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

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means that if the assumption of the Subst rule was valid, then the conclusion is valid as well, i.e. the validity property is preserved. Soundness of Substitution Lemma 318 Subst preserves validity. Proof: We have to show that [B/X](A) is valid, if A is. P.1 Let A be valid, B a formula, ϕ: Vo → {T, F} a variable assignment, and ψ := ϕ, [Iϕ(B)/X]. P.2 then Iϕ([B/X](A)) = I ϕ,[Iϕ(B)/X](A) = T, since A is valid. P.3 As the argumentation did not depend on the choice of ϕ, [B/X](A) valid and we have proven the assertion. c○: Michael Kohlhase 186 The next theorem shows that the implication connective and the entailment relation are closely related: we can move a hypothesis of the entailment relation into an implication assumption in the conclusion of the entailment relation. Note that however close the relationship between implication and entailment, the two should not be confused. The implication connective is a syntactic formula constructor, whereas the entailment relation lives in the semantic realm. It is a relation between formulae that is induced by the evaluation mapping. The Entailment Theorem Theorem 319 If H, A |= B, then H |= (A ⇒ B). Proof: We show that Iϕ(A ⇒ B) = T for all assignments ϕ with Iϕ(H) = T whenever H, A |= B P.1 Let us assume there is an assignment ϕ, such that Iϕ(A ⇒ B) = F. P.2 Then Iϕ(A) = T and Iϕ(B) = F by definition. P.3 But we also know that Iϕ(H) = T and thus Iϕ(B) = T, since H, A |= B. P.4 This contradicts our assumption Iϕ(B) = T from above. P.5 So there cannot be an assignment ϕ that Iϕ(A ⇒ B) = F; in other words, A ⇒ B is valid. c○: Michael Kohlhase 187 Now, we complete the theorem by proving the converse direction, which is rather simple. The Entailment Theorem (continued) Corollary 320 H, A |= B, iff H |= (A ⇒ B) Proof: In the light of the previous result, we only need to prove that H, A |= B, whenever H |= (A ⇒ B) P.1 To prove that H, A |= B we assume that Iϕ(H, A) = T. P.2 In particular, Iϕ(A ⇒ B) = T since H |= (A ⇒ B). P.3 Thus we have Iϕ(A) = F or Iϕ(B) = T. P.4 The first cannot hold, so the second does, thus H, A |= B. 99
c○: Michael Kohlhase 188 The entailment theorem has a syntactic counterpart for some calculi. This result shows a close connection between the derivability relation and the implication connective. Again, the two should not be confused, even though this time, both are syntactic. The main idea in the following proof is to generalize the inductive hypothesis from proving A ⇒ B to proving A ⇒ C, where C is a step in the proof of B. The assertion is a special case then, since B is the last step in the proof of B. The Deduction Theorem Theorem 321 If H, A ⊢ B, then H ⊢ A ⇒ B Proof: By induction on the proof length P.1 Let C1, . . . , Cm be a proof of B from the hypotheses H. P.2 We generalize the induction hypothesis: For all l (1 ≤ i ≤ m) we construct proofs H ⊢ A ⇒ Ci. (get A ⇒ B for i = m) P.3 We have to consider three cases P.3.1 Case 1: Ci axiom or Ci ∈ H: P.3.1.1 Then H ⊢ Ci by construction and H ⊢ Ci ⇒ A ⇒ Ci by Subst from Axiom 1. P.3.1.2 So H ⊢ A ⇒ Ci by MP. P.3.2 Case 2: Ci = A: P.3.2.1 We have already proven ∅ ⊢ A ⇒ A, so in particular H ⊢ A ⇒ Ci. (more hypotheses do not hurt) P.3.3 Case 3: everything else: P.3.3.1 Ci is inferred by MP from Cj and Ck = Cj ⇒ Ci for j, k < i P.3.3.2 We have H ⊢ A ⇒ Cj and H ⊢ A ⇒ Cj ⇒ Ci by IH P.3.3.3 Furthermore, (A ⇒ Cj ⇒ Ci) ⇒ (A ⇒ Cj) ⇒ A ⇒ Ci by Axiom 2 and Subst P.3.3.4 and thus H ⊢ A ⇒ Ci by MP (twice). P.4 We have treated all cases, and thus proven H ⊢ A ⇒ Ci for (1 ≤ i ≤ m). P.5 Note that Cm = B, so we have in particular proven H ⊢ A ⇒ B. c○: Michael Kohlhase 189 In fact (you have probably already spotted this), this proof is not correct. We did not cover all cases: there are proofs that end in an application of the Subst rule. This is a common situation, we think we have a very elegant and convincing proof, but upon a closer look it turns out that there is a gap, which we still have to bridge. This is what we attempt to do now. The first attempt to prove the subst case below seems to work at first, until we notice that the substitution [B/X] would have to be applied to A as well, which ruins our assertion. The missing Subst case Oooops: The proof of the deduction theorem was incomplete (we did not treat the Subst case) 100
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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
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 and 80: 2.5 Boolean Algebra We will now loo
Page 81 and 82: What a mess! Iϕ((x1 + x2) + (x1
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: H 0 axioms are valid Lemma 316 The
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
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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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