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

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The Main Theorem for TCN again P.3 Given two (n + 1)-bit two’s complement numbers a and b. The above theorem tells us that the result s of an (n + 1)-bit adder is the proper sum in two’s complement representation iff the last two carries are identical. If not, a and b were too large or too small. In the case that s is larger than 2 n − 1, we say that an overflow occurred.In the opposite error case of s being smaller than −2 n , we say that an underflow occurred. c○: Michael Kohlhase 262 3.2.3 Towards an Algorithmic-Logic Unit The most important application of the main TCN theorem is that we can build a combinational circuit that can add and subtract (depending on a control bit). This is actually the first instance of a concrete programmable computation device we have seen up to date (we interpret the control bit as a program, which changes the behavior of the device). The fact that this is so simple, it only runs two programs should not deter us; we will come up with more complex things later. Building an Add/Subtract Unit Idea: Build a Combinational Circuit that can add and subtract (sub = 1 subtract) If sub = 0, then the circuit acts like an adder (a ⊕ 0 = a) If sub = 1, let S := 〈〈a〉〉 2s n + 〈〈bn−1, . . . , b0〉〉 2s n + 1 (a ⊕ 0 = 1 − a) For s ∈ R 2s n the TCN main theorem and the TCN structure theorem together guarantee s = 〈〈a〉〉 2s n + 〈〈bn−1, . . . , b0〉〉 2s n + 1 = 〈〈a〉〉 2s n − 〈〈b〉〉 2s n − 1 + 1 145 a b b n−1 0 n A s n n+1 n sub
Summary: We have built a combinational circuit that can perform 2 arithmetic operations depending on a control bit. Idea: Extend this to a arithmetic logic unit (ALU) with more operations (+, -, *, /, n-AND, n-OR,. . . ) c○: Michael Kohlhase 263 In fact extended variants of the very simple Add/Subtract unit are at the heart of any computer. These are called arithmetic logic units. 146
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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
Page 101 and 102: which we can always do, since we ha
Page 103 and 104: e quite difficult to establish in g
Page 105 and 106: H 0 axioms are valid Lemma 316 The
Page 107 and 108: c○: Michael Kohlhase 188 The enta
Page 109 and 110: The deduction theorem and the entai
Page 111 and 112: Inference with local hypotheses [A
Page 113 and 114: 2.7 Machine-Oriented Calculi Now we
Page 115 and 116: Thus the tableau procedure can be u
Page 117 and 118: A ∨ BT AT BT A ⇒ BT AF BT
Page 119 and 120: Proof: P.1 It is easy to see tahat
Page 121 and 122: (P ⇒ Q ⇒ R) ⇒ (P ⇒ Q) ⇒ P
Page 123 and 124: ⎧ ⎪⎨ We represented the maze
Page 125 and 126: defined a directed graph to be a se
Page 127 and 128: Paths in Graphs Definition 373 Giv
Page 129 and 130: This allows us to view Boolean expr
Page 131 and 132: Computing with Combinational Circui
Page 133 and 134: Corollary 399 A fully balanced tree
Page 135 and 136: X 1 X 2 X 3 X n = if L i =X i if L
Page 137 and 138: 3.2 Arithmetic Circuits 3.2.1 Basic
Page 139 and 140: S S ψ ψ −1 fS = ψ −1 ◦ fT
Page 141 and 142: The Full Adder Definition 415 The
Page 143 and 144: first summand 3 4 7 9 8 3 4 7 9 2 s
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 an + bn + (icn(a, b, c)) = 〈
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 and 194: State Machine: 1 1,R 1 1,R 1 1,L 1
Page 195 and 196: The coded description acts as a pro
Page 197 and 198: the turing function uses will_halt
Page 199 and 200: Terabyte (T B) 1,000,000,000,000 by
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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
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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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