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

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etter: [suck, right, if dirt then suck] (decide whether in 6 or 8 using local sensing) c○: Michael Kohlhase 408 In the contingency version of P a solution is the following: [suck({5, 7}), right → ({6, 8}), suck → ({6, 8})], [suck({5, 7})], etc. Of course, local sensing can help: narrow {6, 8} to {6} or {8}, if we are in the first, then suck. Single-state problem formulation Defined by the following four items 1. Initial state: (e.g. Arad) 2. Successor function S: (e.g. S(Arad) = {〈goZer, Zerind〉, 〈goSib, Sibiu〉, . . .}) 3. Goal test: (e.g. x = Bucharest (explicit test) noDirt(x) (implicit test) 4. Path cost (optional): (e.g. sum of distances, number of operators executed, etc.) Solution: A sequence of operators leading from the initial state to a goal state c○: Michael Kohlhase 409 “Path cost”: There may be more than one solution and we might want to have the “best” one in a certain sense. Selecting a state space Abstraction: Real world is absurdly complex State space must be abstracted for problem solving (Abstract) state: Set of real states (Abstract) operator: Complex combination of real actions Example: Arad → Zerind represents complex set of possible routes (Abstract) solution: Set of real paths that are solutions in the real world c○: Michael Kohlhase 410 “State”: e.g., we don’t care about tourist attractions found in the cities along the way. But this is problem dependent. In a different problem it may well be appropriate to include such information in the notion of state. “Realizability”: one could also say that the abstraction must be sound wrt. reality. Example: The 8-puzzle 235 )
States integer locations of tiles Actions left, right, up, down Goal test = goal state? Path cost 1 per move c○: Michael Kohlhase 411 How many states are there? N factorial, so it is not obvious that the problem is in NP. One needs to show, for example, that polynomial length solutions do always exist. Can be done by combinatorial arguments on state space graph (really ?). Example: Vacuum-cleaner Example: Robotic assembly 4.1.2 Search States integer dirt and robot locations Actions left, right, suck, noOp Goal test notdirty? Path cost 1 per operation (0 for noOp) c○: Michael Kohlhase 412 States real-valued coordinates of robot joint angles and parts of the object to be assembled Actions continuous motions of robot joints Goal test assembly complete? Path cost time to execute Tree search algorithms c○: Michael Kohlhase 413 Simulated exploration of state space in a search tree by generating successors of alreadyexplored states (Offline Algorithm) 236
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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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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
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
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: Single-state Problem: Start in 5
Page 245 and 246: Implementation: States vs. nodes A
Page 247 and 248: Breadth-First Search c○: Michael
Page 249 and 250: Breadth-First Search: Romania c○:
Page 251 and 252: Note: Equivalent to breadth-first s
Page 253 and 254: A B C D E F G H I J K L M N O Depth
Page 255 and 256: A B C D E F G H I J K L M N O Depth
Page 257 and 258: Iterative deepening search Depth-l
Page 259 and 260: Breadth-first search Iterative deep
Page 261 and 262: Sibiu 253 Greedy Search: Romania Ar
Page 263 and 264: P.2 Let n be an unexpanded node on
Page 265 and 266: A ∗ search: Properties Complete Y
Page 267 and 268: Definition 618 (n-queens problem) P
Page 269 and 270: escape certain odd phenomena occurr
Page 271 and 272: GAs = evolution: e.g., real genes e
Page 273 and 274: Definition 632 A query is a list of
Page 275 and 276: (autoload ’run-prolog "prolog" "S
Page 277 and 278: Example 638 Computing the the n th
Page 279 and 280: Deduction: knowledge extension Abd
Page 281 and 282: [Koh06] Michael Kohlhase. OMDoc - A
Page 283 and 284: astarSearch search, 255 asymmetric-
Page 285 and 286: infinite, 32 counter program, 152,
Page 287 and 288: functional variable, 170 gate, 122
Page 289 and 290: media access control address, 194 M
Page 291 and 292: 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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