Slide 1How convincing is our Halting Problem proof?DrSchemeSolutionsModeling ComputationWhat is a model?How should we model a Computer?“Computers” before WWIIModeling ComputersModeling InputTuring’s “Computer”Simplest InputModeling OutputModeling Processing (Brains)Modeling ProcessingSlide 16Finite State MachinesHmmm…maybe we don’t need those infinite tapes after all?How many states do we need?Finite State MachineTuring’s ExplanationFSM + Infinite TapeTuring MachineMatching ParenthesesSlide 25Slide 26Slide 27Exam 2David Evanshttp://www.cs.virginia.edu/evansCS150: Computer ScienceUniversity of VirginiaComputer ScienceLecture 36: Lecture 36: Modeling Modeling ComputingComputing2Lecture 36: Modeling ComputingHow convincing is our Halting Problem proof? (define (contradict-halts x) (if (halts? contradict-halts) (loop-forever) #t))contradicts-halts cannot exist. Everything we used to make it except halts? does exist, therefore halts? cannot exist.This “proof” assumes Scheme exists and is consistent!3Lecture 36: Modeling ComputingDrSchemeIs DrScheme a proof that Scheme exists?(define (make-huge n) (if (= n 0) null (cons (make-huge (- n 1)) (make-huge (- n 1)))))(make-huge 10000)Scheme/Charme/Python/etc. all fail to evaluate some program!4Lecture 36: Modeling ComputingSolutions•Option 1: Prove “Scheme” does exist–Show that we could implement all the evaluation rules (if we had “Python”, our Charme interpreter would be a good start, but we don’t have “Python”)•Option 2: Find a simpler computing model–Define it precisely–Show that “contradict-halts” can be defined in this model5Lecture 36: Modeling ComputingModeling Computation•For a more convincing proof, we need a more precise (but simple) model of what a computer can do•Another reason we need a model:Does complexity really make sense without this? (how do we know what a “step” is? are they the same for all computers?)6Lecture 36: Modeling ComputingWhat is a model?7Lecture 36: Modeling ComputingHow should we model a Computer?Apollo Guidance Computer (1969)Colossus (1944)IBM 5100 (1975)Cray-1 (1976)Turing invented the model we’ll use today in 1936. What “computer” was he modeling?8Lecture 36: Modeling Computing“Computers” before WWII9Lecture 36: Modeling ComputingModeling Computers•Input–Without it, we can’t describe a problem•Output–Without it, we can’t get an answer•Processing–Need some way of getting from the input to the output•Memory–Need to keep track of what we are doing10Lecture 36: Modeling ComputingModeling InputEngelbart’s mouse and keypadPunch CardsAltair BASIC Paper Tape, 197611Lecture 36: Modeling ComputingTuring’s “Computer”“Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book.” Alan Turing, On computable numbers, with an application to the Entscheidungsproblem, 193612Lecture 36: Modeling ComputingSimplest Input•Non-interactive: like punch cards and paper tape•One-dimensional: just a single tape of values, pointer to one square on tape0 0 1 1 0 0 1 0 0 0How long should the tape be?Infinitely long! We are modeling a computer, not building one. Our model should not have silly practical limitations (like a real computer does).13Lecture 36: Modeling ComputingModeling Output•Blinking lights are cool, but hard to model•Output is what is written on the tape at the end of a computationConnection Machine CM-5, 199314Lecture 36: Modeling ComputingModeling Processing (Brains)•Rules for steps•Remember a little“For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.”Alan Turing15Lecture 36: Modeling ComputingModeling Processing•Evaluation Rules–Given an input on our tape, how do we evaluate to produce the output•What do we need:–Read what is on the tape at the current square–Move the tape one square in either direction–Write into the current square0 0 1 1 0 0 1 0 0 0Is that enough to model a computer?16Lecture 36: Modeling ComputingModeling Processing•Read, write and move is not enough•We also need to keep track of what we are doing:–How do we know whether to read, write or move at each step?–How do we know when we’re done?•What do we need for this?17Lecture 36: Modeling ComputingFinite State Machines1Start2HALT101#018Lecture 36: Modeling ComputingHmmm…maybe we don’t need those infinite tapes after all?1Start2HALT(not a paren)#not a paren)ERRORWhat if thenext input symbolis ( in state 2?19Lecture 36: Modeling ComputingHow many states do we need?1Start2HALT(not a paren)#not a paren)ERROR3not a paren()4()not a paren(20Lecture 36: Modeling ComputingFinite State Machine•There are lots of things we can’t compute with only a finite number of states•Solutions:–Infinite State Machine•Hard to describe and draw–Add an infinite tape to the Finite State Machine21Lecture 36: Modeling ComputingTuring’s Explanation“We have said that the computable numbers are those whose decimals are calculable by finite means. ... For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.”22Lecture 36: Modeling ComputingFSM + Infinite Tape•Start: –FSM in Start State–Input on Infinite Tape–Pointer to start of input•Step:–Read one input symbol from tape–Write symbol on tape, and move L or R one square–Follow transition rule from current state•Finish:–Transition to halt state23Lecture 36: Modeling ComputingTuring Machine1Start2Input: #Write: #Move: # 1 0 1 1 0 1 1......1 0 1 1 0 1 1 1 #Input: 1Write: 0Move: Input: 1Write: 1Move: Input: 0Write: 0Move: 3Input: 0Write: #Move: 24Lecture 36: Modeling ComputingMatching Parentheses•Find the leftmost )–If you don’t find one, the parentheses match, write a 1 at the tape head and halt.•Replace it with an X•Look left for the first (–If you find it, replace it with an X (they matched)–If you don’t find it, the parentheses didn’t match – end write a 0 at the tape head and halt25Lecture 36: Modeling ComputingMatching Parentheses1: look for )StartHALTInput: )Write: XMove: L), X, L2: look for (X, X, RX, X, L(, X, R#, 0, ##, 1, #(, (, RWill this report thecorrect result for (()?26Lecture 36: Modeling ComputingMatching Parentheses1StartHALT), X, L2: look for ((, (, R(, X, R#, 0, ##,