CS1010: Theory of Computation - Brown University · 2020. 9. 10. · theory of computation: –Automata and Languages –Computability Theory –Complexity Theory This course is about

CS1010:TheoryofComputation

LorenzoDeStefaniFall2020

Lecture0a:Introduction

Whatisthiscourseabout?

• Canwedefinecomputationwithoutreferringtoamoderncomputer?

• Canwedefineacomputermathematically?– YesàTuringmachines

• Iscomputation definableindependentofpresent-dayengineeringlimitations,understandingofphysics,etc.?

• Canacomputersolveanyproblem,givenenoughtimeanddisk-space?

• Oraretherefundamentallimitstocomputability?

Whatis“Computation”?

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Whatisthiscourseabout?

Thiscoursecovers3areas,whichmakeupthetheoryofcomputation:– AutomataandLanguages– ComputabilityTheory– ComplexityTheory

Thiscourseisaboutthefundamentalcapabilitiesandlimitationsofcomputers/computation

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AreasofTheoryofComputation

• Whatcanbecomputed?• Canacomputersolveanyproblem,

givenenoughtimeanddisk-space?

• Howfastcanwesolveaproblem?• Howlittledisk-spacecanweuseto

solveaproblem

• Whatproblemscanwesolvegivenreallyverylittle(constant)space?

COMPUTABILITY

AUTOMATA

COMPLEXITY

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AreasofTheoryofComputation

• Notallproblems!!!• Eg.GivenaC-program,wecannotcheckifitwillnoteventuallycrash

• Verificationofcorrectnessofprogramsishenceimpossible!

COMPUTABILITY

AUTOMATA

COMPLEXITY

Whatproblemscanacomputersolve?

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AreasofTheoryofComputation

• EvencheckingwhetheraC-programwillhalt/terminate isnotpossible!

Program(n)Inputintegern>1while(n!=1)

if(niseven)returnProgram(n/2)

elsereturnProgram(3n+1)

E.g.,forinputn=34:17,52,26,13,40,20,10,5,16,8,4,2,1.

COMPUTABILITY

AUTOMATA

COMPLEXITY

Whatproblemscanacomputersolve?

Impossibletodecidewhetherthisprogramterminatesforallpossibleinputvalues

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AreasofTheoryofComputation

• Certainproblemscannotbesolvedbycomputers– I.e.,theyhavenoalgorithmicsolution

• ATuringmachine,asimplemodelforacomputer,candoeverythingthatacomputercando(Church-Turingthesis)

• WecanuseaTuringmachinetodeterminewhatacomputercanandcan’tdo(i.e.,compute)

COMPUTABILITY

AUTOMATA

COMPLEXITY

Whatproblemscanacomputersolve?

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AreasofTheoryofComputation

• Polynomialtimecomputable• Non-det PolyTime(NP)• Approximation,Randomization

Functionsthatcannotbecomputedfast:• Applicationstosecurity• Encryptfast,decryptioncannotbedonefast

– RSAcryptography– webapplications

COMPUTABILITY

AUTOMATA

COMPLEXITY

Howfastcanwecomputeafunction?Howmuchmemoryspacedowerequire?

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AreasofTheoryofComputation

• Examples:– trafficsignals,– vendingmachines– hardwarecircuits– …

• Applications:– patternmatching,– modeling,– verificationofhardware,– …

COMPUTABILITY

AUTOMATA

COMPLEXITY

Whatcanwedowithmachinewithfinitememoryspace?

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Models

• Automata:– DFA/NFA– Foundationsofcomputing– Mathematicalmethodsofargument– Simplesetting

TURINGMACHINES

AUTOMATA

CONTEX-FREELANGUAGES

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Models

• Context-freelanguages– Grammars,parsing– Finitestatemachineswithrecursion(orstack)

– Stillasimplesettingbutinfinitespace

TURINGMACHINES

AUTOMATA

CONTEX-FREELANGUAGES

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Models

• Turingmachines(1940s):– Themostgeneralnotionofcomputing

– TheChurch-Turingthesis• Allowtostudylimitsofcomputability– Canwesolveanymathematicalquestionmethodically?

– Gödel’sTheorem:NO!!!– “Eventhemostpowerfulmachinescannotsolvesomeproblems”

TURINGMACHINES

AUTOMATA

CONTEX-FREELANGUAGES

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Models• AlanTuring

– “OnComputableNumbers,withan ApplicationtotheEntscheidungs problem”(1936)

– Proveduncomputable functions exist(“haltingproblem”)

– Church-Turingthesis– CryptanalysisworkbreakingEnigmainWW2

• NoamChomsky– “LogicalStructureofLinguisticTheory”(1957)– Introducedthenotionofformallanguagesarguing

generativegrammarsareatthebaseofnaturallanguages

– Hierarchyofformallanguagesthatcoincideswithcomputation

• RabinandScott– “Finiteautomataandtheirdecisionproblems”(1959)– Introducednon-deterministicautomataandthe

formalismwestillusetoday

TURINGMACHINES

AUTOMATA

CONTEX-FREELANGUAGES

COMPLEXITY

FORMALIZATION

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AboutthisCourse

• TheoryofComputationtraditionallyconsideredchallenging– Iexpect(andhope)thatyouwillfindthistobetrue=)

• Averydifferentkindofcourse– Inmanyways,apuretheorycourse

• Butverygrounded(themodelsofcomputationarenotabstractatall)– Proofsareanintegralpartofthecourse,althoughIandthetextbothrely

oninformalproofs• Butthereasoningmuststillbeclear

• Theonlywaytolearnthismaterialisbydoingproblems– Youshouldexpecttospendseveralhoursperweekonhomework– Youshouldexpecttoreadpartsofthetext2-3times– Donotgiveupafter5minutesifyouarestumpedbyaproblem!

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CourseStaff

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http://cs.brown.edu/courses/csci1010/index.html

Textbookandcoursematerials• Textbook:MichaelSipserIntroductiontoTheoryofComputation(2nd ed.recommended;1st ed maybeok)

• Coursewebsite:https://cs.brown.edu/courses/csci1010/– UsethecourseSyllabusforreference– LectureSlidesavailableonthewebsiteafterclass– Homeworkassignmentspublishedonthewebsite– Solutionswillbeposted2daysafterthedeadlines

• Lectures:– ZoomlinkstolikelecturesandvideosonCanvas(dodistribute

them)

• Piazza:– GreatwaytointeractwithTAsandpeers!Useitalot!

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HomeworkandMidterms

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• Homework(ormidterm)everyweek:Postingdatesandhand-backdatespostedonthewebsite.Oneweektotaltimeforeachhomework.– Workinagroup,butthinkandtrytosolveeachproblemyourself!– Listyoucollaboratorsonyoursubmission– Alwayswriteyourownsolutions– Donot“distributetheproblemswithinthegroup”– Usepiazza

• 2take-homeMidtermseachcoveringthefirst(resp.,second)partoftheclass.– Oneweekforeachmidtermassignment– Nocollaborationisallowed!– TAswillonlyanswerquestionsregardingclarificationsontheassignments– QuestionsshouldbeaskedprivatelyonPiazza

HomeworkandMidterms

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• AssignmentssubmittedusingGradescope.• Berigorousbutnotexcessivelyverbose• Simplelatepolicy:

– A “budget”of4totaldays,– Youcanuseupto2daysforanassignments– Requestsforextensiondueatleast24hoursbeforedeadlineusing

Googleformavailableonline(https://docs.google.com/forms/d/e/1FAIpQLScBRrM_B4HjJMp5m-rq9Ud9vRa9JU-y5i8ylMgOma3SudWYLw/viewform)

• Atthebeginningofthecourse,allstudentsarerequiredtosubscribetothecollaborationpolicy(https://forms.gle/1KzmBgLqGxWZ4gv86)

Howtodowell• Youmustunderstandtheconceptswell!

– Ifyoudonot,thereisalmostnochanceofsuccess=(• Ifyoudounderstandtheconcepts,thereisverylittleelsetolearn!

– Youcandoreallywell=D• Youmustdoproblems.There’snoreplacementforthis.

– Discussproblemsandsolutionsingroupbutalwayswriteyourownsolutions• Attendinglecturesishighlyadvised!• Don’tpostponelearning;youwillnotbeableto“makeup”later.

– Startthehomeworkearlyon!• Topicsgethardquickly.• Cometoofficehours!• Participateinclass!!!

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