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

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2.6.3 Logical Systems and Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.6.4 Proof Theory for the Hilbert Calculus . . . . . . . . . . . . . . . . . . . . . 97 2.6.5 A Calculus for Mathtalk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.7 Machine-Oriented Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7.1 Calculi for Automated Theorem Proving: Analytical Tableaux . . . . . . . 106 Analytical Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Practical Enhancements for Tableaux . . . . . . . . . . . . . . . . . . . . . 109 Soundness and Termination of Tableaux . . . . . . . . . . . . . . . . . . . . 111 2.7.2 Resolution for Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . 112 3 How to build <strong>Computer</strong>s and the Internet (in principle) 115 3.1 Combinational Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.1.1 Graphs and Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.1.2 Introduction to Combinatorial Circuits . . . . . . . . . . . . . . . . . . . . . 122 3.1.3 Realizing Complex Gates Efficiently . . . . . . . . . . . . . . . . . . . . . . 124 Balanced Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Realizing n-ary Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2 Arithmetic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2.1 Basic Arithmetics with Combinational Circuits . . . . . . . . . . . . . . . . 130 Positional Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Adders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.2.2 Arithmetics for Two’s Complement Numbers . . . . . . . . . . . . . . . . . 139 3.2.3 Towards an Algorithmic-Logic Unit . . . . . . . . . . . . . . . . . . . . . . . 145 3.3 Sequential Logic Circuits and Memory Elements . . . . . . . . . . . . . . . . . . . 147 3.3.1 Sequential Logic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.3.2 Random Access Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.4 Computing Devices and Programming Languages . . . . . . . . . . . . . . . . . . . 152 3.4.1 How to Build and Program a <strong>Computer</strong> (in Principle) . . . . . . . . . . . . 152 3.4.2 A Stack-based Virtual Machine . . . . . . . . . . . . . . . . . . . . . . . . . 157 A Stack-based Programming Language . . . . . . . . . . . . . . . . . . . . . 158 Building a Virtual Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.4.3 A Simple Imperative Language . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.4.4 Basic Functional Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A Virtual Machine with Procedures . . . . . . . . . . . . . . . . . . . . . . 170 Compiling Basic Functional Programs . . . . . . . . . . . . . . . . . . . . . 180 3.4.5 Turing Machines: A theoretical View on Computation . . . . . . . . . . . . 183 3.5 The Information and Software Architecture of the Internet and World Wide Web . 190 3.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 3.5.2 Internet Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3.5.3 Basic Concepts of the World Wide Web . . . . . . . . . . . . . . . . . . . . 200 Addressing on the World Wide Web . . . . . . . . . . . . . . . . . . . . . . 201 Running the World Wide Web . . . . . . . . . . . . . . . . . . . . . . . . . 202 Multimedia Documents on the World Wide Web . . . . . . . . . . . . . . . 205 3.5.4 Web Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 3.5.5 Introduction to Web Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3.5.6 Security by Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.5.7 An Overview over XML Technologies . . . . . . . . . . . . . . . . . . . . . . 223 3.5.8 The Semantic Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 v
4 Search and Declarative Computation 231 4.1 Problem Solving and Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.1.1 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.1.2 Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 4.1.3 Uninformed Search Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.1.4 Informed Search Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 4.1.5 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 4.2 Logic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 4.2.1 Introduction to Logic Programming and PROLOG . . . . . . . . . . . . . . 264 4.2.2 Programming as Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4.2.3 Logic Programming as Resolution Theorem Proving . . . . . . . . . . . . . 271 vi
Page 1 and 2: General Computer Science 320201 Gen
Page 3 and 4: Java programmer” on the practical
Page 5: Contents 1 Preface i 2 Representati
Page 9 and 10: Fundamental Algorithms and Data str
Page 11 and 12: To earn an audit you have to take t
Page 13 and 14: i.e. to function as a member of the
Page 15 and 16: a factor two in speed. This ability
Page 17 and 18: “Applets, Not Craplets tm ” (-
Page 19 and 20: c○: Michael Kohlhase 17 Can be re
Page 21 and 22: Example 10 (Kruskal’s algorithm,
Page 23 and 24: n 100n µs 7n 2 µs 2 n µs 1 100
Page 25 and 26: 2.2 Elementary Discrete Math 2.2.1
Page 27 and 28: Axiom 19 (P 1) “ ” (aka. “zer
Page 29 and 30: Theorem 36 is a very useful fact to
Page 31 and 32: Definition 43 The unary product ope
Page 33 and 34: y stating element-hood (a ∈ S) or
Page 35 and 36: Idea: We need a notion of “counti
Page 37 and 38: Example 64 On sets of persons, the
Page 39 and 40: Definition 82 We say that a set A i
Page 41 and 42: clean enough to learn important con
Page 43 and 44: Definition 90 anonymous variables (
Page 45 and 46: c○: Michael Kohlhase 66 Defining
Page 47 and 48: - fun both_plus (x:int,y:int) = fn
Page 49 and 50: +(〈n, o〉) = n (base) and +(〈m
Page 51 and 52: the axiom says that any object that
Page 53 and 54: epresent them as a data type, where
Page 55 and 56: 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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first summand 3 4 7 9 8 3 4 7 9 2 s
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c○: Michael Kohlhase 249 The anal
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Problems of Sign-Bit Systems Gener
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generate the n-bit binary number re
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and an + bn + (icn(a, b, c)) = 〈
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Summary: We have built a combinatio
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To understand the operation of the
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Example 443 (Address decoder logic
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3.4 Computing Devices and Programmi
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Our notion of time is in this const
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instructions LOADIN 1 and LOADIN 2
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Definition 457 An ASM program VM is
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instruction effect VPC peek i push
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c○: Michael Kohlhase 289 With the
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Imperative Stack Operations: peek l
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A SW program (see the next slide fo
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arguments to arithmetic operations
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µML, a very simple Functional Prog
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call pushes the return address (of
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[proc 2 26, con 0, arg 2, leq, cjp
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[proc 2 26, con 0, arg 2, leq, cjp
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eturn takes the current frame from
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label instruction effect comment
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Compiling µML Expressions (Continu
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c○: Michael Kohlhase 325 We want
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State Machine: 1 1,R 1 1,R 1 1,L 1
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The coded description acts as a pro
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the turing function uses will_halt
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Terabyte (T B) 1,000,000,000,000 by
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Layers in TCP/IP: TCP/IP uses encap
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name comment 4 version IPv4 or IPv6
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Domain names must be registered to
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That was almost all, but we close t
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c○: Michael Kohlhase 353 Note tha
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Definition 529 HTTP is used by a cl
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structure html,head, body metadata
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Server-Side Scripting: Programming
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c○: Michael Kohlhase 367 Indeed,
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presentation tools), but can also i
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1. reads web page 2. reports it hom
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Definition 570 Let A be a web page
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can combine both to falsify communi
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Candidates for one-way/trapdoor fun
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on a UNIX system, you can create a
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conceptually: transfer of directed
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Definition 591 The XML path languag
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Resources: Globally Identified by U
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R⌉}〉∫⊔⌉∇⌉⌈√⊣∇
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Problem solving Problem: Find algo
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Single-state Problem: Start in 5
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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
Page 293 and 294:
function, 30 spanning tree, 13 spro
Page 295:
name, 70 variable functional, 170 v
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General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

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