Sebastián Bravo López

Transcription

1 Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that we have a task whose solution involves an infinite number of steps. With the omputational power of transfinite Turing mahines [6] it is possible to ompute the task. The study of more powerful models of omputation have impliations in mathematis, physis, omputer siene and even philosophy [1]. The theoretial models of transfinite Turing mahines an be applied to the real world to study physis. Quantum mehanis give us the opportunity to apply the new models to the real world. Transfinite Turing mahines [6] extend the Turing mahine model. This approah extends the operation of ordinary Turing mahines into transfinite ordinal time. This new model an omplete algorithms that require an infinite amount of steps to be ompleted. The steps work following ordinal numbers. If the mahine does not halt in finite amount of time it arrives to the next infinite stage and so on [5]. Transfinite ordinal numbers represent a sequene of inreasing ordinals in whih is possible to find an ordinal number greater than the rest. The onstrution of a new model of infinite omputability is the key to understand and find algorithms that require an infinite number of steps to ompute the solution. The tasks solved by transfinite Turing mahines are alled supertask [6]. Supertasks are solved in a finite interval of time but that require a infinite number of steps to be finished. This paradox an help the reader to understand the supertask onept. Zeno of Elea (a. 450 B.C.) had a theory that it is impossible to go from one plae to another one. Before arriving, one must first get halfway there, and before that one must get halfway to the 1

2 halfway point, till infinite. Based on this argument all motion is impossible, but this kind of supertask ould be solved with the use of the new model. The main type of transfinite Turing mahine is the infinite time Turing mahine [5]. The main property of the infinite time Turing mahine relies on the management of an infinite number of steps. Infinite time Turing mahines are able to arry out and omplete algorithms involving infinitely many omputational steps. The main priniple to do it is based on the division of the task into infinite stages based on ordinal numbers. Aelerated Turing mahines [5] an inrease the omputation speed in every step. Division of the infinite steps into several finite stages is one of the main properties to understand the method. Other examples are the infinite state Turing mahines [6] or the quantum Turing mahines [2][3][1]. The organization of this paper is the following: in Setion 2 the transfinite ordinal numbers will be explained. This topi is very important to understand how transfinite ordinal time is represented. The main types of transfinite Turing mahines, infinite time Turing mahines, infinite state Turing mahines, aelerated Turing mahines and other approahes are explained in Setion 3. Finally at setion 4 there are some thoughts about the power of transfinite Turing mahines and the problems that an be solved with the use of the new models. Final onlusions are drawn in the Setion 5. 2 Transfinite ordinal numbers Transfinite Turing mahines extend the lassial Turing mahine into transfinite ordinal time. This implies that the mahines need infinite number of omputational steps. Unsolvable problems would need infinite omputation. Some problems an be solved in ountable infinite time and by using transfinite numbers [5] we an measure the omplexity of infinite omputations. The omputational steps rely on ordinal numbers. An ordinal number is defined as the order type of a well ordered set. Totally ordered set is defined by the following onditions: 1. Antisymmetry: (a b) and (b a) implies a = b. 2. Transitivity: (a b) and (b ) implies (a ). 3. Comparability (trihotomy law): For any a,b in S, either (a b) or (b a). 2

3 Two totally ordered sets (A, ) and (B, ) are order isomorphi if there is a bijetion f : A B suh that all α 1, α 2 A, α 1 α 2 iff f(α 1 ) f(α 2 ). Finite ordinal numbers are denoted by arabi numerals and transfinite ordinals are denoted using lower ase Greek letters. Every finite totally ordered set is well ordered. Any two totally ordered sets with k elements, where k is a non negative integer, are order isomorphi and they have the same order type. The ordinals for an infinite set are denoted by 1, 2, 3,, ω. The number series is divided into finite stages 1, 2, 3,, ω. The number denoted by ω is the order type of a set of nonnegative integers. ω represents the first of the Cantor s transfinite numbers. ω is the smallest ordinal number but higher than the natural numbers. When we arrive to the stage ω, the series ontinues with the stages (ω + 1), (ω + 2),, (ω + ω),. New ωs are added an infinite number of times. By ω s it is possible to define all ountable numbers [5], some examples of large ordinal numbers are: (ω 2 ) or (ω ω ). One of the main properties of the transfinite numbers is the ability to interhange operations: x + 1 = 1 + x, n x = x n. We an define the suessors adding numbers to the stage: (1 + ω) is the suessor of ω. For transfinite numbers the order of operation determines the meaning: 1. (1 + ω) = (n + ω) = 2 ω = n ω = ω, but 2. (ω +1) = ω s suessor,(ω +2) = (ω +1) s suessor, (ω 2) = (ω +ω), et. 3 More powerful Turing mahines Transfinite Turing mahines [6] are based on the lassial Turing mahines. It is possible to inrease the power of standard Turing mahines by allowing the mahine to have infinite time to run, aelerated speed, infinite number of states or quantum states. 3.1 Classi Turing mahines The standard Turing mahine onsists on several parts [5]. There is a head moving bak and forth reading and writing haraters on a tape aording to the rigid instrutions. The haraters depend on the alphabet used and generally Turing mahines an use any set of haraters. It is ommon the use of the alphabet 0, 1. All other haraters an be represented by strings 3

4 of 0 s ans 1 s. For example we ould group 4 bits to reate a harater. For 4 bits we have 16 different ombinations that ould represent numbers (0,, 9) and some operations (, +,, /). Classi Turing mahines have also a ontrol unit whih defines the transitions between the different states based on the input. They have also tapes in whih the information is stored. The input and output are read and written in those tapes. An algorithm for a Turing mahine is defined by the state transitions. The heads an move along the tapes and read and write values aording to the urrent state. When the algorithm finds the solution the urrent state is the final state and the Turing mahine halts. The Transfinite Turing mahines use the same priniple but with the addition of some properties. CONTROL UNIT Tape 1 a a a a a T b Tape 2 a a a a a T b Tape N a a a a a T b Figure 1: Turing Mahine 3.2 Infinite time Turing mahines Infinite time Turing mahines [5] an ompute an infinite number of steps in a finite amount of time. The tapes on whih the infinite time Turing 4

5 mahines writes are infinite and there is no limit in the spae to use. Sine infinite time Turing mahines an use the entirety of their tapes during their exeution, it is possible to use a more wide range of algorithms with them. The mahine an perform several infinite sequenes of steps and repeat the proess infinitely. The number of steps an be seen as a sequene of ordinal numbers. For example the infinite sequene of infinite sequenes of steps is denoted by ω 2. The infinite time Turing mahine is an extension of the Turing mahine into transfinite ordinal time. At a limit in the ordinal time the mahine s atual state is based on the previous states. The mahine an ompute any reursive funtion in less than ω steps and evaluate the funtion in every ω steps [5] obtaining data already alulated. At every step ω its possible to onsult the partial results. Even if the task is not finished, there is information already omputed that an be obtained as a partial result. This allows to obtain results even if the mahine does not halt. This results an be wrong or have some solution and they an be used to know the state of the mahine. 3.3 Infinite state Turing mahines Infinite state Turing mahines [6] are standard Turing mahines with infinite number of states. Although this an be implemented with an infinite transition table, it is possible to do in pratie with the use of states based on funtions or some other properties. The tape is only used for input and output data and the main work is done by the use of infinite number of states. This leads to better results for questions of omputational omplexity. As we already have an infinite number of states, we have no need for a tape, we an simply inorporate this into the states. The number of possible transitions is also infinite but only a finite number of states an be reahed from a given one. One obvious way of speifying a state is by an infinite string of 0 s and 1 s. This gives an unountable number of states. We simply require that every string at time t determines another string at time t + 1. It is possible even to have all the information in the states, and with the hange of the states, the input data and the output of any funtion an be speified and different algorithms an be used. In this example the tape is only used for input and output but it is also possible to define inputs in the states and avoid ompletely the tape. If we are in the state 1 and we add 1 to our result we move to the state 2, and the state is the result itself. 5

6 For example, following the funtion that adds the number 1 to the urrent number the next step is urrent state+1, the previous step is urrent state 1. The state depends on a given funtion and the result an be obtained by the urrent state. If urrent state is n, the resulting number is n. 3.4 Aelerated Turing Mahines Aelerated Turing Mahines [6] an inrease the omputation speed in every step. Sine < 2, any task an be done in less than two time units. There is no differene between standard Turing mahines and the aelerated ones other than the speed of operation whih is the same in the first step and exponentially inreased in the next steps. Let us study the following example [6]: Let A to be an aelerated Turing mahine that hanges the value of a tape position from 0 to 1 when a given Turing mahine B halts. If the Turing mahine B does not halt, A leaves the original value 0, but after 2 time units the position on A holds the value of the halting funtion value for this Turing mahine B and its input beause after 2 time units the mahine finish regard of if it halts or not. This aelerated Turing mahine only omputes funtions N {0, 1} but it an be extended to N N by designating the odd positions to begin with 0 and only being hanged at most one. Natural numbers and even real numbers an be written in binary on the speial squares after two time units. 3.5 Other Turing mahines The addition of the random fator is an interesting property for the Turing mahines. The probabilisti Turing mahines [7] an have two different and valid transitions from the urrent state. In eah state the mahine selets between the valid transitions giving them equal probabilities to be seleted. Using this model the Turing mahine an ompute reursive funtions on real numbers and natural numbers. Probabilisti Turing mahines an give different results for different exeutions. Quantum Turing mahines [2][3][1] are beoming very popular. In a quantum Turing mahine read, write, and shift operations are all aomplished by quantum interations. The tape itself exists in a quantum state as does the head. In partiular, in the plae of the Turing tape position that ould hold either 0 or 1, in quantum Turing mahine there is a qubit [3][1], whih an hold a value between 0 and 1. 6

7 Qbits an have different states 0, 1 and 0+1 (state between 0 and 1). Qbits are often represented by an sphere with an arrow inside. Arrow up means 1, arrow down means 0, any other arrow state is alled arrow phase. Arrow phase is an intermediate state that provides the quantum Turing mahines an interesting property, the apaity to selet the transitions from non well defined value between 0 and 1. The tape of the quantum Turing mahine is made of qbits. The mahine evolves in many different diretions simultaneously. After some time t its state is a superposition of all states that an be reahed from the initial ondition in that time. Although the quantum Turing mahines an be emulated with standard Turing mahines, the quantum Turing mahines omplexity appears in the layout of transitions of the quantum states. The quantum Turing mahine an enode many inputs to a problem simultaneously, and then it an perform alulations on all the inputs at the same time. This is alled quantum parallelism. Although the quantum Turing mahines are faster than Turing mahines, they an not solve more omplex problems, they have the same power. 4 Power of transfinite Turing mahines The transfinite Turing mahines an solve the same range of problems than the lassial Turing mahines and muh more [5], but we must take into aount the onepts of infinite number of steps. The main onept in the new model is the infinite omputational power of those mahines. The transfinite Turing mahines have infinite time to ompute an algorithm. Another property is the ability of the infinite time Turing mahines to ompute partial results at eah step. When the transfinite Turing mahine arrives to step ω its possible to obtain the state of the mahine and results. Sine the oneption of the Turing mahines it has been known that there exist funtions not omputable with them. It is lear when we onsider an unountable set of funtions from N N and ompare it with the ountable set of the Turing mahines. Halting problem is unsolvable [8] on standard Turing mahines. In the lassial Turing mahines, the halting problem represents the question of whether a given program p halts on a given input n in finitely many steps. Halting problem is one of the first problems whih was proved to be unde- 7

8 ideable. If a solution to a new problem is found it an be used to solve an undeidable problem. That is made by transform instanes of the undeidable problem to instanes of the new problem. If it is not possible to solve the old problem, then is not possible to solve the new one. One onsequene of the halting problem is that is not possible to reate an algorithm that finds if a statement about natural numbers is true or false. The proposition that states that a ertain algorithm will halt with some input an be onverted to another statement about natural numbers. An algorithm that solves any statement about natural numbers an solve also the halting problem, but that determine when the original program halts, that is impossible, so halting problem is undeidable. Halting problem of the lassi Turing mahines is solvable with the use transfinite Turing mahines after ω (infinite ordinal stage) steps. But transfinite Turing mahines have also their own halting problem unsolvable by using transfinite Turing mahines. 5 Conlusions In this paper I have overed the basis of the transfinite Turing mahines. I have desribed different approahes of the transfinite Turing mahines. Quantum Turing mahines represents a new model apable of very powerful alulation like physial world simulation. With the use of transfinite Turing mahines its possible to extend the lassi Turing mahines into transfinite ordinal time. Transfinite Turing mahines provide a natural new model of infinitely omputability and a good point of view for setting the analysis of the power and limits of supertask algorithms (i.e. algorithms that involves infinitely many steps). In this report I did not presented all the different extended Turing mahines. For example asynhronous networks of Turing mahines [6] or error prone Turing mahines [6] are very interesting approahes, but there exist some similarities with the already mentioned Turing mahines variants. There exist several types of quantum Turing mahines like the bulk quantum Turing mahines [2][3][1]. Although bulk quantum Turing mahines are not more powerful than quantum Turing mahines, they provide a better way to define the problems. Both are related with spatial mahines [4], an attempt to reate more powerful models with the use of 3 dimensional spae 8

9 as a limit for the develop of new Turing mahines models. Referenes [1] Paul Benioff. Models of quantum turing mahines. Fortshritte der Physik, 46:423, [2] Gilles Brassard. Quantum omputing: the end of lassial ryptography? SIGACT News, 25(4):15 21, [3] Maro Carpentieri. On the simulation of quantum turing mahines. Theory of Computer Siene, 304(1-3): , [4] Yosee Feldman and Ehud Shapiro. Spatial mahines: a more realisti approah to parallel omputation. Communiations of ACM, 35(10):60 73, [5] Joel David Hamkins. Infinite time Turing mahines. Minds and Mahines, 12(4): , [6] Toby Ord. Hyperomputation: omputing more than the Turing mahine. CoRR, Department of Computer Siene, University of Melbourne, [7] Alon Orlitsky, Narayana P. Santhanam, and Junan Zhang. Always good Turing: Asymptotially optimal probability estimation. fos 03: Proeedings of the 44th annual ieee symposium on foundations of omputer siene, vol 302, no pages , [8] Alan Turing. Halting problem of one binary horn lause is undeidable. pages Ser. 2, Vol. 42,