10.2 Turr,Ilrc MecHtut:s wIrH MoRn Colvtplnx SroRlGu
261
Remember that the various models of Thring ma,t:hinesare considered equivalerrt only with respect to their ability to do things, not witlt respect to eilsc of progra,mmirrg or auy othtlr efficiency mcasure we rnight consider. Wc will return to this importtrrrt point in Chnpter 14.
TuringMochines Multidimensionol A multidirrtensiona,l T\rrittg machine is otre in which the tape ca,rrbe viewed as extcrrditrg infiniterly itr tnore tharr one dimensiotr, A ditlgrart1 of a, twtr dimensiotnl Turing rnachine is shown in Figure 10'12. T[e formal rJe:firritionof a, two-dimensional Ttrring machitre involvt:s a transition function tI of the fbrm
, ): Q x f - I x I ' x { . L R , ,U,D}, where [/ anrl D specify rnuvement of t]rc read-writs ]read up tr,nd down, rcspectively. To simqlate this machitte on a stanrlrrrd't\rring machine, we (iaII use the two-tra,ck moclel depictcd in Figure 10.13. First, wc associa,teorr ordering or acldrnsswith the r:clls of the twtrditneusiona.l tape. This t:atr Lredone irr a lumher of ways, fbr example, in the two-dimt:rrsioual ftr,shion indica,ttxl in Figurc 10.12. Thc two-tra,ck tape of the silrulating rnirchiue will use clne track to stortl cell conttlnts and the other one to kcep the a^rsociated a,dclress. In thc scheme of Figure 10.12, the configrrration in whictr cell (1,2) contains a, ancl cell (10, -3) contains b is shown iil F'igure 10'13' Note orrclcornplica,tiorr:the cell a,tlrlresscan invcllve arbitrnrily large intcgers, so the address track caunot trse a fixed-sizc field to sttlre addresstls. Instead, we must rrstt a variable field-size a,rrrlrrgetnettt,using some spt:c:ialsymbols to delimit the fields, tr,sshowu itt thc picture' Lelt us assumethat, at tlre sta,rt of the simulation of eat:hrrrove,the rcadwrite heaci of thrl two-clirnensional ma,chintl M and ttre read-writcl head of the simrrlatirrg ma,chinc -ffi ut* u]{tw. oll corre$poncling cells. To simrrlate a move. the simulating tlachine M first comprrtgs the addressof the tnll to
Figure 10.12
Two-dimensiond addressscheme