Outline. Unlimited Register Machine. Intuitive Definition. What is Effective Procedure. Basic Concepts Computable Function

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1 Outline UnlimitedRegisterMachine Xiaofeng Gao Department of Computer Science and Engineering Shanghai Jiao Tong University, P.R.China CS363-Computability Theory SpecialthanksisgiventoProf.YuxiFuforsharinghisteachingmaterials. CSC363-Computability Theory@SJTU Xiaofeng Gao 1/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 2/45 What is Effective Procedure Intuitive Methods for addition, multiplication Givenn,findingthenthprimenumber. Differentiating a polynomial. FindingthehighestcommonfactoroftwonumbersHCF(x,y) Euclidean algorithm Giventwonumbersx,y,decidingwhetherxisamultipleofy. An algorithm or effective procedure is a mechanical rule, or automatic method, or programme for performing some mathematical operations. Blackbox: input output Their implementation requires no ingenuity, intelligence, inventiveness. CSC363-Computability Theory@SJTU Xiaofeng Gao 4/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 5/45

2 What is effective procedure? Algorithm : Consider the function g(n) defined as follows: g(n) = Question: Is g(n) effective? 1, ifthereisarunofexactlynconsecutive7 s inthedecimalexpansionof π, 0, otherwise. Theanswerisunknown theanswerisnegative. Analgorithmisaprocedurethatconsistsofafinitesetofinstructions which,givenaninputfromsomesetofpossibleinputs,enablesusto obtain an output through a systematic execution of the instructions that terminates in a finite number of steps. Other Examples: Theorem Proving is in general not effective/algorithmic. Proof Verification is effective/algorithmic. CSC363-Computability Theory@SJTU Xiaofeng Gao 6/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 7/45 When an algorithm or effective procedure is used to calculate the value of a numerical function then the function in question is effectively calculable(or algorithmically computable, effectively computable, computable). Examples: HCF(x, y) is computable; g(n) is non-computable. An (URM) is an idealized computer. Nolimitationinthesizeofthenumbersitcanreceiveasinput. No limitation in the amount of working space available. Inputs and outpus are restricted to natural numbers.(coding for others) From Shepherdson& Sturgis[1963] s description. Shepherdson, J. C.& Sturgis, H.E., Computability of Recursive Functions, Journal of Association for Computing Machinery (Journal of ACM), 10, , CSC363-Computability Theory@SJTU Xiaofeng Gao 9/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 11/45

3 Register Program AURMhasaninfinitenumberofregisterlabeledR 1,R 2,R 3,... R 1 R 2 R 3 R 4 R 5 R 6 R 7 r 1 r 2 r 3 r 4 r 5 r 6 r 7 Every register can hold a natural number at any moment. The registers can be equivalently written as for example AURMalsohasaprogram,whichisafinitelistofinstructions. An instruction is a recognized simple operations(calculation with numbers)toalterthecontentsoftheregisters.(i 1,,I s ) [r 1 r 2 r 3 ] 3 1 [r 4] 4 4 [r 5r 6 r 7...] 5 or simply [r 1,r 2,r 3 ] 3 1 [r 4] 4 4 [r 5,r 6,r 7 ] 7 5. CSC363-Computability Theory@SJTU Xiaofeng Gao 12/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 14/45 Configuration and s Example: The initial registers are: Type Response of the URM Zero Z(n) Replacer n by0.(0 R n,orr n :=0) Successor S(n) Add1tor n.(r n +1 R n,orr n :=r n +1) Transfer T(m,n) Copyr m tor n.(r m R n,orr n :=r m ) Jump J(m,n,q) Ifr m =r n,gototheq-thinstruction; otherwise go to the next instruction. Z(n), S(n), T(m, n) are arithmetic instructions. The program is: I 1 : J(1,2,6) I 2 : S(2) I 3 : S(3) I 4 : J(1,2,6) I 5 : J(1,1,2) I 6 : T(3,1) R 1 R 2 R 3 R 4 R 5 R 6 R CSC363-Computability Theory@SJTU Xiaofeng Gao 15/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 17/45

4 Configuration and Computation OperationofURMunderaprogramP P = {I 1,I 2,,I s } URM Configuration: the contents of the registers + the current instruction number. Initial configuration, computation, final configuration. URMstartsbyobeyinginstructionI 1 WhenURMfinishesobeyingI k,itproceedstothenext instruction in the computation, ifi k isnotajumpinstruction,thenthenextinstructionisi k+1 ; ifi k =J(m,n,q)thennextinstructionis(1)I q,ifr m =r n ;or(2) I k+1,otherwise. ComputationstopswhenthenextinstructionisI v,wherev>s. ifk =s,andi s isanarithmeticinstruction; ifi k =J(m,n,q),r m =r n andq >s; ifi k =J(m,n,q),r m r n andk =s. CSC363-Computability Theory@SJTU Xiaofeng Gao 18/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 19/45 Flow Diagram Some Notation J(m,m,q)is alertunconditional jump Computations that never stop SupposePistheprogramofaURManda 1,a 2,a 3,...arethe numbers stored in the registers. P(a 1,a 2,a 3,...)istheinitialconfiguration. P(a 1,a 2,a 3,...) meansthatthecomputationconverges. P(a 1,a 2,a 3,...) meansthatthecomputationdiverges. P(a 1,a 2,...,a m )isp(a 1,a 2,...,a m,0,0,...). CSC363-Computability Theory@SJTU Xiaofeng Gao 20/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 21/45

5 WhatdoesitmeanthataURMcomputesa(partial)n-aryfunctionf? LetPbetheprogramofaURManda 1,...,a n,b N.When computationp(a 1,...,a n )convergestobifp(a 1,...,a n ) and r 1 =binthefinalconfiguration.wewritep(a 1,...,a n ) b. PURM-computesf if,foralla 1,...,a n,b N, P(a 1,...,a n ) bifff(a 1,...,a n ) =b Function f is URM-computable if there is a program that URM-computes f. (We abbreviate URM-computable to computable ) CSC363-Computability Theory@SJTU Xiaofeng Gao 23/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 24/45 s Examples ConstructaURMthatcomputesx+y. Let C bethesetofcomputablefunctionsand C n bethesetofn-arycomputablefunctions. I 1 : J(3,2,5) I 2 : S(1) I 3 : S(3) I 4 : J(1,1,1) CSC363-Computability Theory@SJTU Xiaofeng Gao 25/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 26/45

6 Examples ConstructaURMthatcomputesx 1 = I 1 : J(1,4,8) I 2 : S(3) I 3 : J(1,3,7) I 4 : S(2) I 5 : S(3) I 6 : J(1,1,3) I 7 : T(2,1) { x 1, ifx >0, 0, ifx =0. Examples ConstructaURMthatcomputesx 2 = I 1 : J(1,2,6) I 2 : S(3) I 3 : S(2) I 4 : S(2) I 5 : J(1,1,1) I 6 : T(3,1) { x/2, ifxiseven, undefined, ifxisodd. CSC363-Computability Theory@SJTU Xiaofeng Gao 27/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 28/45 Function Defined by Program Predicate and Decision Problem GivenanyprogramPandn 1,bythinkingoftheeffectofPon initialconfigurationsoftheforma 1,,a n,0,0,,thereisa uniquen-aryfunctionthatpcomputes,denotedbyf (n) P. { f (n) b, P (a ifp(a1,...,a 1,...,a n ) = n ) b, undefined, ifp(a 1,...,a n ). Thevalueofapredicateiseither true or false. Theanswerofadecisionproblemiseither yes or no. Example:Giventwonumbersx,y,checkwhetherxisamultipleofy. Input:x,y; Output: Yes or No. The operation amounts to calculation of the function { 1, ifxisamultipleofy, f(x,y) = 0, if otherwise. Thusthepropertyorpredicate xisamultipleofy isalgorithmically or effectively decidable, or just decidable if function f is computable. CSC363-Computability Theory@SJTU Xiaofeng Gao 29/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 31/45

7 Decidable Predicate and Decidable Problem Computability on other Domains SupposethatM(x 1,...,x n )isann-arypredicateofnaturalnumbers. Thecharacteristicfunctionc M (x),wherex =x 1,...,x n,isgivenby { f (n) 1, ifm(x)holds, P (a 1,...,a n ) = 0, if otherwise. ThepredicateM(x)isdecidableifc M iscomputable;itisundecidable otherwise. SupposeDisanobjectdomain.AcodingofDisanexplicitand effectiveinjection α :D N.Wesaythatanobjectd Discoded by the natural number α(d). Afunctionf :D Dextendstoanumericfunctionf : N N.We saythatf iscomputableiff iscomputable. f = α f α 1 CSC363-Computability Theory@SJTU Xiaofeng Gao 32/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 33/45 Example Example(Continued) Considerthedomain Z.Anexplicitcodingisgivenbythefunction α where { 2n, ifn 0, α(n) = 2n 1, ifn<0. Then α 1 isgivenby α 1 (m) = { m, ifmiseven, (m+1), ifmisodd. Considerthefunctionf(x) =x 1onZ,thenf : N Nisgivenby 1 ifx =0(i.e.x=α(0)), f (x) = x 2 ifx >0andxiseven(i.e.x = α(n),n >0), x+2 ifxisodd (i.e.x=α(n),n <0). Itisaroutineexercisetowriteaprogramthatcomputesf,hence x 1isacomputablefunctionon Z. CSC363-Computability Theory@SJTU Xiaofeng Gao 34/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 35/45

8 Remark Finiteness s are more advanced than Turing Machines. Models can be classified into three groups: CM(Counter Machine Model). RAM(Random Access Machine Model). RASP(Random Access Stored Program Machine Model). EveryURMusesonlyafixedfinitenumberofregisters,nomatter howlargeaninputnumberis. ThisisafinepropertyofCounterMachineModel. The Model belongs to the CM class. CSC363-Computability Theory@SJTU Xiaofeng Gao 37/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 38/45 Sequential Composition Lemma GivenProgramsPandQ,howdoweconstructthesequential composition P; Q? ThejumpinstructionsofPandQmustbemodified. StandardForm:AprogramP=I 1,...,I s isinstandardformif,for everyjumpinstructionj(m,n,q)wehaveq s+1. ForanyprogramPthereisaprogramP instandardformsuchthat anycomputationunderp isidenticaltothecorresponding computationunderp.inparticular,foranya 1,,a n,b, P(a 1,,a n ) bifandonlyifp (a 1,,a n ) b, andhencef (n) P =f (n) P foreveryn>0. CSC363-Computability Theory@SJTU Xiaofeng Gao 40/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 41/45

9 Proof Join/Concatenation SupposethatP=I 1,I 2,,I s.putp =I1,I 2,,I s where ifi k isnotajumpinstruction,thenik =I k; { ifi k isnotajumpinstruction,thenik = Ik ifq s+1, J(m,n,s+1) ifq >s+1. LetPandQbeprogramsoflengthss,trespectively,instandardform. ThejoinorconcatenationofPandQ,writtenPQor P Q,isaprogram I 1,I 2,,I s,i s+1,,i s+t wherep=i 1,,I s andtheinstructions I s+1,,i s+t aretheinstructionsofqwitheachjumpj(m,n,q) replacedbyj(m,n,s+q). CSC363-Computability Theory@SJTU Xiaofeng Gao 42/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 43/45 Program as Subroutine ThenotationP[l 1,...,l n l]standsforthefollowingprogram: I 1 : T(l 1,1) SupposetheprogramPcomputesf. Letρ(P)betheleastnumberisuchthattheregisterR i isnotusedby the program P.. I n : T(l n,n) I n+1 : Z(n+1). I ρ(p) : Z(ρ(P)) _ : P _ : T(1,l) CSC363-Computability Theory@SJTU Xiaofeng Gao 44/45 CSC363-Computability Theory@SJTU Xiaofeng Gao 45/45