Introduction to Complexity and Computability

116 Pages·2017·6.68 MB·English

by  Jirka Fink

Introduction to Complexity and Computability
NTIN090
JirkaFink
https://ktiml.mff.cuni.cz/˜fink/
DepartmentofTheoreticalComputerScienceandMathematicalLogic
FacultyofMathematicsandPhysics
CharlesUniversityinPrague
Wintersemester2017/18
LastchangeonJanuary11,2018
License:CreativeCommonsBY-NC-SA4.0
JirkaFink IntroductiontoComplexityandComputability 1
Jirka Fink: Introduction to Complexity and Computability
Generalinformation
E-mail fi[email protected]
Homepage https://ktiml.mff.cuni.cz/˜fink/
Consultations Thursday,17:15,officeS305
Examination
Homeworks(theoretical)
Passtheexam
JirkaFink IntroductiontoComplexityandComputability 2
Content
1 Computability
2 Complexity
JirkaFink IntroductiontoComplexityandComputability 3
Literature
Sipser,M.IntroductiontotheTheoryofComputation. Vol. 2. Boston: Thomson
CourseTechnology,2006.
Garey,Johnson: Computersandintractability-aguidetothetheoryof
NP-completeness,W.H.Freeman1978
SoareR.I.: Recursivelyenumerablesetsanddegrees. Springer-Verlag,1987
OdifreddiP.: Classicalrecursiontheory,North-Holland,1989
AroraS.,BarakB.: ComputationalComplexity: AModernApproach. Cambridge
UniversityPress2009.
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Outline
1 Computability
2 Complexity
JirkaFink IntroductiontoComplexityandComputability 5
Post correspondence problem (Post, 1946)
Input
Everydominocontainstwostrings,oneoneachside;e.g. a
ab
(cid:26) (cid:27)
Aninputtotheproblemisacollectionofdominoes;e.g. b , a , ca , abc
ca ab a c
Match
Amatchaisalistofthesedominoes(repetitionspermitted)sothatthestringonthe
topisthesameasthestringonthebottom;e.g. a b ca a abc
ab ca a ab c
Problem
Findanalgorithmwhichdetermineswhetherforagivencollectionofdominoesthere
existsamatch.
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Hilbert’s tenth problem (1900)
Originalstatement(translatedfromGerman)
GivenaDiophantineequationwithanynumberofunknownquantitiesandwithrational
integralnumericalcoefficients: Todeviseaprocessaccordingtowhichitcanbe
determinedinafinitenumberofoperationswhethertheequationissolvableinrational
integers.
Reformulation
Findanalgorithmwhichforagivenmulti-variablepolynomialwithrationalcoefficients
hasintegralroot.
Exampleofaninstanceoftheproblem
Isthereanintergralsolutionofequation6x3yz2+3xy2−x3−10=0?
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Halting problem
Haltingproblem
Isthereanalgorithmwhichforagivenprogramanditsinputdetermineswhetherthe
programonthatinputterminates?
Exampleofaprogram
int main(int argc, char *argv[]) {
int n, total, x, y, z;
scanf("%d", &n);
total=3;
for(;;) {
for(x=1; x<=total-2; ++x) {
for(y=1; y<=total-x-1; ++y) {
z=total-x-y;
if(exp(x,n)+exp(y,n)==exp(z,n))
return 0;
}
}
++total;
}
}
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Random access machine
Components
Memorydividedintocells
Cellsareindexedby(positive)integers
Everycellcanstoreainteger
Programcomposedofinstructions
Arithmeticalinstructions:additions,subtraction,multiplication,division,modulo
Logicalinstructions:conjunction,disjunction,shift
Controlinstructions:conditionalandunconditionaljump,halt
Inputandoutput
Specialinstructionforreadingandwriting,or
storinginmemoryinpre-definedcells
Specialregisters(e.g. programcounter)
Argumentsofinstructionsandmemoryindexing
Integer,e.g. 3
Directaddressing,e.g. [4]
Indirectaddressing,e.g. [[5]]
Exampleofinstruction: [[5]]:=3+[4]
Variablescanbeusedtoimprovereadability
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Random access machine
Example: Factorial
Calculatefactorialoftheinputstoredat[1]andsavetheresultsin[2].
[2] := 1
loop:
[2] := [1] * [2]
[1] := [1] - 1
jump_if_positive [1], loop
halt
VariantsofRandomaccessmachines
Memorypotentiallyinfiniteonlyinonedirection
Cellswithboundednumberofbits
Moreinstructions,e.g. functioncalls
Moreindirectionsinmemoryaccessing
Etc.
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