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

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PROLOG: Are there Fallible Greeks? Program: human(sokrates). human(leibniz). greek(sokrates). fallible(X) :- human(X). Example 635 (Query) ?-fallible(X),greek(X). Answer substitution: [sokrates/X] 4.2.2 Programming as Search c○: Michael Kohlhase 508 In this subsection, we want to really use ProLog as a programming language, so let use first get our tools set up. We will now discuss how to use a ProLog interpreter to get to know the language. The SWI ProLog interpreter can be downloaded from http://www.swi-prolog.org/. To start the ProLog interpreter with pl or prolog or swipl from the shell. The SWI manual is available at http: //gollem.science.uva.nl/SWI-Prolog/Manual/ We will introduce working with the interpreter using unary natural numbers as examples: we first add the fact 1 to the knowledge base unat(zero). which asserts that the predicate unat 2 is true on the term zero. <strong>General</strong>ly, we can add a fact to the knowledge base either by writing it into a file (e.g. example.pl) and then “consulting it” by writing one of the following commands into the interpreter: [example] consult(’example.pl’). or by directly typing assert(unat(zero)). into the ProLog interpreter. Next tell ProLog about the following rule assert(unat(suc(X)) :- unat(X)). which gives the ProLog runtime an initial (infinite) knowledge base, which can be queried by ?- unat(suc(suc(zero))). Yes Running ProLog in an emacs window is incredibly nicer than at the command line, because you can see the whole history of what you have done. Its better for debugging too. If you’ve never used emacs before, it still might be nicer, since its pretty easy to get used to the little bit of emacs that you need. (Just type “emacs \&” at the UNIX command line to run it; if you are on a remote terminal like putty, you can use “emacs -nw”.). If you don’t already have a file in your home directory called “.emacs” (note the dot at the front), create one and put the following lines in it. Otherwise add the following to your existing .emacs file: 1 for “unary natural numbers”; we cannot use the predicate nat and the constructor functions here, since their meaning is predefined in ProLog 2 for “unary natural numbers”. 267
(autoload ’run-prolog "prolog" "Start a Prolog sub-process." t) (autoload ’prolog-mode "prolog" "Major mode for editing Prolog programs." t) (setq prolog-program-name "swipl") ; or whatever the prolog executable name is (add-to-list ’auto-mode-alist ’("\\pl$" . prolog-mode)) The file prolog.el, which provides prolog-mode should already be installed on your machine, otherwise download it at http://turing.ubishops.ca/home/bruda/emacs-prolog/ Now, once you’re in emacs, you will need to figure out what your “meta” key is. Usually its the alt key. (Type “control” key together with “h” to get help on using emacs). So you’ll need a “meta-X” command, then type “run-prolog”. In other words, type the meta key, type “x”, then there will be a little window at the bottom of your emacs window with “M-x”, where you type run-prolog 3 . This will start up the SWI ProLog interpreter, . . . et voilà! The best thing is you can have two windows “within” your emacs window, one where you’re editing your program and one where you’re running ProLog. This makes debugging easier. Depth-First Search with Backtracking So far, all the examples led to direct success or to failure. (simpl. KB) Search Procedure: top-down, left-right depth-first search Work on the queries in left-right order. match first query with the head literals of the clauses in the program in top-down order. if there are no matches, fail and backtrack to the (chronologically) last point. otherwise backchain on the first match , keep the other matches in mind for backtracking. (backtracking points) c○: Michael Kohlhase 509 Note: We have seen before that depth-first search has the problem that it can go into loops. And in fact this is a necessary feature and not a bug for a programming language: we need to be able to write non-terminating programs, since the language would not be Turing-complete otherwise. The argument can be sketched as follows: we have seen that for Turing machines the halting problem is undecidable. So if all ProLog programs were terminating, then ProLog would be weaker than Turing machines and thus not Turing complete. Backtracking by Example has_wheels(mytricycle,3). has_wheels(myrollerblade,3). has_wheels(mybmw,4). has_motor(mybmw). car(X):-has_wheels(X,3),has_motor(X). % cars sometimes have 3 wheels car(X):-has_wheels(X,4),has_motor(X). ?- car(Y). ?- has_wheels(Y,3),has_motor(Y). % backtrack point 1 Y = mytricycle % backtrack point 2 ?- has_motor(mytricycle). FAIL % fails, backtrack to 2 Y = myrollerblade % backtrack point 2 ?- has_motor(myrollerblade). FAIL % fails, backtrack to 1 ?- has_wheels(Y,4),has_motor(Y). Y = mybmw ?- has_motor(mybmw). Y=mybmw Yes 3 Type “control” key together with “h” then press “m” to get an exhaustive mode help. 268
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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
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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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Page 225 and 226: can combine both to falsify communi
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Page 231 and 232: conceptually: transfer of directed
Page 233 and 234: Definition 591 The XML path languag
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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 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
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Page 271 and 272: GAs = evolution: e.g., real genes e
Page 273: Definition 632 A query is a list of
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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