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

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as they are in UCS and BFS. For completeness, we need repeated state checking as the example shows. This enforces complete enumeration of state space (provided that it is finite), and thus gives us completeness. Note that nothing prevents from all nodes nodes being searched in worst case; e.g. if the heuristic function gives us the same (low) estimate on all nodes except where the heuristic misestimates the distance to be high. So in the worst case, greedy search is even worse than BFS, where d (depth of first solution) replaces m. The search procedure cannot be optional, since actual cost of solution is not considered. For both, completeness and optimality, therefore, it is necessary to take the actual cost of partial solutions, i.e. the path cost, into account. This way, paths that are known to be expensive are avoided. A ∗ search Idea: Avoid expanding paths that are already expensive (make use of actual cost) The simplest way to combine heuristic and path cost is to simply add them. Definition 603 The evaluation function for A ∗ -search is given by f(n) = g(n) + h(n), where g(n) is the path cost for n and h(n) is the estimated cost to goal from n. Thus f(n) is the estimated total cost of path through n to goal Definition 604 Best-First-Search with evaluation function g + h is called astarSearch search. c○: Michael Kohlhase 478 This works, provided that h does not overestimate the true cost to achieve the goal. In other words, h must be optimistic wrt. the real cost h ∗ . If we are too pessimistic, then non-optimal solutions have a chance. A ∗ search: Admissibility Definition 605 (Admissibility of heuristic) h(n) is called admissible if (0 ≤ h(n) ≤ h ∗ (n)) for all nodes n, where h ∗ (n) is the true cost from n to goal. (In particular: h(G) = 0 for goal G) Example 606 Straight-line distance never overestimates the actual road distance (triangle inequality) Thus hSLD(n) is admissible. A ∗ Search: Admissibility c○: Michael Kohlhase 479 Theorem 607 A ∗ search with admissible heuristic is optimal Proof: We show that sub-optimal nodes are never selected by A ∗ P.1 Suppose a suboptimal goal G has been generated then we are in the following situation: n start O G 255
P.2 Let n be an unexpanded node on a path to an optimal goal O, then f(G) = g(G) since h(G) = 0 g(G) > g(O) since G suboptimal g(O) = g(n) + h ∗ (n) n on optimal path g(n) + h ∗ (n) ≥ g(n) + h(n) since h is admissible g(n) + h(n) = f(n) P.3 Thus, f(G) > f(n) and astarSearch never selects G for expansion. A ∗ Search Example A ∗ Search Example Sibiu 393=140+253 A ∗ Search Example Arad 646=280+366 Fagaras 415=239+176 A ∗ Search Example Arad 646=280+366 Fagaras 415=239+176 c○: Michael Kohlhase 480 Arad 366=0+366 c○: Michael Kohlhase 481 Arad Timisoara 447=118+329 c○: Michael Kohlhase 482 Arad Sibiu Timisoara Oradea 671=291+380 R. Vilcea 413=220+193 447=118+329 c○: Michael Kohlhase 483 Arad Sibiu Timisoara Oradea 671=291+380 Craiova 526=366+160 R. Vilcea 256 Pitesti 417=317+100 447=118+329 Sibiu 553=300+253 Zerind 449=75+374 Zerind 449=75+374 Zerind 449=75+374
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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
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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
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
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: Sibiu 253 Greedy Search: Romania Ar
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
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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