General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

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Post production systems ([Post]) Turing machines ([Turing]) Random-access machine Conjecture 482 ([Church/Turing]) (unprovable, but accepted) Anything that can be computed at all, can be computed by a Turing Machine Definition 483 We will call a computational system Turing complete, iff it can compute what a Turing machine can. c○: Michael Kohlhase 331 Note that the Church/Turing hypothesis is a very strong assumption, but it has been born out by experience so far and is generally accepted among computer scientists. The Church/Turing hypothesis is strengthened by another concept that Alan Turing introduced in [Tur36]: the universal turing machine – a Turing machine that can simulate arbitrary Turing machine on arbitrary input. The universal Turing machine achieves this by reading both the Turing machine specification T as well as the I input from its tape and simulates T on I, constructing the output that T would have given on I on the tape. The construction itself is quite tricky (and lengthy), so we restrict ourselves to the concepts involved. Some researchers consider the universal Turing machine idea to be the origin of von Neumann’s architecture of a stored-program computer, which we explored in Subsection 3.4.0. Universal Turing machines Note: A Turing machine computes a fixed partial string function. In that sense it behaves like a computer with a fixed program. Idea: we can encode the action table of any Turing machine in a string. try to construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and then computes the tape that the encoded Turing machine would have computed. Theorem 484 Such a Turing machine is indeed possible (e.g. with 2 states, 18 symbols) Definition 485 Call it a universal Turing machine (UTM). (it can simulate any TM) UTM accepts a coded description of a Turing machine and simulates the behavior of the machine on the input data. 187
The coded description acts as a program that the UTM executes, the UTM’s own internal program is fixed. The existence of the UTM is what makes computers fundamentally different from other machines such as telephones, CD players, VCRs, refrigerators, toaster-ovens, or cars. c○: Michael Kohlhase 332 Indeed the existence of UTMs is one of the distinguishing feature of computing. Whereas other tools are single purpose (or multi-purpose at best; e.g. in the sense of a Swiss army knife, which integrates multiple tools), computing devices can be configured to assume any behavior simply by supplying a program. This makes them universal tools. Note: that there are very few disciplines that study such universal tools, this makes <strong>Computer</strong> <strong>Science</strong> special. The only other discipline with “universal tools” that comes to mind is Biology, where ribosomes read RNA codes and synthesize arbitrary proteins. But for all we know at the moment, RNA codes is linear and therefore Turing completeness of the RNA code is still hotly debated (I am skeptical). Even in our limited experience from this course, we have seen that we can compile µML to L(VMP) and SW to L(VM) both of which we can interpret in ASM. And we can write an SML simulator of the REMA that closes the circle. So all these languages are equivalent and inter-simulatable. Thus, if we can simulate any of them in Turing machines, then we can simulate any of them. Of course, not all programming languages are inter-simulatable, for instance, if we had forgotten the jump instructions in L(VM), then we could not compile the control structures of SW or µML into L(VM) or L(VMP). So we should read the Church/Turing hypothesis as a statement of equivalence of all non-trivial programming languages. Question: So, if all non-trivial programming languages can compute the same, are there things that none of them can compute? This is what we will have a look at next. Is there anything that cannot be computed by a TM Theorem 486 (Halting Problem [Tur36]) No Turing machine can infallibly tell if another Turing machine will get stuck in an infinite loop on some given input. The problem of determining whether a Turing machine will halt on an input is called the halting problem. Proof: Coded description of some TM Input for TM Loop−detector Turing Machine "yes, it will halt" "no, it will not halt" P.1 let’s do the argument with SML instead of a TM assume that there is a loop detector program written in SML 188
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General Computer Science 320201 Gen
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Java programmer” on the practical
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Contents 1 Preface i 2 Representati
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4 Search and Declarative Computatio
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Fundamental Algorithms and Data str
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To earn an audit you have to take t
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i.e. to function as a member of the
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a factor two in speed. This ability
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“Applets, Not Craplets tm ” (-
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c○: Michael Kohlhase 17 Can be re
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Example 10 (Kruskal’s algorithm,
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n 100n µs 7n 2 µs 2 n µs 1 100
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2.2 Elementary Discrete Math 2.2.1
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Axiom 19 (P 1) “ ” (aka. “zer
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Theorem 36 is a very useful fact to
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Definition 43 The unary product ope
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y stating element-hood (a ∈ S) or
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Idea: We need a notion of “counti
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Example 64 On sets of persons, the
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Definition 82 We say that a set A i
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clean enough to learn important con
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Definition 90 anonymous variables (
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c○: Michael Kohlhase 66 Defining
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- fun both_plus (x:int,y:int) = fn
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+(〈n, o〉) = n (base) and +(〈m
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the axiom says that any object that
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epresent them as a data type, where
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Example 127 〈{N}, {[o: N], [s: N
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The central idea here is what we ha
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Substitutions Definition 149 Let A
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Idea: Well-formed parts of construc
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Other programming languages chose a
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generally: fn+1 := fn + fn−1 plus
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input/output: the interesting bit a
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exception NaN; (* Not a Number *) f
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Example 191 If A = {a, b, c}, then
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Example 210 The Morse Code in the t
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The first 32 characters are control
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Idea: Unicode supports multiple enc
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2.5 Boolean Algebra We will now loo
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What a mess! Iϕ((x1 + x2) + (x1
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c○: Michael Kohlhase 140 The defi
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(f ≤a g), iff there is an n0 ∈
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P.1.2.3 then there are ei ∈ Ebool
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2.5.4 The Quine-McCluskey Algorithm
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the disjunctive normal form, and th
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Proof: by contradiction: let p /∈
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So, the minimal polynomial of f is
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2.6 Propositional Logic 2.6.1 Boole
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Idea: Import semantics from Boolean
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which we can always do, since we ha
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e quite difficult to establish in g
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H 0 axioms are valid Lemma 316 The
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c○: Michael Kohlhase 188 The enta
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The deduction theorem and the entai
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Inference with local hypotheses [A
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2.7 Machine-Oriented Calculi Now we
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Thus the tableau procedure can be u
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A ∨ BT AT BT A ⇒ BT AF BT
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Proof: P.1 It is easy to see tahat
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(P ⇒ Q ⇒ R) ⇒ (P ⇒ Q) ⇒ P
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⎧ ⎪⎨ We represented the maze
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defined a directed graph to be a se
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Paths in Graphs Definition 373 Giv
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This allows us to view Boolean expr
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Computing with Combinational Circui
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Corollary 399 A fully balanced tree
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X 1 X 2 X 3 X n = if L i =X i if L
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3.2 Arithmetic Circuits 3.2.1 Basic
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S S ψ ψ −1 fS = ψ −1 ◦ fT
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The Full Adder Definition 415 The
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Page 145 and 146: c○: Michael Kohlhase 249 The anal
Page 147 and 148: Problems of Sign-Bit Systems Gener
Page 149 and 150: generate the n-bit binary number re
Page 151 and 152: and an + bn + (icn(a, b, c)) = 〈
Page 153 and 154: Summary: We have built a combinatio
Page 155 and 156: To understand the operation of the
Page 157 and 158: Example 443 (Address decoder logic
Page 159 and 160: 3.4 Computing Devices and Programmi
Page 161 and 162: Our notion of time is in this const
Page 163 and 164: instructions LOADIN 1 and LOADIN 2
Page 165 and 166: Definition 457 An ASM program VM is
Page 167 and 168: instruction effect VPC peek i push
Page 169 and 170: c○: Michael Kohlhase 289 With the
Page 171 and 172: Imperative Stack Operations: peek l
Page 173 and 174: A SW program (see the next slide fo
Page 175 and 176: arguments to arithmetic operations
Page 177 and 178: µML, a very simple Functional Prog
Page 179 and 180: call pushes the return address (of
Page 181 and 182: [proc 2 26, con 0, arg 2, leq, cjp
Page 183 and 184: [proc 2 26, con 0, arg 2, leq, cjp
Page 185 and 186: eturn takes the current frame from
Page 187 and 188: label instruction effect comment
Page 189 and 190: Compiling µML Expressions (Continu
Page 191 and 192: c○: Michael Kohlhase 325 We want
Page 193: State Machine: 1 1,R 1 1,R 1 1,L 1
Page 197 and 198: the turing function uses will_halt
Page 199 and 200: Terabyte (T B) 1,000,000,000,000 by
Page 201 and 202: Layers in TCP/IP: TCP/IP uses encap
Page 203 and 204: name comment 4 version IPv4 or IPv6
Page 205 and 206: Domain names must be registered to
Page 207 and 208: That was almost all, but we close t
Page 209 and 210: c○: Michael Kohlhase 353 Note tha
Page 211 and 212: Definition 529 HTTP is used by a cl
Page 213 and 214: structure html,head, body metadata
Page 215 and 216: Server-Side Scripting: Programming
Page 217 and 218: c○: Michael Kohlhase 367 Indeed,
Page 219 and 220: presentation tools), but can also i
Page 221 and 222: 1. reads web page 2. reports it hom
Page 223 and 224: Definition 570 Let A be a web page
Page 225 and 226: can combine both to falsify communi
Page 227 and 228: Candidates for one-way/trapdoor fun
Page 229 and 230: on a UNIX system, you can create a
Page 231 and 232: conceptually: transfer of directed
Page 233 and 234: Definition 591 The XML path languag
Page 235 and 236: Resources: Globally Identified by U
Page 237 and 238: R⌉}〉∫⊔⌉∇⌉⌈√⊣∇
Page 239 and 240: Problem solving Problem: Find algo
Page 241 and 242: Single-state Problem: Start in 5
Page 243 and 244: States integer locations of tiles A
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Implementation: States vs. nodes A
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Breadth-First Search c○: Michael
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Breadth-First Search: Romania c○:
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Note: Equivalent to breadth-first s
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A B C D E F G H I J K L M N O Depth
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A B C D E F G H I J K L M N O Depth
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Iterative deepening search Depth-l
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Breadth-first search Iterative deep
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Sibiu 253 Greedy Search: Romania Ar
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P.2 Let n be an unexpanded node on
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A ∗ search: Properties Complete Y
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Definition 618 (n-queens problem) P
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escape certain odd phenomena occurr
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GAs = evolution: e.g., real genes e
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Definition 632 A query is a list of
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(autoload ’run-prolog "prolog" "S
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Example 638 Computing the the n th
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Deduction: knowledge extension Abd
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[Koh06] Michael Kohlhase. OMDoc - A
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astarSearch search, 255 asymmetric-
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infinite, 32 counter program, 152,
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functional variable, 170 gate, 122
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media access control address, 194 M
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procedure abstract, 54 procedure de
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function, 30 spanning tree, 13 spro
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name, 70 variable functional, 170 v
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General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

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