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

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Definition 236 We will write (−A) as A and (A∗B) as A ∗ B (and similarly for +). Furthermore we will write variables such as x71 as x71 and elide brackets for sums and products according to their usual precedences. Example 237 σ(((−(x1∗x2))+(x3∗x4))) = x1 ∗ x2 + x3 ∗ x4 : Do not confuse + and ∗ (Boolean sum and product) with their arithmetic counterparts. (as members of a formal language they have no meaning!) c○: Michael Kohlhase 134 Now that we have defined the formal language, we turn the process of giving the strings a meaning. We make explicit the idea of providing meaning by specifying a function that assigns objects that we already understand to representations (strings) that do not have a priori meaning. The first step in assigning meaning is to fix a set of objects what we will assign as meanings: the “universe (of discourse)”. To specify the meaning mapping, we try to get away with specifying as little as possible. In our case here, we assign meaning only to the constants and functions and induce the meaning of complex expressions from these. As we have seen before, we also have to assign meaning to variables (which have a different ontological status from constants); we do this by a special meaning function: a variable assignment. Boolean Expressions: Semantics via Models Definition 238 A model 〈U, I〉 for Ebool is a set U of objects (called the universe) together with an interpretation function I on A with I(Cbool) ⊆ U, I(F 1 bool ) ⊆ F(U; U), and I(F 2 bool ) ⊆ F(U 2 ; U). Definition 239 A function ϕ: V → U is called a variable assignment. Definition 240 Given a model 〈U, I〉 and a variable assignment ϕ, the evaluation function Iϕ : Ebool → U is defined recursively: Let c ∈ Cbool, a, b ∈ Ebool, and x ∈ V , then Iϕ(c) = I(c), for c ∈ Cbool Iϕ(x) = ϕ(x), for x ∈ V Iϕ(a) = I(−)(Iϕ(a)) Iϕ(a + b) = I(+)(Iϕ(a), Iϕ(b)) and Iϕ(a ∗ b) = I(∗)(Iϕ(a), Iϕ(b)) U = {T, F} with 0 ↦→ F, 1 ↦→ T, + ↦→ ∨, ∗ ↦→ ∧, − ↦→ ¬. U = Eun with 0 ↦→ /, 1 ↦→ //, + ↦→ div, ∗ ↦→ mod, − ↦→ λx.5. U = {0, 1} with 0 ↦→ 0, 1 ↦→ 1, + ↦→ min, ∗ ↦→ max, − ↦→ λx.1 − x. c○: Michael Kohlhase 135 Note that all three models on the bottom of the last slide are essentially different, i.e. there is no way to build an isomorphism between them, i.e. a mapping between the universes, so that all Boolean expressions have corresponding values. To get a better intuition on how the meaning function works, consider the following example. We see that the value for a large expression is calculated by calculating the values for its subexpressions and then combining them via the function that is the interpretation of the constructor at the head of the expression. Evaluating Boolean Expressions Example 241 Let ϕ := [T/x1], [F/x2], [T/x3], [F/x4], and I = {0 ↦→ F, 1 ↦→ T, + ↦→ ∨, ∗ ↦→ ∧, − ↦→ ¬}, then 73
What a mess! Iϕ((x1 + x2) + (x1 ∗ x2 + x3 ∗ x4)) = Iϕ(x1 + x2) ∨ Iϕ(x1 ∗ x2 + x3 ∗ x4) = Iϕ(x1) ∨ Iϕ(x2) ∨ Iϕ(x1 ∗ x2) ∨ Iϕ(x3 ∗ x4) = ϕ(x1) ∨ ϕ(x2) ∨ ¬(Iϕ(x1 ∗ x2)) ∨ Iϕ(x3 ∗ x4) = (T ∨ F) ∨ (¬(Iϕ(x1) ∧ Iϕ(x2)) ∨ (Iϕ(x3) ∧ Iϕ(x4))) = T ∨ ¬(¬(Iϕ(x1)) ∧ ϕ(x2)) ∨ (ϕ(x3) ∧ ϕ(x4)) = T ∨ ¬(¬(ϕ(x1)) ∧ F) ∨ (T ∧ F) = T ∨ ¬(¬(T) ∧ F) ∨ F = T ∨ ¬(F ∧ F) ∨ F = T ∨ ¬(F) ∨ F = T ∨ T ∨ F = T c○: Michael Kohlhase 136 A better mouse-trap: Truth Tables Truth tables to visualize truth functions: · T F F T ∗ T F T T F F F F + T F T T T F T F If we are interested in values for all assignments (e.g. of x123 ∗ x4 + x123 ∗ x72) Boolean Algebra assignments intermediate results full x4 x72 x123 e1 := x123 ∗ x72 e2 := e1 e3 := x123 ∗ x4 e3 + e2 F F F F T F T F F T F T F T F T F F T F T F T T T F F F T F F F T F T T F T F T T T T T F F T F T T T T T F T T c○: Michael Kohlhase 137 Definition 242 A Boolean algebra is Ebool together with the models 〈{T, F}, {0 ↦→ F, 1 ↦→ T, + ↦→ ∨, ∗ ↦→ ∧, − ↦→ ¬}〉. 〈{0, 1}, {0 ↦→ 0, 1 ↦→ 1, + ↦→ max, ∗ ↦→ min, − ↦→ λx.1 − x}〉. BTW, the models are equivalent (0 ˆ=F, 1 ˆ=T) Definition 243 We will use B for the universe, which can be either {0, 1} or {T, F} Definition 244 We call two expressions e1, e2 ∈ Ebool equivalent (write e1 ≡ e2), iff Iϕ(e1) = Iϕ(e2) for all ϕ. Theorem 245 e1 ≡ e2, iff (e1 + e2) ∗ (e1 + e2) is a theorem of Boolean Algebra. c○: Michael Kohlhase 138 As we are mainly interested in the interplay between form and meaning in Boolean Algebra, we will often identify Boolean expressions, if they have the same values in all situations (as specified 74
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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
Page 29 and 30: Theorem 36 is a very useful fact to
Page 31 and 32: Definition 43 The unary product ope
Page 33 and 34: y stating element-hood (a ∈ S) or
Page 35 and 36: Idea: We need a notion of “counti
Page 37 and 38: Example 64 On sets of persons, the
Page 39 and 40: Definition 82 We say that a set A i
Page 41 and 42: clean enough to learn important con
Page 43 and 44: Definition 90 anonymous variables (
Page 45 and 46: c○: Michael Kohlhase 66 Defining
Page 47 and 48: - fun both_plus (x:int,y:int) = fn
Page 49 and 50: +(〈n, o〉) = n (base) and +(〈m
Page 51 and 52: the axiom says that any object that
Page 53 and 54: epresent them as a data type, where
Page 55 and 56: Example 127 〈{N}, {[o: N], [s: N
Page 57 and 58: The central idea here is what we ha
Page 59 and 60: Substitutions Definition 149 Let A
Page 61 and 62: Idea: Well-formed parts of construc
Page 63 and 64: Other programming languages chose a
Page 65 and 66: generally: fn+1 := fn + fn−1 plus
Page 67 and 68: input/output: the interesting bit a
Page 69 and 70: exception NaN; (* Not a Number *) f
Page 71 and 72: Example 191 If A = {a, b, c}, then
Page 73 and 74: Example 210 The Morse Code in the t
Page 75 and 76: The first 32 characters are control
Page 77 and 78: Idea: Unicode supports multiple enc
Page 79: 2.5 Boolean Algebra We will now loo
Page 83 and 84: c○: Michael Kohlhase 140 The defi
Page 85 and 86: (f ≤a g), iff there is an n0 ∈
Page 87 and 88: P.1.2.3 then there are ei ∈ Ebool
Page 89 and 90: 2.5.4 The Quine-McCluskey Algorithm
Page 91 and 92: the disjunctive normal form, and th
Page 93 and 94: Proof: by contradiction: let p /∈
Page 95 and 96: So, the minimal polynomial of f is
Page 97 and 98: 2.6 Propositional Logic 2.6.1 Boole
Page 99 and 100: 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
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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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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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