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Problem 1.14: Program the function f with f(x) = x 2 on unary natural numbers without using the multiplication function. Solution: We will use the abstract data type mynat datatype mynat = zero | s of mynat fun add(n,zero) = n | add(n,s(m))=s(add(n,m)) fun sq(zero)=zero|sq(s(n))=s(add(add(sq(n),n),n)) 15 15pt
Problem 1.15 (Translating between Integers and Strings) SML has pre-defined types int and string, write two conversion functions: • int2string converts an integer to a string, i.e. int2string(~317) "~317":string • string2int converts a suitable string to an integer, i.e. string2int("444") 444:int. For the moment, we do not care what happens, if the input string is unsuitable, i.e does not correspond to an integer. do not use any built-in functions except elementary arithmetic (which include mod and div BTW), explode, and implode. Solution: (* Note: this implementation does not consider negative numbers *) (*integer to string*) fun dig2chr 0 = #"0" | dig2chr 1 = #"1" | dig2chr 2 = #"2" | dig2chr 3 = #"3" | dig2chr 4 = #"4" | dig2chr 5 = #"5" | dig2chr 6 = #"6" | dig2chr 7 = #"7" | dig2chr 8 = #"8" | dig2chr 9 = #"9"; fun int2lst 0 = [] | int2lst num = int2lst(num div 10) @ [dig2chr(num mod 10)]; fun int2string 0 = "0" | int2string num = implode(int2lst num); (*string to integer*) fun chr2dig #"0" = 0 | chr2dig #"1" = 1 | chr2dig #"2" = 2 | chr2dig #"3" = 3 | chr2dig #"4" = 4 | chr2dig #"5" = 5 | chr2dig #"6" = 6 | chr2dig #"7" = 7 | chr2dig #"8" = 8 | chr2dig #"9" = 9; fun lst2int [] = 0 | lst2int (h::t) = (lst2int t + chr2dig h )*10; fun rev nil = nil | rev (h::t) = rev t @ [h]; fun string2int(s) = lst2int(rev (explode s)) div 10; 16 20pt
Page 1 and 2: General Computer Science 320201 Gen
Page 3 and 4: Contents Preface . . . . . . . . .
Page 5 and 6: 0.2 Motivation and Introduction Pro
Page 7 and 8: Problem 1.2 (Natural numbers) Prove
Page 9 and 10: 1.1.2 Naive Set Theory Problem 1.4:
Page 11 and 12: 1.1.3 Naive Set Theory Problem 1.7:
Page 13 and 14: 1.2 Computing with Functions over I
Page 15 and 16: Problem 1.11: Define functions to z
Page 17: Problem 1.13 (Decompressing binary
Page 21 and 22: Problem 1.17: Write a non-recursive
Page 23 and 24: Problem 1.19 (List functions via fo
Page 25 and 26: 1.2.2 Inductively Defined Sets and
Page 27 and 28: Problem 1.23: In class, we have bee
Page 29 and 30: Problem 1.25: Declare a data type m
Page 31 and 32: Problem 1.27 (Your own lists) Defin
Page 33 and 34: Problem 1.29 (Nary Multiplication)
Page 35 and 36: Problem 1.31: Translate the given S
Page 37 and 38: A First Abstract Interpreter Proble
Page 39 and 40: Problem 1.35: Consider the followin
Page 41 and 42: Definition 2 We call a substitution
Page 43 and 44: fun substApply (subst_in,term_in) =
Page 45 and 46: Problem 1.40: Explain the concept o
Page 47 and 48: Problem 1.42: Give the recursion re
Page 49 and 50: 1.2.6 Even more SML: Exceptions and
Page 51 and 52: Problem 1.45 (List Functions with E
Page 53 and 54: Problem 1.47 (Simple SML data conve
Page 55 and 56: if is_negative then whole_r - fract
Page 57 and 58: Problem 1.50 (List evaluation) Writ
Page 59 and 60: Problem 1.51 (String parsing) Write
Page 61 and 62: Problem 1.52 (SML File IO) Write an
Page 63 and 64: Problem 1.54: Translate the given S
Page 65 and 66: 1.3.2 A First Abstract Interpreter
Page 67 and 68: Problem 1.58: Consider the followin
Page 69 and 70:
Definition 4 We call a substitution
Page 71 and 72:
fun substApply (subst_in,term_in) =
Page 73 and 74:
1.3.5 Evaluation Order and Terminat
Page 75 and 76:
Problem 1.65: Give the recursion re
Page 77 and 78:
Problem 1.67 (Operations with Excep
Page 79 and 80:
Problem 1.69 (Transformations with
Page 81 and 82:
Problem 1.71 (Strings and numbers)
Page 83 and 84:
Problem 1.72 (Recursive evaluation)
Page 85 and 86:
val test7 = eql ( evaluate_list [va
Page 87 and 88:
val test4 = eql (evaluate_str ["~1.
Page 89 and 90:
1.5 Encoding Programs as Strings 1.
Page 91 and 92:
Problem 1.78: Let A := {a, h, /, #,
Page 93 and 94:
1.5.2 Elementary Codes Problem 1.80
Page 95 and 96:
Problem 1.82: Let A := {a, b, c, d,
Page 97 and 98:
Problem 1.84 (Morse Code again) Wit
Page 99 and 100:
1.5.3 Character Codes in the Real W
Page 101 and 102:
1.6 Boolean Algebra 1.6.1 Boolean E
Page 103 and 104:
Problem 1.88 (Partial orders in a B
Page 105 and 106:
Problem 1.90: Given the SML data ty
Page 107 and 108:
val (p,s)=find_sign(lst,0,1) (* we
Page 109 and 110:
Problem 1.93: Is the expression e :
Page 111 and 112:
Problem 1.95 (Boolean Equivalence)
Page 113 and 114:
Problem 1.97 (CNF and DNF) Write th
Page 115 and 116:
Problem 1.99 (Relations among polyn
Page 117 and 118:
Problem 1.101 (Upper and lower boun
Page 119 and 120:
Problem 1.103 (Proof of Membership
Page 121 and 122:
Problem 1.105: Use the algorithm of
Page 123 and 124:
Problem 1.106 (Quine-McCluskey with
Page 125 and 126:
1.6.5 A simpler Method for finding
Page 127 and 128:
Problem 1.110 (CNF with Karnaugh-Ve
Page 129 and 130:
Problem 1.112 (Don’t-Care Minimiz
Page 131 and 132:
1.7.2 Logical Systems and Calculi P
Page 133 and 134:
Problem 1.116 (Almost a Proof) Plea
Page 135 and 136:
Problem 1.118 (Alternative Calculus
Page 137 and 138:
Problem 1.120 (Hilbert Calculus) Pr
Page 139 and 140:
1.7.4 The Calculus of Natural Deduc
Page 141 and 142:
Problem 1.123: Prove the associativ
Page 143 and 144:
Problem 1.125 (Refutation and model
Page 145 and 146:
Problem 1.127 (A Nor Tableau Calcul
Page 147 and 148:
Problem 1.129 (Automated Theorem Pr
Page 149 and 150:
G 1.8.2 Resolution for Propositiona
Page 151 and 152:
Problem 1.133 (Basics of Resolution
Page 153 and 154:
Problem 1.135: Use the resolution m
Page 155 and 156:
We note that for the formula in DNF
Page 157 and 158:
Problem 2.2 (Node Connectivity Rela
Page 159 and 160:
Problem 2.4: Draw examples of 1. a
Page 161 and 162:
Problem 2.6 (Parse trees and isomor
Page 163 and 164:
Problem 2.8 (Graph basics) For each
Page 165 and 166:
Problem 2.10 (Graph representation
Page 167 and 168:
Problem 2.12 (Undirected tree prope
Page 169 and 170:
Problem 2.14 (Parse Tree) Given the
Page 171 and 172:
2.1.2 Introduction to Combinatorial
Page 173 and 174:
Problem 2.17 (Combinational Circuit
Page 175 and 176:
Problem 2.19 (Is XOR universal?) Im
Page 177 and 178:
2.1.3 Realizing Complex Gates Effic
Page 179 and 180:
Problem 2.23 (Length of the inner p
Page 181 and 182:
Realizing n-ary Gates No problems s
Page 183 and 184:
Problem 2.26 (Mapping between Posit
Page 185 and 186:
Problem 2.28 (Playing with bases) C
Page 187 and 188:
Adders Problem 2.30 (Cost and depth
Page 189 and 190:
Problem 2.32 (Carry Chain Adder and
Page 191 and 192:
Problem 2.34 (2-Stage Adder) Design
Page 193 and 194:
Problem 2.36 (Sign-and-Magnitude Ad
Page 195 and 196:
Problem 2.38: Given the following i
Page 197 and 198:
Problem 2.40 (2s Complement Convers
Page 199 and 200:
Problem 2.42: Compute the intermedi
Page 201 and 202:
Problem 2.44 (Carry Chain Adder and
Page 203 and 204:
Problem 2.46 (Reading from and writ
Page 205 and 206:
Problem 2.47 (Event Detection with
Page 207 and 208:
Problem 2.49 (Displaying a two-bit
Page 209 and 210:
Problem 2.50 (Making a speedometer)
Page 211 and 212:
Problem 2.52 (Multiplication) Write
Page 213 and 214:
Problem 2.54 (Simulating a Register
Page 215 and 216:
Problem 2.55 (sorting-by-selection)
Page 217 and 218:
2.4.2 A Stack-based Virtual Machine
Page 219 and 220:
Problem 2.59 (Fibonacci Numbers) As
Page 221 and 222:
Problem 2.61 (Static procedure for
Page 223 and 224:
Problem 2.63 (Simple While program
Page 225 and 226:
Problem 2.65 (Duplicate identifiers
Page 227 and 228:
App("IsPrime", [Con 119]) ); Anothe
Page 229 and 230:
Problem 2.68 (Turing Machine) Given
Page 231 and 232:
Problem 2.70 (Boolean Equivalence)
Page 233 and 234:
Problem 2.71 (Halting Reductions) T
Page 235 and 236:
Problem 2.73 (TM and languages) Des
Page 237 and 238:
Problem 2.74 (TM and TCN numbers) G
Page 239 and 240:
2.5 The Information and Software Ar
Page 241 and 242:
10. 11. 12. 13. 14. 15. 16.
Page 243 and 244:
Problem 2.79 (Web browsers) • Wha
Page 245 and 246:
You are not required to handle link
Page 247 and 248:
2.5.6 Security by Encryption Nothin
Page 249 and 250:
Problem 3.2 (Define Problem Formula
Page 251 and 252:
Problem 3.4 (Problem formulation) Y
Page 253 and 254:
Problem 3.6 (Search of the max elem
Page 255 and 256:
Problem 3.8 (Moving a Knight) Consi
Page 257 and 258:
8 EdN:8 Problem 3.10 (Sudoku) This
Page 259 and 260:
Problem 3.12 (Actions with Negative
Page 261 and 262:
Problem 3.14 (Implementing Search)
Page 263 and 264:
9 EdN:9 Problem 3.15 (A Trip Throug
Page 265 and 266:
Problem 3.17 (Search Strategy Compa
Page 267 and 268:
Problem 3.19: Write the next functi
Page 269 and 270:
Problem 3.20 (Interpreting Search R
Page 271 and 272:
Problem 3.22 (Power Source Search)
Page 273 and 274:
3.1.4 Informed Search Strategies Pr
Page 275 and 276:
Problem 3.26 (A variant of A ∗ )
Page 277 and 278:
Problem 3.28 (Sudoku Revisited) Rem
Page 279 and 280:
Problem 3.30 (A Good Old Friend, th
Page 281 and 282:
in end; tie(h1::t1, h2::t2) = (h1,h
Page 283 and 284:
Problem 3.33 (Relations between sea
Page 285 and 286:
3.1.5 Local Search Problem 3.35 (Lo
Page 287 and 288:
Problem 3.37 (Local Beam Search) Wh
Page 289 and 290:
Sample run: val matrix = [[(false,t
Page 291 and 292:
val init4 = [(man 2,woman 1),(man 3
Page 293 and 294:
Problem 3.40 (Implementing simulate
Page 295 and 296:
Problem 3.41 (Simulated annealing s
Page 297 and 298:
3.2 Logic Programming 3.2.1 Introdu
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