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

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The capacity of RAM chips doubles approximately every year. c○: Michael Kohlhase 271 One aspect of this is particularly interesting – and user-visible in the sense that the division of storage addresses is divided into a high- and low part of the address. So we we will briefly discuss it here. Layout of Memory Chips To take advantage of the two-dimensional nature of the chip, storage elements are arranged on a square grid. (columns and rows of storage elements) For example, a 1 Megabit RAM chip has of 1024 rows and 1024 columns. identify storage element by its row and column “coordinates”. (AND them for addressing) Hence, to select a particular storage location the address information must be translated into row and column specification. The address information is divided into two halves; the top half is used to select the row and the bottom half is used to select the column. c○: Michael Kohlhase 272 151
3.4 Computing Devices and Programming Languages The main focus of this sectionis a discussion of the languages that can be used to program register machines: simple computational devices we can realize by combining algorithmic/logic circuits with memory. We start out with a simple assembler language which is largely given by the ALU employed and build up towards higher-level, more structured programming languages. We build up language expressivity in levels, first defining a simple imperative programming language SW with arithmetic expressions, and block-structured control. One way to make this language run on our register machine would be via a compiler that transforms SW programs into assembler programs. As this would be very complex, we will go a different route: we first build an intermediate, stack-based programming language L(VM) and write a L(VM)-interpreter in ASM, which acts as a stack-based virtual machine, into which we can compile SW programs. The next level of complexity is to add (static) procedure calls to SW, for which we have to extend the L(VM) language and the interpreter with stack frame functionality. Armed with this, we can build a simple functional programming language µML and a full compiler into L(VM) for it. We conclude this sectionby an investigation into the fundamental properties and limitations of computation, discussing Turing machines, universal machines, and the halting problem. Acknowledgement: Some of the material in this sectionis inspired by and adapted from Gert Smolka excellent introduction to Computer Science based on SML [Smo11]. 3.4.1 How to Build and Program a Computer (in Principle) In this subsection, we will combine the arithmetic/logical units from Section 3.1 with the storage elements (RAM) from Subsection 3.3.1 to a fully programmable device: the register machine. The “von Neumann” architecture for computing we use in the register machine, is the prevalent architecture for general-purpose computing devices, such as personal computers nowadays. This architecture is widely attribute to the mathematician John von Neumann because of [vN45], but is already present in Konrad Zuse’s 1936 patent application [Zus36]. REMA, a simple Register Machine Take an n-bit arithmetic logic unit (ALU) add registers: few (named) n-bit memory cells near the ALU program counter (P C) (points to current command in program store) accumulator (ACC) (the a input and output of the ALU) add RAM: lots of random access memory (elsewhere) program store: 2n-bit memory cells (addressed by P : N → B 2n ) data store: n-bit memory cells (words addressed by P : N → B n ) add a memory management unit(MMU) (move values between RAM and registers) program it in assembler language (lowest level of programming) c○: Michael Kohlhase 273 We have three kinds of memory areas in the REMA register machine: The registers (our architecture has two, which is the minimal number, real architectures have more for convenience) are just simple n-bit memory cells. The programstore is a sequence of up to 2 n memory 2n-bit memory cells, which can be accessed (written to and queried) randomly i.e. by referencing their position in the sequence; we do not have 152
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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
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 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: Example 443 (Address decoder logic
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
Page 201 and 202: Layers in TCP/IP: TCP/IP uses encap
Page 203 and 204: name comment 4 version IPv4 or IPv6
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Page 207 and 208: 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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