sense. Note that machne halt n general sense equals classcal TM halt [3, 4. The concept corresponds to machne halt at the sense of convergence of computatonal hstores. For example, a machne s a non-haltable machne f the number of resource dvsons tends nfnte. It has not acceptve and rejectve states n general sense snce these states cannot be acheved wthn a fnte tme. However, machne halt n convergent meanng of computatonal hstores s to llustrates some mportant problems n CTMs. Theorem guarantees that the dvson of resources does not ncrease the power of TMs. Importantly, any TM wth resource dvson has an equvalent ordnary TM no matter how many tapes t has and t s determnstc or nondetermnstc. Ths equvalency necessarly satsfes machne non-haltable n theoretcal sense. Actually, the equvalency cannot be guaranteed n real-world hardware envronments. For example, nfnte dvson of resources s not possble by the bt restrcton of computers. Hence, a more general stuatons should be consdered. For example, we deal wth the machnes that are smlar to Msd, whch can machne halt n general sense. These machnes can acheve acceptve or rejectve states. The judgments of these states should based on the length of computatonal hstores snce any smple computaton s nfnte f tends nfnte. In other words, we consder the approxmate calculaton of Msd. The classcal computatonal hstory of a TM s usually consdered as the confguratons of the TM. However, f real number s taced, any confguraton may has an nfnte length. Hence, classcal computatonal hstory cannot be appled to descrbe the dynamc behavor of the TM wth real numbers. Hence, a class of computatonal hstory wth respect to state transtons s proposed. Defnton.A strng denoted by ω h = j s sad to be a computatonal hstory f the strng composed of the subscrpts of the exstent states, where every two adjacent characters and j represents that the machne restores the computaton path from state q to q j. In ths case, the computatonal hstory of an non-haltable TM s a strng wth nfnte length, even f the TM has fnte states. For example, the states and transtons shown n Fg. (c) can generate computaton hstory wth nfnte length. Obvously, the length of computatonal hstores s fnte f the machnes are haltable snce the states of them are necessarly acheve acceptve or rejectve states. Defnton.A computatonal hstory of M s sad to be convergent f there exsts an absolute dfference between the output of M and another output of M wth the length l of computatonal hstory, whch s less than any gven ε. Defnton 3.HALT T M = { M K,ω M K s a TM that the length of ts computatonal hstory s fnte or ts computatonal hstory s convergent}. Theorem.TM M halt has an equvalent sngle-tape determnstc TM M ordnary (ordnary means that t s a haltable TM). Proof. The fact that tends nfnte means that the number of tapes tends nfnte n unt pece. If the computatonal hstory of Mhalt s convergent, ts approxmate computatons allows that there exsts a reasonable length l of computatonal hstory for transton functons at the sense of convergency. The evoluton of Mhalt acheves termnal states and the machne outputs approxmate computatonal results f the number of state transtons comes to l. By usng Theorem, we can fnd a sngle-tape determnstc TM M sd and mae the length of ts tape tends nfnte. The sngle-tape s dvded nto nfnte number of wor-groups. Actually, t s just attach the longtudnal tapes n Mhalt to the sngle-tape of M sd such that M sd s equvalent to Mhalt. In ths case, f the computatonal hstory of Mhalt s convergent, M sd s haltable. The haltable M sd at the sense of convergency s denoted by M ordnary. Hence, M ordnary and Mhalt have the same computatonal results. Theorem 3.M sd has not an equvalent TM M halt. Proof. By usng Theorems and, we only need to proof that M sd s not equvalent to M ordnary. M sd cannot acheve acceptve or rejectve states snce M sd cannot ensure machne halt. However, M ordnary has termnal states by consderng the approxmaton. Hence, M sd cannot smulate the termnal states of M ordnary. Actually, M sd can be regarded as the lmtng state of the computatons of M ordnary. They are not equvalent n the sense of convergence of computatonal hstory. Theorems and guarantee the ratonalty of the extenson from TMs to CTMs such that CTMs can be formally defned. In ths paper, sngle-tape determnstc CTMs (ordnary CTMs) and the ordnary CTMs at the sense of convergence n computatonal hstory are defned. The former are models of computatonal theory for real numbers and the latter are theoretcal models of real functonal approxmaton. Other classes of CTMs are equvalent to them. Defnton 4.An ordnary CTM s a 7-tuple, (Q,Σ,Γ, α, q,q LIM accept,q LIM re ject ), where. Q s a fnte and non-empty set of states.. Σ s the set of nput alphabets. It comes fromr + and s expressed n blocs. 3. Γ = Σ {,, } s the set of tape alphabets, where s a symbol of wor-space delmter, s a symbol of vrtual head, and s blan. 4. α : Q Γ Q Γ {L,R,S} s the transton functon, where α s transfer quantty that s defned as a postve ratonal number. c 04 NSP
Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) / www.naturalspublshng.com/journals.asp 4 5. q,q LIM re ject Q are the ntal state, acceptve state and rejectve state, respectvely. accept,q LIM If we have a CTM and want to construct a CTM algorthm wth the form 0 that s smlar to 0 n TM, we only need to guarantee that Σ of the CTM comes from the range(0, (a subset ofr + ). α s defned as a postve ratonal number snce any transfer quantty of transton functons s stored n the controllers of CTMs wth the form of data table. In other words, α s a fxed value and s not a varable. Hence, the contnuty of CTMs are ensured snce the data of computatons of CTMs are real numbers. Defnton 5.A haltable CTM at the sense of computatonal hstory convergency s a 7-tuple, (Q,Σ,Γ, α, q,q LIM accept,q LIM re ject ), where. Q s a fnte and non-empty set of states.. Σ s the set of nput alphabets. It comes fromr + and s expressed n blocs. 3. Γ = Σ {,, } s the set of tape alphabets, where s a symbol of wor-space delmter, s a symbol of vrtual head, and s blan. 4. α : Q Γ Q Γ {L,R,S} s the transton functon, where α s transfer quantty that s defned as a postve ratonal number. 5. q,q m accept,q m re ject Q s the ntal state, acceptve state and rejectve state when the length of computatonal hstory s m, respectvely. The length m follows the hardware restrctons or the computatonal precson demands. By consderng Defntons 4 and 5, the machnes Msd s an ordnary CTM and Mhalt s a haltable CTM at the sense of convergency. A confguraton of CTMs s smlar to that n TMs, whch contans three tems: the current state, tape contents, and head poston. It s represented by the strng uqv, where the current state s q, the current tape contents are u and v and the current head poston s the frst symbol of v. Any symbol n u and v comes from R +. A confguraton s a descrpton of computatons of a TM at some moment. The number of confguratons n a haltable TM s fnte. However, any change of states may contan nfnte confguratons n a CTM by the nfluence of the contnuous. Hence, the confguratons n TMs does not sut to represent the evoluton of the computatons n CTMs. Importantly, state graphs cannot express computatons of CTMs n detal. Hence, the concept of confguraton evoluton graphs (CEGs) s proposed to better descrbe and analyze the confguraton change of CTMs. Usng CEGs to descrbe some specal CTMs wthout consderng the complex CTM constructon algorthms s hence avalable. Defnton 6.The CEG of a CTM (M CT M,C q 0 ) s a dgraph CEG (M CT M,C q 0 ) =(V,E), where Cq 0 means the ntal confguraton of M CT M, V={C C s a confguraton of M CT M } and E={(C q j, j α,(cq j ) ) C q j,(c q j ) V,C α respectvely, where (C q j > C } are sets of vertexes and edges ) s a successor of C q j. The CEG for CTM M (M comes from M ) s shown n Fg. 6. The nfnte confguratons and state transton processes can be expressed n smple form wth fnte elements (vertex, edge, and arc). The ntal confguraton s denoted by C 0 = [ 0 q, where q means that C 0 s just n the state q, the number of both sdes of represents the total amount of the resources n the two tapes. The range of changng of them sr +. q q [ # 0 [ - α # α q [ 0# 3 α q [ # [ [ α α [ α [ [ c α α [ m [ c [ [ α α [ m q [ 0# Fg. 6: CEG for CTM M (M,, M s shown n Fg. (b)) However, the moments when dfferent classes of transton functons are executed are only recorded as the vertexes of the CEG. The weghts on arcs means transfer qualtes executed accordng to transton functons. The amount of confguratons n any two adjacent vertexes wth the transfer qualty α s nfnte. The sgnfcance of CEGs for CTMs are as follows: Infnte confguratons are expressed by fnte vertexes. The transton functon grd n detal from the graph can be obtaned. The state graphs by foldng vertexes and arcs accordng to superscrpt of vertexes can be obtaned. 3 CTM computable functons A CTM computes a functon by addng the nputs of the functon to ts tape and haltng wth the outputs of the functon on the same tape at the sense of computatonal hstory convergency. Defnton 7.A functon f : ΣCT M Σ CT M s a CTM computable functon f some CTMs for every nputs ω can halt wth just ( f(ω)) on ts tape at the sense of computatonal hstory convergency, where ( f(ω)) denotes precse values or theoretcal values. c 04 NSP
4 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... Defnton 8.A functon f : ΣCT M Σ CT M s a CTM handleable functon f some CTMs for every nputs can halt wth the unque fnte length computatonal hstory ( f(ω)) m on ts tape by approxmaton, where ( f(ω)) m denotes the approxmate values of length m of computatonal hstory. The unque n Defnton 8 les on hardware restrctons or approxmate precsons. CTM computable functons contan classc TM computable functons snce the termnal states of TMs are specal stuatons of the lmtng forms of CTMs. Dscrete characterstcs of TMs mae them to acheve the termnal states n fnte tmes. Any CTM computable functon s necessarly a CTM handleable functon by Defnton 7 and 8. In other words, any CTM computable functon can fnd an approxmaton functon n certan hardware restrcton. Verse s not true. The purpose to defne CTM handleable functons s to descrbe the numercal and functon approxmaton problems n R +. For example, by consderng teratve methods for matrx egenvalues calculaton, t may possble for a computer to get an approxmate value but a precse value. Computers are hardware lmted. It can machne halt through the overflow. However, the computatons of the theoretcal machne CTM s platform rrelevant, whch may not lead to machne halt snce the length of computatonal hstory may be nfnte. Hence, we cannot to say that the computatonal processes of egenvalues are CTM computable snce the machne cannot halt n precse value, and we cannot also to say t s not CTM computable snce computatons always access to precse value. Consequently, the defnton of CTM handleable s necessary. If a functon s CTM handleable, ts lmtng evolved states are CTM computable. The concept s specal occurs n CTMs and not appear n TMs, whch wll consder n detal n next secton. CTM computable has two meanngs: usual arthmetc operatons on reals and the transformatons of machne coded descrptons. On the former, the nputs can acheve (not approxmaton) the outputs accordng to the computatons of CTMs at the sense of convergency. For example, we construct a CTM that taes an nput < r,r > r,r R + and returns r r. Note that the substracton s assumed as true substracton: { r r r r = r r 0 r < r Let M substract be a CTM that can do ths wor. Its CEG can be shown n Fg. 7, where α (0,max(r,r )) and related transton functons are as follows: :(q,m,n)=(q +,m,n,r,r) m (0,r,n (0,r :(q,0,n)=(q re ject,0,0,r,r) n (0,r r 3 :(q,m,0)=(q accept,r r,0,l,l) m (0,r r Accordng to transton drectons (the drectons of arcs) n Fg. 7, the CTM M substract can acheve termnal states n ts lmts. Although( f(ω)) as a CTM pecewse functon has two possble values r r and 0, t wll halt wth just the value of( f(ω)) on ts tape. Hence, functon f s CTM computable accordng to Defnton 7. α r α [ [ r r 0 [ [ r-α r α q [ r r # 0 q # q r [ r [ # r α [ - 3 [ r r q q 0# [ r r#0 accept q [ 0#0 reject Fg. 7: Confguraton evoluton graph of CT M substract. On the latter, A TM can get ts machne coded descrpton < T M > and can mae computatons for coded descrpton of other machnes by self reference and recurson theorem [3 n classc computaton theory. CTMs can also get coded descrptons by smlar methods. A CTM computable functon s a class of transformaton of CTM coded descrptons. For example, a CTM computable functon f taes an nput ω =< M > and returns another coded descrpton < M >, whch s ept as f :< M > >< M >, where M and M recognze the same language but locate on dfferent confguratons. Therefore, whether a functon s CTM computable can be proved by constructng a CTM to compute t and returnng unque coded strngs at the sense of computatonal hstory convergency. Coded descrptons can greatly enhance the descrbed ablty of machnes. For example, f we proof that a complex functon ϕ = n = (R r ),R, r R + s CTM computable, we only need to proof functon f :< r,r > >< r r > s CTM computable (the concluson s proved by the constructon of M substruct ), then we proof the functon f :< M substruct,m substruct > >< M substruct + M substruct > s CTM computable (t s easly to construct M add snce addng and subtracton are smlar). Fnally, we construct coded functon ϕ :< f, f >. If there exsts a CTM wth the nputs of < f, f > that can outputs the unque strng by the transton functons of ϕ, functon ϕ s CTM computable. Actually, substracton of real s CTM computable, addton s the opposte of the operaton of subtracton, and multplcaton s the result of contnuous addtons. Hence, functon ϕ s CTM computable. There are many wors that are not CTM computable. Dvergent teratve process s an example. c 04 NSP
Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) / www.naturalspublshng.com/journals.asp 43 Mappng reducble at the sense of CTMs s consdered whch can extend the proof drectons of CTM computable. Algorthm for CTM M f constructon: CTM M f := On nput ω = x : Output x, then accept. Defnton 9.CTM < M A > s mappng reducble to CTM < M B >, wrtten < M A > m < M B >, f there s a CTM computable functon f : ΣCT M Σ CT M such that for every ω, < M A,ω > nput ω < M B, f(ω) > nput f(ω) The functon f s called the reducton from machne < M A > to < M B >, where the meanng of < M A,ω > nput ω s that the coded machne < M A > computes the strng ω (ω can be a functon) and generates a new strng ω on ts tape when < M A > halt at the sense of computatonal hstory convergency. M A and M A locate n dfferent confguratons but recognze the same language. Actually, Defnton 9 s recursve. It s possble to construct a CTM to compute or decde a reductonal functon f. For example, < M A > and < M B > n Defnton 9 may be the machnes to computng another reductonal functons. If a CTM computable problem s reducble to another problem, whch s proved f the orgnal problem s CTM computable. The followng theorem can llustrates ths deal. Theorem 4.Let < M A > and < M B > be machnes to compute functons f and f, denoted as < M A, f > and < M A, f >, respectvely. Functon f s CTM < M A > computable f < M A > m < M B > and < M B > s CTM computable. Proof. We assume that < M B, f > s CTM < M > computable and functon f s the reducton from < M A > to < M B >. We construct a new CTM N to compute < M A, f > such that machne N can halt wth just code << M B, f > > on ts tape. Algorthm for CTM N constructon: CTM N:= On nput ω =< M A, f >: step: Accordng to reductonal functon f, we have f(ω)=< M B, f >. Run machne < M B, f > and output << M B, f > > f < M B, f > machne halt at the sense of lmt. step: Run machne < M > wth the nput << M B, f > > and output whatever < M > outputs. Obvously, f < M A > m < M B > and f s CTM < M B > computable, then f s M A computable. An example of the proof for CTM computable by usng reducton s proposed. By consderng a functon f (x) = x, x [,, we construct CTM M f as follow: Obvously, f s a CTM computable functon. Let another functon be f (y) = cos(y), y [0,π whch computable property s not clear and the wor of constructng a CTM to compute cos s also complcated. We assume such a CTM s exstent and s denoted by M f. The mappng from f to f s clearly exst by ther geometrc meanngs such that for any y [0,π, we have < M f,y > nput y < M f, f (y) > nput f (y) Then, functon f (y) = cos(y), y [0,π s also CTM computable accordng to Theorem 4. 4 Iteratve technology based on CTMs The concept of teratons appears n computatonal mathematcs and the theory of programmng. Almost all hgh-level programmng languages support teratons. The soluton of many mathematc approxmaton problems needs teratons. However, teratons n R are not TM computable. In real-world, computers can handle approxmate and teratve computatons nr. Hence, TMs should have an ablty to do and do better the wors snce TMs are a class of platform-ndependent theoretcal models. However, ther descrpton abltes do not match usual computers. It s unreasonable that TMs are models of ths class of computatons. The proposed CTMs are the expansons of TMs, whch can deal wth these ssues. Therefore, the computable propertes of teraton problems under CTMs are dscussed n ths secton, whch can as a classcal example of CTM applcatons. The process of teraton n numercal calculatons s consdered. It taes an ntal pont and an teratve formula. Then, let the obtaned soluton be the next ntal pont and the process eeps teraton untl the fxed pont s approxmated. The computaton ends when the adjacent approxmatve solutons satsfy a precson requrement. Ths secton ntroduces CTM to compute the teraton problems of system lnear equatons, whch can mae better understandng for the sgnfcance of CTM computable compared wth consderng smple teraton problems. The ey problems to prove teraton computablty s that the concept of convergence and a type of fxed-pont theorem can be descrbed by CTMs. On the former, we defne n-dmensonal real value vector as the nput of CTMs. All these CTMs consttute a space. Then, the concepts of dstance, convergence can be defned. On the latter, we can use CTM recurson theorem to get the defntons. It s necessary to ensure convergence and fxed-pont theorem. If the computatons of a CTM M teraton are convergence and satsfy the fxed-pont theorem, the teraton process s CTM M teraton handleable. Its lmt case s CTM M teraton computable. Otherwse, the teraton process s nether M teraton computable nor M teraton handleable. c 04 NSP
44 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... Theorem 5.(Recurson theorem n CTM) Let T be a CTM that computes a functon t : ΣCT M Σ CT M Σ CT M, then there exsts a CTM U that computes a functon u : ΣCT M ΣCT M for every ω, such that u(ω)= t(< U,ω >) The proof s s abbrevated snce t s smlar to the recurson theorem n TMs [3. Recurson theorem ndcates that CTMs can output the descrptons of themselves and contnuous perform a computaton by these descrptons. Hence, any complex CTM can be descrbed by recursve coded methods. Theorem 5 s the bass of the followng defntons. ω s used to denote a possble strng n a CTM. For any x = (x,x,...,x n ) R n, there exsts a CTM M that taes ω = x x...x n as the nput, whch s wrtten as < M,ω >. All of these CTMs consttute a new complete space that s called CT M R n space. The concept of dstance n CT M R n space s smlar to Eucldean dstance. For any strngs ω = x x...x n and ω = y y...y n, we can obtan < M,ω >,< M,ω > CT M R n. The dstance n CT M R n space s defned as follows: ρ(< M,ω >,< M,ω >)=< M sqrt,< M sum(= to n),< M square,< M sub,x y >>>>. If x,y R, we have the dstance ρ(< M,ω >,< M,ω >)= < M abs,< M sub,x y>>. It s easy to prove that they satsfy dstance axoms. Defnton 0.Let CT M R n be a metrc space and < M,ω n >, n =,,... be CTMs n CT M R n space. A CTM < M,ω n > s convergent to< M,ω >, wrtten as lm < M,ω n >=< M,ω >, n f ρ(< M,ω n >,< M,ω >) < M,0> Defnton.Let CT M R n be a metrc space. Functon < M T,ω >: CT M R n CT M R n s a CTM wth the ablty of contracton (t s smlar to the contracton operator n mathematc) f θ, 0 θ <, < M,ω >,< M,ω > CT M R n, we have ρ(< M T,< M,ω >>,< M T,< M,ω >>) < M mul,θ ρ(< M,ω >,< M,ω >)> Defnton.Let CT M R n be a metrc space. < M T,ω >: CT M R n CT M R n. CTM < M,ω > s called the fxedpont of < M T,ω >, f there exsts a CTM < M,ω > CT M R n such that < M,ω >=< M T,< M,ω >> Theorem 6.(Fxed-pont theorem n CTM) Let CT M R n be a complete metrc space, <M T,ω > CT M R n, CTM < M T,ω > possesses unque fxed-pont such that < M,ω >=< M T,< M,ω >> The proof s abbrevated snce t s smlar to that of the fxed-pont theorem n mathematc. Ths secton ntroduces an example of solvng a lnear system equaton n j= a j x j = b, =,,...,n by constructng a CTM M teraton to compute t. The equaton can be denoted by the form x = n j= ( j α j )x j + b, =,,...,n. If m j = j α j s hold, the teratve scheme can s represented as x (+) = n j= m j x () j + b, =,,...,n, =0,,,... The CTM M teraton can be constructed recursvely by Theorem 5. Frstly, we construct CT M sub (t s smlar to M substract ) to compute m j. Second, CT M sub s constructed to compute m j x () j. Fnally, CT M sum s ntroduced to deal wth n j= m jx () j + b. Machne M teraton can be executed by nputtng the coded descrptons of CT M sub, CT M sub, CT M sum, and ther correspondng nputs on the tapes of M teraton. Fg. 8 shows the structure of M teraton. Proposton.Iteratve computaton to solve system lnear equatons s CTM handleable f t satsfes Theorem 6 (Fxed-pont theorem n CTM). Proof. If the teratve computaton does not satsfy fxed-pont theorem, then the computaton may dvergent wthn two stuatons. Frst, the computatonal hstory may be fnte by embeddng a controllable CTM such that M teraton can machne halt at a gven length. However, the fact volates the meanng of approxmaton. Second, many convergent values lead to dfferent approxmate values, whch can generate dfferent computatonal hstores. Hence, the teratve computaton s not CTM handleable by Defnton 8. The proposton s proofed. Actually, f the teratve computaton satsfes fxed-pont theorem, related results are necessarly convergent, unqueness, and reasonable approxmaton. Accordng to algorthm 6, f haltng judgment condton ε s gven, M teraton s necessarly machne halt and the length of the unque computatonal hstory s fnte. Accordng to Defnton 8 (CTM handleable), teratve computaton to solve system lnear equaton s M teraton handleable. c 04 NSP
Appl. Math. Inf. Sc. 8, No. 5, 405-46 (04) / www.naturalspublshng.com/journals.asp 45 n Algorthm for M teraton constructon: CTM M teraton := On nput ω = ω,ω,ω 3,B,ω 5, ω 6,ω 7, B,ω 9 ( B means no ntal nput on the tape): step: Copy In/Output tape to d-tape3 (save x () ), delete checed symbol. step: Format c-tape4, copy In/Output tape to c-tape4, delete checed symbol, add checed symbol on x ( for mddle computaton). step3: In c-tape, run machne < CT M sub > wth the data of c-tape and d-tape by checed symbol (compute m j ). Copy the result < CT M sub > and current x j n c-tape4 to c-tape and separate them wth symbol. step4: In c-tape, run machne < CT M mul > wth the rght nformaton (compute m j x () j ). Copy the result < CT M mul > j to c-tape3, the subscrpt j comes from x j. Intal c-tape. step5: If not move n In/Output tape to blan, go to step6, else go to step7. step6: If n c-tape4 does not removed to blan, then consder c-tape, c-tape4, and d-tape, move one step to the rght, go to step3, else copy b n d-tape3 to c-tape3 by checng. In c-tape3, run machne < M sum > wth the rght nformaton (compute x (+) = n j= m jx () j + b ). Replace current x n In/Output tape wth the result < CT M sum >. Move all current one step to the rght. Intal c-tape3. The symbols n c-tape and d-tape are moved one step to the rght and locate on the rght frst symbol of $. Go to step. step7: Copy In/Output tape and d-tape3 to c-tape5 and run< M jud >. If the result< M jud > satsfes the precson requrement ε, output In/Output tape (dsplay approxmaton x (+) ), then machne M teraton halt, else ntal all tapes except In/Output tape, go to step. Tape functon [ m [ mj ( ) j x j ( ) [ m j xj + b j= Mddle computatons [ j b [ [ ( +) x < CTM, ω ω > sub M teraton Controller # # aa an $ aa an $ $ an ' < CTM mul, < CTM sub > # xj x x x j x n n $ n $ $ n b x b x > b j b n x j x n a ' ' ' < CTM, < CTM > < CTM > < CTM > b> c-tape 3 sum mul mul mul n nn nn c-tape c-tape c-tape 4 d-tape d-tape In/Output tape ( ) [ x x x x j x n d-tape 3 Haltng ( + ) judgment < CTM jud, < CTMsub, x # x > #ε > c-tape 5 Fg. 8: Structure of CTM M teraton. Proposton.If teratve computaton to solve system lnear equatons s CTM handleable and M teraton can machne halt wth ε = 0, t s CTM M teraton computable. Proposton s obvous and the proof s abbrevated. 5 Concluson and future wors Ths paper deals wth a class of tme-ndependent contnuous Turng machnes (CTMs) by extendng from homologous dscrete counterpart to contnuous. Some equvalent proofs between CTMs and TMs and the constructve algorthms n the sense of lmtatons are gven. The mportant concepts such as CTM, CTM handleable, CTM computable are defned, whch provde an unfed framewor of computable theory for real numbers (real functons) computatons. If the state graphs of TMs are usng n CTMs, the fact wll generate nfnte confguratons and the ncomplete nformaton descrptons. Hence a graph based representaton methods CEGs are proposed, whch can descrbe nfnte confguratons n CTMs by fnte elements and can effectvely present the behavors of CTMs. On the other hand, machne descrpton wth recursve methods s proposed and a class of machne space CT M R n s defned to dscuss the approxmate problems n R n. Then, some basc concepts of approxmatons such as convergence of machne computaton, dstance n machne space CT M R n, and Fxed-pont theory n CTM etc, are gven. Fnally teraton approxmate computatons n solvng system lnear equatons are proved to be CTM handleable or computable. Future wors wll consder the complexty theory n CTMs. Researchers have dscovered an elegant scheme to classfy a problem accordng to there computatonal dffcult. Hence, we wll use CTMs as measure tools to analyss algorthmc tme complexty and space complexty. Acnowledgements Ths research was supported n part by the Natonal Nature Scence Foundaton of Chna (No. 6743), Fundamental Research Funds for Scence and Technology Department of Schuan Provnce (No. 03JY0089), Fundamental Research Funds for Educatonal Department of Schuan Provnce (No. 08ZA09), ey Scentfc Research Fund of Xhua Unversty(No. z364), and ey Scentfc Research Fund of Xhua Unversty (No. Z63). References [ D.P. Yang and A.S. L, Theory of conputablty. Bejng: Scence press, (975). c 04 NSP
46 X. L. Chen et. al. : Contnuous Turng Machne: Real Functon Computablty... [ Z.Z. Zhao, Introducton to computatonal scence. Bejng: Scence press, (00). [3 M. Spser, Introducton to the Theory of Computaton, Boston: Course Technology Inc, (005). [4 J. Hopcroft, R. Motwan, and J. Ullman, Introducton to Automata Theory, Languages, and Computaton. Boston: Addson Wesley, (006). [5 S. Mazur, A. Grzegorczy, and H. Rasowa, Computable analyss. Pa n stwowe Wydawn. Nauowe, (963). [6 C. Kretz and K. Wehrauch, Theoretcal computer scence, 38, 35-53 (985). [7 K. Wehrauch, Computable analyss: An ntroducton. Sprnger Verlag, (000). [8 A. Edalat, Informaton and Computaton, 0, 3-48 (995). [9 A. Edalat, Theoretcal Computer Scence, 5, 63-93 (995). [0 A. Edalat, Bulletn of Symbolc Logc, 3, 40-45 (997). [ A. Edalat and P. Snderhauf, Theoretcal Computer Scence, 0, 73-98 (999). [ C. Moore, Theoretcal Computer Scence, 6, 3-44 (996). [3 U. Doraszels and M. Satterthwate, The RAND Journal of Economcs, 4, 5-43 (00). [4 L. Blum, F. Cucer, and S. Smale, Internatonal Journal of Bfurcaton and Chaos, 6, 3-6 (996). [5 L. Blum, Complexty and real computaton. New Yor: Sprnger Verlag, (998). [6 P. Hertlng, Annals of Pure and Appled Logc, 3, 7-46 (005). [7 L. Recalde, E. Teruel, and M. Slva, Applcaton and Theory of Petr Nets, 639, 07-6 (999). [8 R. Davd and H. Alla, Dscrete, Contnuous, and Hybrd Petr Nets, Berln: Sprnger-Verlag, (005). Xaolang Chen receved B.S. and M.S. degrees from Xhua Unversty, Chengdu, Chna, n 007 and 00, respectvely. He s currently a Ph.D. student n School of Electro-Mechancal Engneerng, Xdan Unversty, X an, Chna. Hs research nterests nclude supervsor control and fault dagnoss of dscrete event systems. analyss. Wen Song s a professor of Xhua Unversty snce 005. senor member of Petr net Specal Commsson. Hs man research nterests are theory of Petr nets and mathematcal logc. Zexa Huang receved B.S. and M.S. degrees from Xhua Unversty, Chengdu, Chna, n 007 and 00, respectvely. She receved Ph.D n Captal Normal Unversty, Bejng, Chna. Her research nterests nclude functon approxmaton theory and mathematcal Mngwe Tang s an assocate professor wth the School of Mathematcs and Computer Scence Technology of Xhua Unversty. He receved a Ph.D. degree at the School of Computer Scence and Engneerng from Unversty of Electronc Scence and Technology of Chna n 0. Hs current research nterests nclude networ securty and nformaton hdng. c 04 NSP