DiscussionSelf-ReproductionFallacy Resolved: “Blueprint” can involve some computation; need not be an exact copy!High-level description of program that self-reproducesNext several lectures: Computational HardwareLogical ReasoningPropositional Logic: HistoryExampleEd goes to the party if Dan goes or Stella goesBoolean expressionsTruth tableBoolean “algebra”3 equivalent ways of representationBen RevisitedDiscussion Did the “Theory of Everything” article make you look at something in a new way? What is the Church-Turing thesis and how convincing is it to you?Self-ReproductionFallacious argument for impossibility:BlueprintBlueprintBlueprintM.C. EscherPrint GalleryFallacy Resolved: “Blueprint” can involve some computation; need not be an exact copy!Print this sentence twice, the second time in quotes. “Print this sentence twice, the second time in quotes.”High-level description of program that self-reproducesPrint 0Print 1...Print 0. . . . . . . . . . . .. . . . . .. . . . . .Prints binary code of BTakes binary string on tape, and in its place prints (in English) the sequence of statements that produce it, followed by the translation of the binary string intoEnglish.ABNext several lectures: Computational Hardware Boolean logic and Boolean circuits Sequential circuits (circuits with memory) Clocked circuits and Finite State Machines CPUs Operating System Networks, InternetLogical ReasoningBen only rides to class if he overslept, but even then if it is raining he’ll walk and show up late to class (he really hates to bike in the rain). But if there’s an exam that day he’ll bike if he overslept, even in the rain.It is raining today, Ben overslept, and there’s an exam. Will Ben bike today?“Propositional logic.”Propositional Logic: History Aristotle – Law of excluded middle and Law of contradiction. Stoic Philosophers (3rdcentury BC) – Basic inference rules (modus ponens etc.) Some work by medieval philosophers De Morgan and Boole (19thcentury): symbolic logic – “automated”, “mechanical” C. Shannon (1930s) – proposal to use digital hardwareExampleEd goes to the party if Dan does not and Stella does.Associate Boolean variables with 3 eventsE: Ed goes to partyD: Dan goes to partyS: Stella goes to partyEach is either TRUE or FALSEE = S AND (NOT D)Alternatively, E = S AND DEd goes to the party if Dan goes or Stella goesE = D OR SMeans E is TRUE if one or both of D and S are TRUENote:Different from everyday meaning of OR!Example: You can eat an orange or an appleORBoolean expressionsComposed of boolean variables, AND, OR, and NOTExamples:D AND ( P OR (NOT Q))C OR D OR ETruth tableLists the truth value of the boolean expression for allcombinations of values for the variables.Boolean Expression E = S AND D001011110000ESDTruth tableBoolean “algebra”A AND B written as A · BA OR B written as A + B0 + 0=01 + 0 =11 + 1=10 · 0 =00 · 1 =01·1 = 1Funny arithmeticSee assigned reading. (More next time)3 equivalent ways of representationBoolean Expression E = S AND DTruth table – Gives value of E for every possible assignment to D, S.TRUE=1; FALSE= 0.001011110000ESDBoolean CircuitESDBen RevisitedBen only rides to class if he overslept, but even then if it is raining he’ll walk and show up late to class (he really hates to bike in the rain). But if there’s an exam that day he’ll bike if he overslept, even in the rain.B: Ben BikesR: It is rainingE: There is an exam todayO: Ben oversleptGive boolean expression for B in terms of R, E and
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