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1 | dashley | 140 | %$Header$ |

2 | |||

3 | \chapter[\cbalzeroshorttitle{}]{\cbalzerolongtitle{}} | ||

4 | |||

5 | \label{cbal0} | ||

6 | |||

7 | \section{Introduction, Definition, And History Of Boolean Algebra} | ||

8 | %Section tag: INT0 | ||

9 | \label{cbal0:sint0} | ||

10 | |||

11 | \index{Boolean algebra} | ||

12 | In this chapter, we review methods of | ||

13 | simplifying \index{Boolean function}Boolean functions, and | ||

14 | show how such methods can be used to simplify software constructs | ||

15 | or to construct software which is more compact (in ROM or RAM) | ||

16 | or executes more quicky. | ||

17 | |||

18 | Boolean algebra is named after \index{Boole, George}George Boole, | ||

19 | a 19th-century British mathematician. | ||

20 | A fairly concise biography of George Boole comes from | ||

21 | \cite{bibref:w:georgeboolebio01}: | ||

22 | |||

23 | \begin{quote} | ||

24 | George Boole first attended a school in Lincoln, then a commerical school. | ||

25 | His early instruction in mathematics, however, was from his father who | ||

26 | also gave George a liking for constructing optical instruments. George's | ||

27 | interests turned to languages and he received instruction in Latin from a | ||

28 | local bookseller. | ||

29 | |||

30 | By the age of 12 George had become so skilled in Latin that it provoked | ||

31 | an argument. He translated an ode by the Latin poet Horace which his | ||

32 | father was so proud of that he had it published. However the talent was | ||

33 | such that a local schoolmaster disputed that any 12 year old could | ||

34 | have written with such depth. | ||

35 | |||

36 | Boole did not study for an academic degree, but from the age of 16 he was an | ||

37 | assistant school teacher. He maintained his interest in languages | ||

38 | and intended to enter the Church. From 1835, however, he seems to have | ||

39 | changed his mind for he opened his own school and began to study mathematics | ||

40 | on his own. He was later to realize that he had almost wasted five years | ||

41 | in trying to teach himself the subject instead of having a skilled teacher. | ||

42 | |||

43 | At this time Boole studied the works of Laplace and Lagrange, making | ||

44 | notes which would later be the basis for his first mathematics paper. | ||

45 | However he did receive encouragement from Duncan Gregory who at this | ||

46 | time was in Cambridge and the editor of the recently founded | ||

47 | \emph{Cambridge Mathematical Journal}. | ||

48 | |||

49 | Boole was unable to take Duncan Gregory's advice and study courses at | ||

50 | Cambridge as he required the income from his school to look after | ||

51 | his parents. However he began publishing in the \emph{Cambridge | ||

52 | Mathematical Journal} and his interests were also influenced by Duncan Gregory | ||

53 | as he began to study algebra. An application of algebraic methods | ||

54 | to the solution of differential equations was published by Boole in the | ||

55 | \emph{Transactions of the Royal Society} and for this work he received | ||

56 | the Society's Royal Medal. His mathematical work was beginning to bring | ||

57 | him fame. | ||

58 | |||

59 | Boole was appointed to the chair of mathematics at Queens College, | ||

60 | Cork in 1849. He taught there for the rest of his life, gaining a reputation | ||

61 | as an outstanding and dedicated teacher. | ||

62 | |||

63 | In 1854 he published \emph{An investigation into the Laws of Thought, on | ||

64 | Which are founded the Mathematical Theories of Logic and Probabilities}. | ||

65 | Boole approached logic in a new way reducing it to a simple algebra, | ||

66 | incorporating logic into mathematics. He pointed out the analogy | ||

67 | between algebraic symbols and those that represent logical forms. | ||

68 | It began the algebra of logic called Boolean algebra which now finds | ||

69 | application in computer construction, switching circuits etc. | ||

70 | |||

71 | Boole also worked on differential equations, the influential | ||

72 | \emph{Treatise on Differential Equations} appeared in 1859, the calculus of finite | ||

73 | differences, \emph{Treatise on the Calculus of Finite Differences} (1860), | ||

74 | and general methods in probability. He published around 50 papers and | ||

75 | was one of the first to investigate the basic properties of numbers, such | ||

76 | as the distributive property, that underlie the subject of algebra. | ||

77 | |||

78 | Many honours were given to Boole as the genius in his work was recognised. | ||

79 | He received honorary degrees from the universities of Dublin | ||

80 | and Oxford and was elected a Fellow of the Royal Society (1857). | ||

81 | However his career, which was started rather late, came to an unfortunately | ||

82 | early end when he died at the age of 49. The circumstances are described | ||

83 | by Macfarlane in [17]\footnote{As reproduced from | ||

84 | \cite{bibref:w:georgeboolebio01}---this is not a valid reference | ||

85 | number in this work.} as follows: | ||

86 | |||

87 | \begin{quote} | ||

88 | One day in 1864 he walked from his residence to the College, | ||

89 | a distance of two miles, in the drenching rain, and lectured in wet | ||

90 | clothes. The result was a feverish cold which soon fell upon his | ||

91 | lungs and terminated his career \ldots{} | ||

92 | \end{quote} | ||

93 | |||

94 | What Macfarlane fails to say is that Boole's wife | ||

95 | (Mary---niece of Sir George Everest, after whom the mountain is named) believed that a | ||

96 | remedy should resemble the cause. She put Boole to bed and threw | ||

97 | buckets of water over the bed since his illness had been caused by getting | ||

98 | wet. | ||

99 | |||

100 | Hirst described Boole as: | ||

101 | |||

102 | \begin{quote} | ||

103 | \ldots{} evidently an earnest able and at the same time a genial man. | ||

104 | \end{quote} | ||

105 | |||

106 | His work was praised by \index{DeMorgan, Augustus}De Morgan who said: | ||

107 | |||

108 | \begin{quote} | ||

109 | Boole's system of logic is but one of many proofs of genius and patience combined. | ||

110 | \ldots{} That the symbolic processes of algebra, | ||

111 | invented as tools of numerical calculation, should be competent to | ||

112 | express every act of thought, and to furnish the grammar and | ||

113 | dictionary of an all-containing system of logic, would not have | ||

114 | been believed until it was proved. When Hobbes \ldots{} published his | ||

115 | ``Computation or Logique'' he had a remote glimpse of some of the points which | ||

116 | are placed in the light of day by Mr Boole. | ||

117 | \end{quote} | ||

118 | |||

119 | Boolean algebra has wide applications in telephone switching and the | ||

120 | design of modern computers. Boole's work has to be seen as a | ||

121 | fundamental step in today's computer revolution. | ||

122 | (Article by: J. J. O'Connor and E. F. Robertson.) | ||

123 | \end{quote} | ||

124 | |||

125 | In the preface of | ||

126 | \cite{bibref:b:whitesittbooleanalgandapps}, Whitesitt explains the | ||

127 | significance of Boole's work: | ||

128 | |||

129 | \begin{quote} | ||

130 | George Boole (1815-1864) introduced in his book \emph{The Laws Of Thought} | ||

131 | the first systematic treatment of logic and developed for this purpose | ||

132 | the algebraic system now known by his name, Boolean algebra. Few | ||

133 | mathematical works of the last 100 years have had a greater impact | ||

134 | upon mathematics and philosophy than this famous book. The significance | ||

135 | of the work has been well expressed by Augustus De Morgan: | ||

136 | |||

137 | \begin{quote} | ||

138 | That the symbolic processes of algebra, invented as tools of numerical | ||

139 | calculation, should be competent to express every act of thought, and to | ||

140 | furnish the grammar and dictionary of an all-containing system of | ||

141 | logic, would not have been believed until it was proved in | ||

142 | \emph{Laws Of Thought}. | ||

143 | \end{quote} | ||

144 | |||

145 | In addition to its applications in the field of logic, Boolean algebra | ||

146 | has two other important applications. The first of these arises from | ||

147 | the fact that Boolean algebra is the natural algebra with which to | ||

148 | treat the combination of sets of elements under the operations of | ||

149 | intersection and union of sets. With the addition of the idea of | ||

150 | ``number of elements'' in a set, Boolean algebra becomes the | ||

151 | foundation for the theory of probability. The algebra | ||

152 | of sets is also important in many other branches of mathematics. | ||

153 | |||

154 | With the publication of two papers approximately 20 years | ||

155 | ago,\footnote{Whitesitt's book was first published in 1961, | ||

156 | so Whitesitt probably means \emph{20 years ago} with | ||

157 | respect to 1961.} | ||

158 | \index{Shannon, Claude E.}Claude E. Shannon introduced a new | ||

159 | area of application of | ||

160 | Boolean algebra when he showed that the basic properties of | ||

161 | series and parallel combinations of bistable electrical | ||

162 | devices such as relays could be adequately represented | ||

163 | by this algebra. Since this time, Boolean algebra has | ||

164 | played a significant role in the important and complicated | ||

165 | task of designing telephone switching circuits, automatic | ||

166 | control devices, and electronic computers. At the present | ||

167 | time, more interest is centered in this application than | ||

168 | in either of the others. | ||

169 | \end{quote} | ||

170 | |||

171 | Claude E. Shannon's classic 1937 thesis is described separately | ||

172 | in several places. From \cite{{bibref:w:boolealghist01}}: | ||

173 | |||

174 | \begin{quote} | ||

175 | Shannon, whose 1937 thesis centered on the improvement of the | ||

176 | Differential Analyzer, a clunky mechanical gear and shaft addition | ||

177 | machine, realized that Boolean algebraic principles governed its operation. | ||

178 | He proposed that such a machine could be built with circuits using | ||

179 | components that were governed by Boolean principles. Shannon's paper | ||

180 | was regarded as genius and his ideas were almost immediately | ||

181 | incorporated into the telephone system. Shannon took a position at Bell | ||

182 | Labs. Later, of course, Shannon's thesis came to be seen as a focal point | ||

183 | in the development of modern computers. | ||

184 | \end{quote} | ||

185 | |||

186 | In \cite{bibref:w:boolealghist02}, Shannon's thesis is described | ||

187 | very favorably: | ||

188 | |||

189 | \begin{quote} | ||

190 | Around the 1850's, the British mathematician George Boole was | ||

191 | busily inventing a new form of mathematics, in which he represented | ||

192 | logical expressions in a mathematical form now known as | ||

193 | Boolean Algebra. | ||

194 | |||

195 | Unfortunately, with the exception of students of philosophy | ||

196 | and symbolic logic, Boolean Algebra was destined to remain | ||

197 | largely unknown and unused for the better part of a century. | ||

198 | It fact it was not until 1938 that Claude E. Shannon published | ||

199 | an article based on his master's thesis at MIT. (Shannon's | ||

200 | thesis has since been described as: \emph{``Possibly the | ||

201 | most important master's thesis of the twentieth century.''}) | ||

202 | |||

203 | In his paper, which was widely circulated, Shannon showed how | ||

204 | Boole's concepts of TRUE and FALSE could be used to represent | ||

205 | the functions of switches in electronic circuits. It is difficult | ||

206 | to convey just how important this concept was; suffice it to say | ||

207 | that Shannon had provided electronics engineers with the mathematical | ||

208 | tool they needed to design digital electronic circuits, and these | ||

209 | techniques remain the cornerstone of digital electronic | ||

210 | design to this day. | ||

211 | |||

212 | Apropos of nothing at all, Shannon is also credited with the invention | ||

213 | of the rocket-powered Frisbee, and is famous for riding down the | ||

214 | corridors of Bell Laboratories on a unicycle while simultaneously | ||

215 | juggling four balls. | ||

216 | \end{quote} | ||

217 | |||

218 | In summary, in 1854 George Boole published his classic work | ||

219 | \emph{An investigation into the Laws of Thought, on | ||

220 | Which are founded the Mathematical Theories of Logic and Probabilities}, | ||

221 | which laid the foundation of manipulation of | ||

222 | logical ideas using algebraic mechanisms. Boole's ideas | ||

223 | were revived in 1937 and 1938 by Claude E. Shannon | ||

224 | in his master's thesis and a subsequent article, in which | ||

225 | Shannon applied Boole's ideas to the design of electric | ||

226 | switching circuits. Shannon's reapplication of Boole's work | ||

227 | came to be seen as a focal point in the development of modern | ||

228 | computers. | ||

229 | |||

230 | |||

231 | \section[Simplification By Algebraic Manipulation] | ||

232 | {Simplification Of Boolean Functions By Algebraic Manipulation} | ||

233 | %Section Tag: SAM0 | ||

234 | \label{cbal0:ssam0} | ||

235 | |||

236 | We assume that the reader has had previous exposure to | ||

237 | \index{Boolean function!simplification of}simplification of | ||

238 | Boolean functions. For this reason, this section is a review (it is terse | ||

239 | and moves quickly). (For readers without this background, we | ||

240 | recommend \cite{bibref:b:manodigitaldesignseconded}.) | ||

241 | |||

242 | The most direct way to simplify Boolean functions is | ||

243 | \index{Boolean function!algebraic simplification}\emph{algebraic} | ||

244 | transformation---to use transformation postulates and | ||

245 | theorems to transform algebraic | ||

246 | expressions to a more desirable equivalent form. | ||

247 | |||

248 | |||

249 | \subsection{Nomenclature And Operators} | ||

250 | %Subsection Tag: NOM0 | ||

251 | \label{cbal0:ssam0:snom0} | ||

252 | |||

253 | We define the set of two elements which we will use in | ||

254 | two-valued Boolean | ||

255 | algebra as `0' and '1' (or alternatively as | ||

256 | `FALSE' and `TRUE', respectively). | ||

257 | All functions which we will describe subsequently require | ||

258 | inputs which are members of this set and will produce outputs | ||

259 | which are also members of this set. | ||

260 | |||

261 | The \emph{complement} or \emph{negation} operator (or function) | ||

262 | maps from 0$\rightarrow$1 and from 1$\rightarrow$0 (see Table | ||

263 | \ref{tbl:cbal0:ssam0:snom0:01}). | ||

264 | We denote this operator | ||

265 | by an apostrophe | ||

266 | following the variable to be complemented (example: $x'$). | ||

267 | Alternatively, in some circumstances, we may use the | ||

268 | tilde ($\sim{}x$), the overbar ($\overline{x}$), or | ||

269 | the negation symbol ($\neg{}x$). | ||

270 | |||

271 | \begin{table} | ||

272 | \caption{Definition Of Logical Complement, Logical And, Logical Or, And Logical | ||

273 | Exclusive Or Operators} | ||

274 | \label{tbl:cbal0:ssam0:snom0:01} | ||

275 | \begin{center} | ||

276 | \begin{tabular}{|c|c||c|c|c|c|} | ||

277 | \hline | ||

278 | $X$ & $Y$ & $X'$ & $XY$ & $X+Y$ & $X \oplus{} Y$ \\ | ||

279 | \hline | ||

280 | \hline | ||

281 | 0 & 0 & 1 & 0 & 0 & 0 \\ | ||

282 | \hline | ||

283 | 0 & 1 & 1 & 0 & 1 & 1 \\ | ||

284 | \hline | ||

285 | 1 & 0 & 0 & 0 & 1 & 1 \\ | ||

286 | \hline | ||

287 | 1 & 1 & 0 & 1 & 1 & 0 \\ | ||

288 | \hline | ||

289 | \end{tabular} | ||

290 | \end{center} | ||

291 | \end{table} | ||

292 | |||

293 | The \emph{logical AND} (or simply \index{AND}\index{Boolean algebra!AND}\emph{AND}) | ||

294 | operator (or function) | ||

295 | returns TRUE if both of its input arguments are TRUE | ||

296 | (see Table | ||

297 | \ref{tbl:cbal0:ssam0:snom0:01}). | ||

298 | We denote this operator | ||

299 | by the \emph{times} symbol (example: $x \times{} y$), by | ||

300 | the keyword `AND', or by placing the operands adjacent to | ||

301 | each other (examples: $xy$, $x(y)$). | ||

302 | |||

303 | The \emph{logical OR} | ||

304 | (or simply \index{OR}\index{Boolean algebra!OR}\emph{OR}) | ||

305 | operator (or function) | ||

306 | returns TRUE if either of its input arguments are TRUE | ||

307 | (see Table | ||

308 | \ref{tbl:cbal0:ssam0:snom0:01}). | ||

309 | We denote this operator | ||

310 | by the \emph{plus} symbol (example: $x + y$) or by | ||

311 | the keyword `OR'. | ||

312 | |||

313 | The \emph{logical exclusive-OR} | ||

314 | (or simply \index{XOR}\index{Boolean algebra!XOR} \emph{XOR}) | ||

315 | operator (or function) | ||

316 | returns TRUE if either but not both of its input arguments are TRUE | ||

317 | (see Table | ||

318 | \ref{tbl:cbal0:ssam0:snom0:01}). | ||

319 | We denote this operator | ||

320 | by the \emph{circled plus} symbol (example: $x \oplus{} y$), | ||

321 | by the \emph{caret} character (example: $x$\^{}$y$), or by | ||

322 | the keyword `XOR'. | ||

323 | |||

324 | The exclusive-OR function is not a | ||

325 | function traditionally considered in the canonical forms of Boolean | ||

326 | algebra. However, we consider it here because our aims may in | ||

327 | some cases be subtly | ||

328 | different than the aims of classic two-valued Boolean algebra. | ||

329 | The goal of classic Boolean algebra is to minimize the number of | ||

330 | terms which appear in a canonical form. | ||

331 | However, often our goal here is to rearrange Boolean functions into a form | ||

332 | which is optimal for implementation on a computer. Since actual | ||

333 | computers usually provide a bitwise exclusive-OR instruction, there | ||

334 | are some applications where it may be useful to consider the | ||

335 | underlying instruction set of the computer (for example, | ||

336 | see \emph{Vertical Counters}, | ||

337 | Section \ref{cbal0:svct0}). | ||

338 | |||

339 | In general, there are $2^{2^N}$ different Boolean functions of $N$ | ||

340 | input variables, so it isn't practical to name functions involving more | ||

341 | than 2 input variables. There are 16 ($=2^{2^2}$) Boolean functions of | ||

342 | 2 input variables, and these are enumerated in | ||

343 | Table \ref{tbl:cbal0:ssam0:snom0:02}. | ||

344 | |||

345 | \begin{table} | ||

346 | \caption{Sixteen Possible Boolean Functions $f(A,B)$ Of Two Boolean Variables} | ||

347 | \label{tbl:cbal0:ssam0:snom0:02} | ||

348 | \begin{center} | ||

349 | \begin{tabular}{|c||c|c|c|c||l|} | ||

350 | \hline | ||

351 | \small{Func.} & $\scriptstyle{f(1,1)}$ & $\scriptstyle{f(1,0)}$ | ||

352 | & $\scriptstyle{f(0,1)}$ & $\scriptstyle{f(0,0)}$ | ||

353 | & Function Description \\ | ||

354 | \small{Num.} & & & & & \\ | ||

355 | \hline | ||

356 | \hline | ||

357 | 0 & 0 & 0 & 0 & 0 & \small{``Zero'' or ``clear'' function. Always} \\ | ||

358 | & & & & & \small{returns zero regardless of $A$ and $B$ } \\ | ||

359 | & & & & & \small{input values. } \\ | ||

360 | \hline | ||

361 | 1 & 0 & 0 & 0 & 1 & \small{Logical NOR = $(A+B)'$. } \\ | ||

362 | \hline | ||

363 | 2 & 0 & 0 & 1 & 0 & \small{Inhibition = $BA'$. So named because } \\ | ||

364 | & & & & & \small{a TRUE value of $A$ inhibits $B$. Also } \\ | ||

365 | & & & & & \small{equivalent to $B>A$ or to $A<B$. } \\ | ||

366 | \hline | ||

367 | 3 & 0 & 0 & 1 & 1 & \small{NOT A = $A'$. Ignores $B$ and returns } \\ | ||

368 | & & & & & \small{$A'$. } \\ | ||

369 | \hline | ||

370 | 4 & 0 & 1 & 0 & 0 & \small{Inhibition = $AB'$. So named because } \\ | ||

371 | & & & & & \small{a TRUE value of $B$ inhibits $A$. Also } \\ | ||

372 | & & & & & \small{equivalent to $A>B$ or to $B<A$. } \\ | ||

373 | \hline | ||

374 | 5 & 0 & 1 & 0 & 1 & \small{NOT B = $B'$. Ignores $A$ and returns } \\ | ||

375 | & & & & & \small{$B'$. } \\ | ||

376 | \hline | ||

377 | 6 & 0 & 1 & 1 & 0 & \small{Exclusive-OR (XOR) = $A \oplus B$. Also} \\ | ||

378 | & & & & & \small{equivalent to $A \neq B$. } \\ | ||

379 | \hline | ||

380 | 7 & 0 & 1 & 1 & 1 & \small{Logical NAND = $(AB)'$. } \\ | ||

381 | \hline | ||

382 | 8 & 1 & 0 & 0 & 0 & \small{Logical AND = $AB$. } \\ | ||

383 | \hline | ||

384 | 9 & 1 & 0 & 0 & 1 & \small{Equivalence: true if $A=B$. Also} \\ | ||

385 | & & & & & \small{known as exclusive-NOR, i.e. $(A \oplus B)'$.} \\ | ||

386 | \hline | ||

387 | 10 & 1 & 0 & 1 & 0 & \small{Copy $B$. Ignores $A$ and returns $B$. } \\ | ||

388 | \hline | ||

389 | 11 & 1 & 0 & 1 & 1 & \small{Implication: $B \rightarrow A$ (if $B$ } \\ | ||

390 | & & & & & \small{then $A$). Also equivalent to $B \geq A$.} \\ | ||

391 | \hline | ||

392 | 12 & 1 & 1 & 0 & 0 & \small{Copy $A$. Ignores $B$ and returns $A$. } \\ | ||

393 | \hline | ||

394 | 13 & 1 & 1 & 0 & 1 & \small{Implication: $A \rightarrow B$ (if $A$ } \\ | ||

395 | & & & & & \small{then $B$). Also equivalent to $A \geq B$.} \\ | ||

396 | \hline | ||

397 | 14 & 1 & 1 & 1 & 0 & \small{Logical OR = $A+B$. } \\ | ||

398 | \hline | ||

399 | 15 & 1 & 1 & 1 & 1 & \small{``One'' or ``set'' function. Always } \\ | ||

400 | & & & & & \small{returns one regardless of $A$ and $B$ } \\ | ||

401 | & & & & & \small{input values. } \\ | ||

402 | \hline | ||

403 | \end{tabular} | ||

404 | \end{center} | ||

405 | \end{table} | ||

406 | |||

407 | \emph{Any} Boolean function can be synthesized using AND gates and inverters | ||

408 | (i.e. NOT gates), OR gates and inverters, NAND gates alone, or NOR | ||

409 | gates alone. However, not all Boolean functions can be synthesized | ||

410 | using AND gates alone or OR gates alone. | ||

411 | |||

412 | |||

413 | \subsection{Postulates And Theorems} | ||

414 | %Subsection Tag: POS0 | ||

415 | \label{cbal0:ssam0:spos0} | ||

416 | |||

417 | Mathematicians tend to be very picky about what exactly a | ||

418 | \emph{postulate} is versus what a \emph{theorem} is. We are | ||

419 | far less picky. Either one (a postulate or a theorem) is a | ||

420 | rule that can be used to transform a Boolean algebraic expression | ||

421 | into another expression which is equivalent but for one reason or another | ||

422 | more desirable. For our purposes here, postulates and theorems | ||

423 | are interchangeable. | ||

424 | |||

425 | Table \ref{tbl:cbal0:ssam0:spos0:01} | ||

426 | (from \cite{bibref:b:manodigitaldesignseconded}) | ||

427 | supplies the important postulates and theorems which are useful in | ||

428 | algebraically simplifying Boolean functions. | ||

429 | |||

430 | In Table \ref{tbl:cbal0:ssam0:spos0:01}, we should explain what | ||

431 | is meant by the \emph{Dual Postulate Or Theorem}. To | ||

432 | quote \cite{bibref:b:manodigitaldesignseconded}, p. 41: | ||

433 | |||

434 | \begin{quote} | ||

435 | \ldots This important property of Boolean algebra is called | ||

436 | the \index{Boolean algebra!duality principle}\index{duality principle}% | ||

437 | \emph{duality principle}. | ||

438 | It states that every algebraic expression deducible from the postulates | ||

439 | of Boolean algebra remains valid if the operators and identity | ||

440 | elements are interchanged. In a two-valued Boolean algebra, | ||

441 | the identity elements and the elements of the set $B$ are the | ||

442 | same: 1 and 0. The duality principle has many applications. | ||

443 | If the \emph{dual} of an algebraic expression is desired, we | ||

444 | simply interchange OR and AND operators and replaces 1s by | ||

445 | 0s and 0s by 1s. | ||

446 | \end{quote} | ||

447 | |||

448 | \begin{table} | ||

449 | \caption{Important Postulates And Theorems Of Two-Valued Boolean Algebra} | ||

450 | \label{tbl:cbal0:ssam0:spos0:01} | ||

451 | \begin{center} | ||

452 | \begin{tabular}{|c||c|c||l|} | ||

453 | \hline | ||

454 | \small{Postulate} & \small{Postulate} & \small{Dual} & \small{Remarks} \\ | ||

455 | \small{Or} & \small{Or} & \small{Postulate} & \\ | ||

456 | \small{Theorem} & \small{Theorem} & \small{Or} & \\ | ||

457 | \small{Number} & & \small{Theorem} & \\ | ||

458 | \hline | ||

459 | 1 & $x + 0 = x$ & $x \cdot 1 = 1$ & \\ | ||

460 | \hline | ||

461 | 2 & $x + x' = 1$ & $x \cdot x' = 0$ & \\ | ||

462 | \hline | ||

463 | 3 & $x + x = x$ & $x \cdot x = x$ & \\ | ||

464 | \hline | ||

465 | 4 & $x + 1 = 1$ & $x \cdot 0 = 0$ & \\ | ||

466 | \hline | ||

467 | 5 & $(x')' = x$ & & \small{Involution.} \\ | ||

468 | \hline | ||

469 | 6 & $x + y = y + x$ & $x \cdot y = y \cdot x $ & \small{Commutative property.} \\ | ||

470 | \hline | ||

471 | 7 & $x + (y + z)$ & $x(yz) = (xy)z$ & \small{Associative property.} \\ | ||

472 | & $= (x + y) + z$ & & \\ | ||

473 | \hline | ||

474 | 8 & $x (y + z)$ & $x + yz$ & \small{Distributive property.} \\ | ||

475 | & $= xy + xz$ & $= (x+y)(x+z)$ & \\ | ||

476 | \hline | ||

477 | 9 & $(x+y)' = x'y'$ & $(xy)' = x' + y'$ & \small{DeMorgan's theorem.} \\ | ||

478 | \hline | ||

479 | 10 & $x + xy = x$ & $x(x+y) = x$ & \small{Absorption.} \\ | ||

480 | \hline | ||

481 | \end{tabular} | ||

482 | \end{center} | ||

483 | \end{table} | ||

484 | |||

485 | Note also from Item 8 of | ||

486 | Table \ref{tbl:cbal0:ssam0:spos0:01} that in Boolean algebra | ||

487 | AND distributes over OR and OR distributes over AND (waiting on | ||

488 | newsgroup posters to clarify). This is unlike ``normal'' algebra, where | ||

489 | in general, | ||

490 | |||

491 | \begin{equation} | ||

492 | x + yz \neq (x+y) (x+z). | ||

493 | \end{equation} | ||

494 | |||

495 | The \index{Boolean algebra!operator precedence}% | ||

496 | \index{operator precedence!Boolean algebra}\index{operator precedence}% | ||

497 | operator precedence for evaluating Boolean expressions assumed throughout | ||

498 | this work is: | ||

499 | |||

500 | \begin{quote} | ||

501 | \begin{enumerate} | ||

502 | \item Parenthesis. | ||

503 | \item NOT. | ||

504 | \item AND. | ||

505 | \item OR. | ||

506 | \end{enumerate} | ||

507 | \end{quote} | ||

508 | |||

509 | \noindent{}Note that this operator precedence very closely mirrors | ||

510 | the standard precedence of normal algebraic expressions. | ||

511 | |||

512 | \subsection{Canonical And Standard Forms} | ||

513 | %Subsection Tag: CAS0 | ||

514 | \label{cbal0:ssam0:scas0} | ||

515 | |||

516 | \cite{bibref:b:manodigitaldesignseconded}, p. 49 provides an concise and | ||

517 | excellent definition of \emph{minterms} and \emph{maxterms}. The definition | ||

518 | here is taken primarily from that source. | ||

519 | |||

520 | A binary variable may appear either in its normal form ($x$, for example) | ||

521 | or in its complemented form ($x'$, for example). Now consider two | ||

522 | binary variables $x$ and $y$ combined with an AND | ||

523 | operation. Since each variable may appear in either form, there are four | ||

524 | possible combinations: $x'y'$, $x'y$, $xy'$, and $xy$. These | ||

525 | four AND terms are mutually exclusive and mutually exhaustive---that is, | ||

526 | no combination of values for | ||

527 | $x$ and | ||

528 | $y$ that causes one of these AND terms to be TRUE may cause any of the | ||

529 | others to be TRUE, and every combination of $x$ and $y$ will cause exactly | ||

530 | one of these AND terms to be TRUE. Each of these | ||

531 | four AND terms is called a \index{Boolean algebra!minterm}\index{minterm}\emph{minterm} | ||

532 | or \index{Boolean algebra!standard product}\index{standard product}\emph{standard product}. | ||

533 | In a similar manner, $n$ variables can be combined to form $2^n$ minterms. | ||

534 | |||

535 | In a similar fashion, $n$ variables forming an OR term, with each variable | ||

536 | being primed or unprimed, provide $2^n$ possible combinations, called | ||

537 | \index{Boolean algebra!maxterm}\index{maxterm}\emph{maxterms}, or | ||

538 | \index{Boolean algebra!standard sum}\index{standard sum}\emph{standard sums}. | ||

539 | |||

540 | \begin{table} | ||

541 | \caption{Minterms And Maxterms Of Three Boolean Variables} | ||

542 | \label{tbl:cbal0:ssam0:scas0:01} | ||

543 | \begin{center} | ||

544 | \begin{tabular}{|c|c|c||c|c||c|c|} | ||

545 | \hline | ||

546 | $x$ & $y$ & $z$ & \small{Minterm} & \small{Minterm} & \small{Maxterm} & \small{Maxterm} \\ | ||

547 | & & & \small{Term} & \small{Designation} & \small{Term} & \small{Designation} \\ | ||

548 | \hline | ||

549 | 0 & 0 & 0 & $x'y'z'$ & $m_0$ & $x+y+z$ & $M_0$ \\ | ||

550 | \hline | ||

551 | 0 & 0 & 1 & $x'y'z$ & $m_1$ & $x+y+z'$ & $M_1$ \\ | ||

552 | \hline | ||

553 | 0 & 1 & 0 & $x'yz'$ & $m_2$ & $x+y'+z$ & $M_2$ \\ | ||

554 | \hline | ||

555 | 0 & 1 & 1 & $x'yz$ & $m_3$ & $x+y'+z'$ & $M_3$ \\ | ||

556 | \hline | ||

557 | 1 & 0 & 0 & $xy'z'$ & $m_4$ & $x'+y+z$ & $M_4$ \\ | ||

558 | \hline | ||

559 | 0 & 0 & 1 & $x'y'z$ & $m_5$ & $x+y+z'$ & $M_5$ \\ | ||

560 | \hline | ||

561 | 1 & 1 & 0 & $xyz'$ & $m_6$ & $x'+y'+z$ & $M_6$ \\ | ||

562 | \hline | ||

563 | 1 & 1 & 1 & $xyz$ & $m_7$ & $x'+y'+z'$ & $M_7$ \\ | ||

564 | \hline | ||

565 | \end{tabular} | ||

566 | \end{center} | ||

567 | \end{table} | ||

568 | |||

569 | The two primary canonical forms in Boolean algebra are the | ||

570 | \emph{sum of minterms} (or \emph{sum of products}) form and the | ||

571 | \emph{product of maxterms} (or \emph{product of sums}) form. | ||

572 | |||

573 | The sum of minterms canonical form (which is the most common | ||

574 | canonical form) is sometimes written using a shorthand | ||

575 | notation involving the Greek letter sigma ($\Sigma$). We describe this | ||

576 | shorthand form now. Consider the Boolean function of 3 variables, | ||

577 | |||

578 | \begin{equation} | ||

579 | \label{eq:cbal0:ssam0:scas0:01} | ||

580 | F(A, B, C) = A + B'C. | ||

581 | \end{equation} | ||

582 | |||

583 | \noindent{}Using the postulates and theorems in Table | ||

584 | \ref{tbl:cbal0:ssam0:spos0:01}, it is possible | ||

585 | to expand (\ref{eq:cbal0:ssam0:scas0:01}) into | ||

586 | a sum of minterms form: | ||

587 | |||

588 | \begin{eqnarray} | ||

589 | \label{eq:cbal0:ssam0:scas0:02} | ||

590 | F(A, B, C) & = & A'B'C + AB'C' + AB'C + ABC' + ABC \\ | ||

591 | & = & m_1 m_4 m_5 m_6 m_7.\nonumber | ||

592 | \end{eqnarray} | ||

593 | |||

594 | \noindent{}(\ref{eq:cbal0:ssam0:scas0:02}) is also commonly written | ||

595 | in the shorthand form: | ||

596 | |||

597 | \begin{equation} | ||

598 | \label{eq:cbal0:ssam0:scas0:03} | ||

599 | F(A, B, C) = \Sigma{}(1, 4, 5, 6, 7). | ||

600 | \end{equation} | ||

601 | |||

602 | \noindent{}In the right-hand side of | ||

603 | (\ref{eq:cbal0:ssam0:scas0:03}), each integer | ||

604 | (1, 4, 5, 6, or 7) corresponds to the integer | ||

605 | that would be formed by treating the corresponding | ||

606 | minterm in (\ref{eq:cbal0:ssam0:scas0:02}) as | ||

607 | a binary number, i.e. $A=2^2$, $B=2^1$, and | ||

608 | $C=2^0$. This is the basis of the short hand | ||

609 | notation. | ||

610 | |||

611 | A second but much less commonly used canonical form | ||

612 | is the product of maxterms (or product of sums) form, | ||

613 | which uses the Greek letter pi ($\Pi$) for its shorthand | ||

614 | form. Without further explanation, we present the | ||

615 | product of maxterms form of $\Sigma{}(1,4,5,6,7)$ | ||

616 | (Eqns. \ref{eq:cbal0:ssam0:scas0:01} through | ||

617 | \ref{eq:cbal0:ssam0:scas0:03}) as | ||

618 | (\ref{eq:cbal0:ssam0:scas0:04}) | ||

619 | through (\ref{eq:cbal0:ssam0:scas0:07}) | ||

620 | (Figure \ref{fig:cbal0:ssam0:scas0:01}). | ||

621 | |||

622 | \begin{figure} | ||

623 | \begin{eqnarray} | ||

624 | \label{eq:cbal0:ssam0:scas0:04} | ||

625 | F(A, B, C) & = & \Sigma{}(1, 4, 5, 6, 7) \\ | ||

626 | \label{eq:cbal0:ssam0:scas0:05} | ||

627 | & = & \Pi{}(0, 2, 3) \\ | ||

628 | \label{eq:cbal0:ssam0:scas0:06} | ||

629 | & = & M_0 M_2 M_3 \\ | ||

630 | \label{eq:cbal0:ssam0:scas0:07} | ||

631 | & = & (x+y+z)(x+y'+z)(x+y'+z') | ||

632 | \end{eqnarray} | ||

633 | \caption{Product Of Maxterms Form Of $\Sigma{}(1,4,5,6,7)$} | ||

634 | \label{fig:cbal0:ssam0:scas0:01} | ||

635 | \end{figure} | ||

636 | |||

637 | |||

638 | Because there are so many excellent digital logic books on the market | ||

639 | that contain examples and exercises of algebraic simplification | ||

640 | (we recommend \cite{bibref:b:manodigitaldesignseconded}), we don't | ||

641 | dwell on algebraic simplification or provide examples. Again, we | ||

642 | assume that nearly all of our readers have had a course in digital | ||

643 | logic or symbolic logic. | ||

644 | |||

645 | |||

646 | \section[Karnaugh Maps] | ||

647 | {Simplification Of Boolean Functions Using Karnaugh Maps} | ||

648 | %Section tag: SKM0 | ||

649 | \label{cbal0:skm0} | ||

650 | |||

651 | In 1953, M. Karnaugh published the graphical map method of simplifying | ||

652 | combinational logic functions (\cite{bibref:p:kmapclassic01}). This | ||

653 | method has come to be known as the \emph{Karnaugh map} or \emph{K-map} | ||

654 | method of reduction, and is taught to engineers in nearly every | ||

655 | digital logic class. | ||

656 | |||

657 | Although it is possible, starting from an arbitrary Boolean formula, to simplify | ||

658 | the formula through algebraic manipulation, algebraic simplification | ||

659 | has the disadvantage that formulas can have many forms and there is not | ||

660 | an easily remembered, precise, and repeatable procedure for algebraic simplification. | ||

661 | The Karnaugh map method has the advantage that each Boolean function | ||

662 | has only one graphical representation, and that the procedure for simplification | ||

663 | is simple and easily remembered. | ||

664 | |||

665 | In a Karnaugh map, each square corresponds to a possible minterm of the function. | ||

666 | The essence of the method is to populate a grid of squares with outputs of | ||

667 | the function to be simplified. Each minterm that must exist in the | ||

668 | function (i.e. each combination of inputs for which the function | ||

669 | must return TRUE) receives a `1' in the corresponding square. Each | ||

670 | combination of inputs for which the function must return FALSE | ||

671 | receives a `0' in the corresponding square. Each combination of inputs | ||

672 | for which the function may return either TRUE or FALSE receives an | ||

673 | `X' in the corresponding square. | ||

674 | |||

675 | After the Karnaugh map squares are populated with 1's, 0's, and X's | ||

676 | corresponding to the required values of the function for each combination | ||

677 | of inputs, the 1's are graphically combined into the largest rectangle possible. | ||

678 | Each rectangle may include 1's, may include X's where convenient, but | ||

679 | may not include 0's. | ||

680 | Each of these squares is then translated into a product of uncomplemented and | ||

681 | complemented input variables, and these products are summed to | ||

682 | produce the simplified algebraic expression for the function. (We don't | ||

683 | belabor this process, because we assume that most of our readers | ||

684 | understand it very thoroughly.) There are some subtle points | ||

685 | to the process, such as combining corners and the ``folding'' of the | ||

686 | 5- and 6-variable maps, but we don't explain these subtle points here | ||

687 | because they are so thoroughly explained in so many books. | ||

688 | |||

689 | Figures \ref{fig:cbal0:skm0:00} through \ref{fig:cbal0:skm0:04} | ||

690 | supply the canonical forms of two-, three-, four-, five-, and | ||

691 | six-variable Karnaugh maps. | ||

692 | |||

693 | \begin{figure} | ||

694 | \centering | ||

695 | \includegraphics[width=1.25in]{c_bal0/kmap02cf.eps} | ||

696 | \caption{Canonical Form Of Two-Variable Karnaugh Map} | ||

697 | \label{fig:cbal0:skm0:00} | ||

698 | \end{figure} | ||

699 | |||

700 | \begin{figure} | ||

701 | \centering | ||

702 | \includegraphics[height=1.25in]{c_bal0/kmap03cf.eps} | ||

703 | \caption{Canonical Form Of Three-Variable Karnaugh Map} | ||

704 | \label{fig:cbal0:skm0:01} | ||

705 | \end{figure} | ||

706 | |||

707 | \begin{figure} | ||

708 | \centering | ||

709 | \includegraphics[height=2.0in]{c_bal0/kmap04cf.eps} | ||

710 | \caption{Canonical Form Of Four-Variable Karnaugh Map} | ||

711 | \label{fig:cbal0:skm0:02} | ||

712 | \end{figure} | ||

713 | |||

714 | \begin{figure} | ||

715 | \centering | ||

716 | \includegraphics[width=4.25in]{c_bal0/kmap05cf.eps} | ||

717 | \caption{Canonical Form Of Five-Variable Karnaugh Map} | ||

718 | \label{fig:cbal0:skm0:03} | ||

719 | \end{figure} | ||

720 | |||

721 | \begin{figure} | ||

722 | \centering | ||

723 | \includegraphics[width=4.5in]{c_bal0/kmap06cf.eps} | ||

724 | \caption{Canonical Form Of Six-Variable Karnaugh Map} | ||

725 | \label{fig:cbal0:skm0:04} | ||

726 | \end{figure} | ||

727 | |||

728 | \section[Simplification Using The Scheinman Method] | ||

729 | {Simplification Of Boolean Functions Using The Scheinman Method} | ||

730 | %Section Tag: SCM0 | ||

731 | \label{cbal0:sscm0} | ||

732 | |||

733 | Two major disadvantages of the Karnaugh map method of simplifying Boolean | ||

734 | functions are that: | ||

735 | |||

736 | \begin{itemize} | ||

737 | \item It is generally not possible to apply the method to functions | ||

738 | of more than six variables. | ||

739 | \item It is not easy to automate the method (in the form that it is stated) | ||

740 | because it is inherently graphical. | ||

741 | \end{itemize} | ||

742 | |||

743 | In 1962, A. H. Scheinman published a method for the simplification | ||

744 | of Boolean functions (\cite{bibref:p:scheinmanclassic01}) which is more amenable | ||

745 | to automation than the Karnaugh map method. | ||

746 | |||

747 | \section[The Quine-McCluskey Method] | ||

748 | {Simplification Of Boolean Functions Using The Quine-McCluskey Method} | ||

749 | |||

750 | |||

751 | \section{Multiple Functions Of $N$ Boolean Variables} | ||

752 | |||

753 | Need to get back to Dr. Singh to inquire about the multiple Scheinman method. | ||

754 | |||

755 | \section{Vertical Counters} | ||

756 | %Section Tag: VCT0. | ||

757 | \label{cbal0:svct0} | ||

758 | |||

759 | \index{vertical counter}As has been hinted at elsewhere | ||

760 | in this work, the need to generate | ||

761 | firmware which minimizes ROM consumption leads to clever (but perhaps | ||

762 | less-than-intuitive) programming techniques. One such technique | ||

763 | is the construction of \emph{vertical counters}. | ||

764 | |||

765 | A \index{vertical counter}vertical counter | ||

766 | is a set of identical combinational mappings or state machines | ||

767 | where the inputs, state vector, intermediate results, and outputs of each | ||

768 | combinational mapping or state machine are stored as a single bit | ||

769 | in the same position | ||

770 | in each of multiple bytes or words. When inputs, state vectors, | ||

771 | intermediate results, and outputs are stored | ||

772 | in this way, the bitwise logical instructions of the machine | ||

773 | (AND, OR, NOT, XOR) can be used to process several state machines in | ||

774 | parallel. Vertical counters are intimately related to the reduction | ||

775 | of Boolean functions because such reduction must be performed in order | ||

776 | to design the vertical counter. Debouncing and filtering, where the | ||

777 | inputs and outputs naturally tend to be arranged as groups of bits, | ||

778 | are the most common application of vertical counters. | ||

779 | |||

780 | A very helpful resource on the web is the set of web pages | ||

781 | maintained by \index{Dattalo, Scott}Scott Dattalo | ||

782 | \cite{bibref:i:scottdattalo} for the | ||

783 | PIC microcontroller, including | ||

784 | especially \cite{bibref:w:sdattalovc02}. Many of the vertical counter | ||

785 | designs which follow come from Mr. Dattalo's pages. | ||

786 | |||

787 | We abuse the term \emph{counter} somewhat, and we refer to | ||

788 | any bit-wise mapping useful in microcontroller work as | ||

789 | a vertical counter. We categorize vertical counters as | ||

790 | \index{vertical counter!combinational}\emph{combinational} | ||

791 | (not truly a counter---the output is a function of the | ||

792 | inputs only), \index{vertical counter!sequential}\emph{sequential} | ||

793 | (the next state and output may be functions of both the | ||

794 | inputs and the present state---i.e. a proper state machine), | ||

795 | \index{vertical counter!cyclic}\emph{cyclic} (the counter | ||

796 | goes through a series of states and then repeats), or | ||

797 | \index{vertical counter!terminating} (the counter | ||

798 | will reach a terminal state). | ||

799 | |||

800 | \subsection{2-Bit 4-State Cyclic Vertical Counter} | ||

801 | %Subsection Tag: TBF0 | ||

802 | \label{cbal0:svct0:stbf0} | ||

803 | In this discussion of vertical counters, we begin with the simplest | ||

804 | designs, as often the simpler designs appear as part of more complex | ||

805 | designs. | ||

806 | |||

807 | Table \ref{tbl:cbal0:svct0:stbf0:01} supplies the state transition table | ||

808 | for a 2-bit 4-state | ||

809 | cyclic vertical counter. Note that the counter advances through the | ||

810 | bit patterns in the ``proper'' binary integer order. | ||

811 | |||

812 | \begin{table} | ||

813 | \caption{State Transition Table Of 2-Bit 4-State Cyclic Vertical Counter} | ||

814 | \label{tbl:cbal0:svct0:stbf0:01} | ||

815 | \begin{center} | ||

816 | \begin{tabular}{|c|c|c|c|} | ||

817 | \hline | ||

818 | $A_k$ & $B_k$ & $A_{k+1}$ & $B_{k+1}$ \\ | ||

819 | \hline | ||

820 | \hline | ||

821 | 0 & 0 & 0 & 1 \\ | ||

822 | \hline | ||

823 | 0 & 1 & 1 & 0 \\ | ||

824 | \hline | ||

825 | 1 & 0 & 1 & 1 \\ | ||

826 | \hline | ||

827 | 1 & 1 & 0 & 0 \\ | ||

828 | \hline | ||

829 | \end{tabular} | ||

830 | \end{center} | ||

831 | \end{table} | ||

832 | |||

833 | From Table \ref{tbl:cbal0:svct0:stbf0:01} it can be readily verified | ||

834 | even without a Karnaugh map that the least significant bit | ||

835 | ($B$) is always complemented from one state to the next, so that | ||

836 | $B_{k+1} = \neg B_k$. It can also be seen that the most significant | ||

837 | bit ($A$) is complemented from one state to the next only if $B$=1, | ||

838 | so that $A_{k+1} = A_k \oplus B_k$. | ||

839 | |||

840 | The implementation of such a counter in `C' comes immediately, since | ||

841 | `C' directly supports bit-wise complementation and exclusive-OR of | ||

842 | integers: | ||

843 | |||

844 | \begin{verbatim} | ||

845 | A = A ^ B; | ||

846 | B = ~B; | ||

847 | \end{verbatim} | ||

848 | |||

849 | Note in the `C' snippet above, the two operations are ordered so as | ||

850 | not to interfere with each other. Note that reversing the two steps above | ||

851 | would not yield the desired result. | ||

852 | |||

853 | |||

854 | \subsection{3-Bit 8-State Cyclic Vertical Counter} | ||

855 | %Subsection Tag: TBF4 | ||

856 | \label{cbal0:svct0:stbf4} | ||

857 | |||

858 | |||

859 | \subsection[$N$-Bit $2^N$-State Cyclic Vertical Counter] | ||

860 | {\mbox{\boldmath $N$}-Bit \mbox{\boldmath $2^N$}-State Cyclic Vertical Counter} | ||

861 | %Subsection Tag: TBF1 | ||

862 | \label{cbal0:svct0:stbf1} | ||

863 | |||

864 | The design of the 2-bit 4-state vertical counter presented in | ||

865 | Section \ref{cbal0:svct0:stbf0} (immediately above) and be | ||

866 | generalized to an $N$-bit $2^N$-state counter which follows | ||

867 | the traditional binary integer counting sequence. | ||

868 | |||

869 | Note from Section \ref{cbal0:svct0:stbf0} that the least | ||

870 | significant bit of the counter is always complemented | ||

871 | in going from one state to the next, and that each more | ||

872 | significant bit is complemented only if all less significant | ||

873 | bits are 1. If $X_0$ is the least significant bit of the | ||

874 | counter and $X_N$ the most significant bit, | ||

875 | (\ref{eq:cbal0:svct0:stbf1:01}) through (\ref{eq:cbal0:svct0:stbf1:05}) | ||

876 | (Figure \ref{eq:cbal0:svct0:stbf1:01}) | ||

877 | describe how the counter is advanced from one state to the next. | ||

878 | Note that this set of equations gives only a \emph{logical} description | ||

879 | of how to obtain the next state from the present state---these | ||

880 | equations cannot be used as a direct blueprint for implementation because | ||

881 | earlier steps would destroy the results needed for later steps. | ||

882 | |||

883 | \begin{figure} | ||

884 | \begin{eqnarray} | ||

885 | \label{eq:cbal0:svct0:stbf1:01} | ||

886 | X_0(k+1) & = & \neg X_0(k) \\ | ||

887 | \label{eq:cbal0:svct0:stbf1:02} | ||

888 | X_1(k+1) & = & X_1(k) \oplus X_0(k) \\ | ||

889 | \label{eq:cbal0:svct0:stbf1:03} | ||

890 | X_2(k+1) & = & X_2(k) \oplus (X_1(k) X_0(k)) \\ | ||

891 | \label{eq:cbal0:svct0:stbf1:04} | ||

892 | X_3(k+1) & = & X_3(k) \oplus (X_2(k) X_1(k) X_0(k)) \\ | ||

893 | & \ldots & \nonumber \\ | ||

894 | \label{eq:cbal0:svct0:stbf1:05} | ||

895 | X_N(k+1) & = & X_N(k) \oplus (X_{N-1}(k) \ldots X_0(k)) | ||

896 | \end{eqnarray} | ||

897 | \caption{State Transition Equations Of $N$-Bit $2^N$-State Cyclic Vertical Counter} | ||

898 | \label{fig:cbal0:svct0:stbf1:01} | ||

899 | \end{figure} | ||

900 | |||

901 | The form of | ||

902 | (\ref{eq:cbal0:svct0:stbf1:01}) through (\ref{eq:cbal0:svct0:stbf1:05}) | ||

903 | suggests a way to economically implement an $N$-bit $2^N$-state | ||

904 | cyclic vertical counter in software, using two temporary | ||

905 | variables $TEMP_A$ and $TEMP_B$. The general blueprint for | ||

906 | implementation is supplied as | ||

907 | (\ref{eq:cbal0:svct0:stbf1:06}) through (\ref{eq:cbal0:svct0:stbf1:20}) | ||

908 | (Figure \ref{eq:cbal0:svct0:stbf1:02}). | ||

909 | |||

910 | \begin{figure} | ||

911 | \begin{eqnarray} | ||

912 | \label{eq:cbal0:svct0:stbf1:06} | ||

913 | TEMP_A & = & X_0 \\ | ||

914 | \label{eq:cbal0:svct0:stbf1:07} | ||

915 | X_0 & = & \neg X_0 \\ | ||

916 | \label{eq:cbal0:svct0:stbf1:08} | ||

917 | TEMP_B & = & X_1 TEMP_A \\ | ||

918 | \label{eq:cbal0:svct0:stbf1:09} | ||

919 | X_1 & = & X_1 \oplus TEMP_A \\ | ||

920 | \label{eq:cbal0:svct0:stbf1:10} | ||

921 | TEMP_A & = & TEMP_B \\ | ||

922 | \label{eq:cbal0:svct0:stbf1:11} | ||

923 | TEMP_B & = & X_2 TEMP_B \\ | ||

924 | \label{eq:cbal0:svct0:stbf1:12} | ||

925 | X_2 & = & X_2 \oplus TEMP_A \\ | ||

926 | \label{eq:cbal0:svct0:stbf1:13} | ||

927 | TEMP_A & = & TEMP_B \\ | ||

928 | \label{eq:cbal0:svct0:stbf1:14} | ||

929 | TEMP_B & = & X_3 TEMP_B \\ | ||

930 | \label{eq:cbal0:svct0:stbf1:15} | ||

931 | X_3 & = & X_3 \oplus TEMP_A \\ | ||

932 | & \ldots & \nonumber \\ | ||

933 | \label{eq:cbal0:svct0:stbf1:16} | ||

934 | X_{N-2} & = & X_{N-2} \oplus TEMP_A \\ | ||

935 | \label{eq:cbal0:svct0:stbf1:17} | ||

936 | TEMP_A & = & TEMP_B \\ | ||

937 | \label{eq:cbal0:svct0:stbf1:18} | ||

938 | TEMP_B & = & X_{N-1} TEMP_B \\ | ||

939 | \label{eq:cbal0:svct0:stbf1:19} | ||

940 | X_{N-1} & = & X_{N-1} \oplus TEMP_A \\ | ||

941 | \label{eq:cbal0:svct0:stbf1:20} | ||

942 | X_N & = & X_N \oplus X_N TEMP_B | ||

943 | \end{eqnarray} | ||

944 | \caption{Implementation Blueprint Of $N$-Bit $2^N$-State Cyclic Vertical Counter} | ||

945 | \label{fig:cbal0:svct0:stbf1:02} | ||

946 | \end{figure} | ||

947 | |||

948 | Figure \ref{fig:cbal0:svct0:stbf1:01} | ||

949 | supplies a concrete example of a C-language | ||

950 | implementation of a 5-bit 32-state cyclic vertical counter. | ||

951 | Note that the counter will follow the traditional | ||

952 | binary integer counting sequence. Note also that | ||

953 | the blueprint provided by Eqns. | ||

954 | (\ref{eq:cbal0:svct0:stbf1:06}) through (\ref{eq:cbal0:svct0:stbf1:20}) | ||

955 | and Figure \ref{fig:cbal0:svct0:stbf1:01} can | ||

956 | be extended to a counter of any size. | ||

957 | |||

958 | \begin{figure} | ||

959 | \begin{verbatim} | ||

960 | /**************************************************************/ | ||

961 | /* Assume: */ | ||

962 | /* x4 : Byte containing most significant bits of the */ | ||

963 | /* a group of 8 bits. */ | ||

964 | /* x3, */ | ||

965 | /* x2, */ | ||

966 | /* x1 : Bytes containing the intermediate bits. */ | ||

967 | /* x0 : Byte containing the least significant bits. */ | ||

968 | /* temp_a, */ | ||

969 | /* temp_b : Temporary variables, used to accumulate the */ | ||

970 | /* result of AND'ing old counter bit values. */ | ||

971 | /**************************************************************/ | ||

972 | |||

973 | temp_a = x0; | ||

974 | x0 = ~x0; | ||

975 | temp_b = x1 & temp_a; | ||

976 | x1 = x1 ^ temp_a; | ||

977 | temp_a = temp_b; | ||

978 | temp_b = x2 & temp_b; | ||

979 | x2 = x2 ^ temp_a; | ||

980 | temp_a = temp_b; | ||

981 | temp_b = x3 & temp_b; | ||

982 | x3 = x3 ^ temp_a; | ||

983 | x4 = x4 ^ temp_b; | ||

984 | |||

985 | /* End of code. */ | ||

986 | \end{verbatim} | ||

987 | \caption{C-Language Implementation Of 5-Bit 32-State Cyclic Vertical Counter} | ||

988 | \label{fig:cbal0:svct0:stbf1:01b} | ||

989 | \end{figure} | ||

990 | |||

991 | |||

992 | \subsection{3-Bit 5-State Cyclic Vertical Counter} | ||

993 | %Subsection Tag: TBF5 | ||

994 | \label{cbal0:svct0:stbf5} | ||

995 | |||

996 | |||

997 | \subsection{3-Bit 6-State Cyclic Vertical Counter} | ||

998 | %Subsection Tag: TBF6 | ||

999 | \label{cbal0:svct0:stbf6} | ||

1000 | |||

1001 | |||

1002 | \subsection{3-Bit 7-State Cyclic Vertical Counter} | ||

1003 | %Subsection Tag: TBF7 | ||

1004 | \label{cbal0:svct0:stbf7} | ||

1005 | |||

1006 | |||

1007 | \subsection{2-Bit 4-State Terminating Vertical Counter} | ||

1008 | %Subsection Tag: TBF8 | ||

1009 | \label{cbal0:svct0:stbf8} | ||

1010 | |||

1011 | |||

1012 | \subsection{3-Bit 5-State Terminating Vertical Counter} | ||

1013 | %Subsection Tag: TBF9 | ||

1014 | \label{cbal0:svct0:stbf9} | ||

1015 | |||

1016 | |||

1017 | \subsection{3-Bit 6-State Terminating Vertical Counter} | ||

1018 | %Subsection Tag: TBG0 | ||

1019 | \label{cbal0:svct0:stbg0} | ||

1020 | |||

1021 | |||

1022 | \subsection{3-Bit 7-State Terminating Vertical Counter} | ||

1023 | %Subsection Tag: TBG1 | ||

1024 | \label{cbal0:svct0:stbg1} | ||

1025 | |||

1026 | |||

1027 | \subsection{2/3 Debouncing Vertical Counter} | ||

1028 | %Subsection Tag: TTD0. | ||

1029 | \label{cbal0:svct0:sttd0} | ||

1030 | |||

1031 | \index{debouncing}\index{debouncing!2/3}\emph{2/3 debouncing} | ||

1032 | is debouncing where at least two of the most recent | ||

1033 | three samples must be at the same value (either 0 or 1) to cause the output | ||

1034 | to be that same value. | ||

1035 | |||

1036 | Note that \emph{2/3 debouncing} as we describe it here represents a | ||

1037 | purely combinational mapping from the 3 most recent samples to | ||

1038 | the debounced output. When 3 samples are available, 0 of the | ||

1039 | samples or 1 of the samples with value 1 map to a debounced | ||

1040 | output value of 0; while 2 of the samples or 3 of the samples | ||

1041 | with value 1 map to a debounced output value of 1. | ||

1042 | |||

1043 | If $A$ is the most recent sample, $B$ is the next-most-recent sample, and | ||

1044 | $C$ is the oldest sample, Fig. \ref{fig:cbal0:svct0:sttd0:01} | ||

1045 | supplies the Karnaugh map for 2/3 | ||

1046 | debouncing. | ||

1047 | |||

1048 | \begin{figure} | ||

1049 | \centering | ||

1050 | \includegraphics[height=1.25in]{c_bal0/kmap23db.eps} | ||

1051 | \caption{Karnaugh Map Of 2/3 Debouncing} | ||

1052 | \label{fig:cbal0:svct0:sttd0:01} | ||

1053 | \end{figure} | ||

1054 | |||

1055 | It can be seen from the figure that the expression for the output is | ||

1056 | |||

1057 | \begin{equation} | ||

1058 | \label{eq:cbal0:svct0:sttd0:01} | ||

1059 | AB + AC + BC = A (B+C) + BC . | ||

1060 | \end{equation} | ||

1061 | |||

1062 | Intuitively, (\ref{eq:cbal0:svct0:sttd0:01}) makes sense---the output | ||

1063 | will be 1 if any two of the most recent samples are 1 | ||

1064 | ($AB + AC + BC$). Similarly, the output will be 0 if any two of | ||

1065 | the most recent samples are 0. | ||

1066 | |||

1067 | Figure \ref{fig:cbal0:svct0:sttd0:02} supplies the C-language | ||

1068 | code to implement 2/3 debouncing as a vertical mapping. | ||

1069 | A C-compiler will typically implement this code very directly | ||

1070 | using the bitwise logical instructions of the machine. | ||

1071 | |||

1072 | \begin{figure} | ||

1073 | \begin{verbatim} | ||

1074 | /**************************************************************/ | ||

1075 | /* Assume: */ | ||

1076 | /* A : Most recent sample (i.e. at t(0)), arranged as */ | ||

1077 | /* a group of 8 bits. */ | ||

1078 | /* B : Next most recent sample t(-1). */ | ||

1079 | /* C : Oldest sample t(-2). */ | ||

1080 | /* output : Debounced collection of 8 bits presented to */ | ||

1081 | /* software internals. */ | ||

1082 | /**************************************************************/ | ||

1083 | |||

1084 | output = (A & (B | C)) | (B & C); | ||

1085 | |||

1086 | /* End of code. */ | ||

1087 | \end{verbatim} | ||

1088 | \caption{C-Language Implementation Of 2/3 Debouncing} | ||

1089 | \label{fig:cbal0:svct0:sttd0:02} | ||

1090 | \end{figure} | ||

1091 | |||

1092 | \subsection{3/3 Debouncing Vertical Counter} | ||

1093 | %Subsection Tag: TTD1. | ||

1094 | \label{cbal0:svct0:sttd1} | ||

1095 | |||

1096 | \index{debouncing}\index{debouncing!3/3}\emph{3/3 debouncing} | ||

1097 | is debouncing where all three of the most recent | ||

1098 | three samples must be at a value (either 0 or 1) to cause the | ||

1099 | debounced output to transition to that same value. | ||

1100 | |||

1101 | Note that 3/3 debouncing as we present it is a sequential (rather than | ||

1102 | a purely combinational) mapping from the 3 most recent samples to the | ||

1103 | debounced output. In addition to the 3 inputs, the behavior of the | ||

1104 | mapping depends on a single bit of state which is held. (In the | ||

1105 | implementation presented in this section, the state and the debounced | ||

1106 | output are the same bit.) For example, if 2 of the 3 most recent | ||

1107 | input samples are 1, the debounced output value may be either 0 | ||

1108 | or 1. The transition of the debounced output value from | ||

1109 | 0 to 1 or from 1 to 0 can only occur if all 3 most recent samples | ||

1110 | are 1 or are 0, respectively. | ||

1111 | |||

1112 | If $A$ is the most recent sample, $B$ is the next-most-recent sample, | ||

1113 | $C$ is the oldest sample, and $O$ is the output (assumed maintained | ||

1114 | as a RAM location) Fig. \ref{fig:cbal0:svct0:sttd1:01} | ||

1115 | supplies the Karnaugh map for 3/3 | ||

1116 | debouncing. | ||

1117 | |||

1118 | \begin{figure} | ||

1119 | \centering | ||

1120 | \includegraphics[height=2.0in]{c_bal0/kmap33db.eps} | ||

1121 | \caption{Karnaugh Map Of 3/3 Debouncing} | ||

1122 | \label{fig:cbal0:svct0:sttd1:01} | ||

1123 | \end{figure} | ||

1124 | |||

1125 | It can be seen from the figure that the expression for the output is | ||

1126 | |||

1127 | \begin{equation} | ||

1128 | \label{eq:cbal0:svct0:sttd1:01} | ||

1129 | ABC + AO + BO + CO = ABC + O(A + B + C). | ||

1130 | \end{equation} | ||

1131 | |||

1132 | Intuitively, (\ref{eq:cbal0:svct0:sttd1:01}) makes sense---the output | ||

1133 | will be unconditionally 1 if all three of the most recent samples are 1 | ||

1134 | ($ABC$). The output will also be 1 if the previous output was 1 | ||

1135 | and at least one of the most recent samples are 1 [$O(A+B+C)$]---at least | ||

1136 | one true recent sample blocks the output from transition to 0. | ||

1137 | |||

1138 | Figure \ref{fig:cbal0:svct0:sttd1:02} supplies the C-language | ||

1139 | code to implement 3/3 debouncing as a vertical mapping. | ||

1140 | A C-compiler will typically implement this code very directly | ||

1141 | using the bitwise logical instructions of the machine. | ||

1142 | |||

1143 | \begin{figure} | ||

1144 | \begin{verbatim} | ||

1145 | /**************************************************************/ | ||

1146 | /* Assume: */ | ||

1147 | /* A : Most recent sample (i.e. at t(0)), arranged as */ | ||

1148 | /* a group of 8 bits. */ | ||

1149 | /* B : Next most recent sample t(-1). */ | ||

1150 | /* C : Oldest sample t(-2). */ | ||

1151 | /* output : Debounced collection of 8 bits presented to */ | ||

1152 | /* software internals. Note that this is both */ | ||

1153 | /* an input (to the combinational mapping) and */ | ||

1154 | /* the new result. */ | ||

1155 | /**************************************************************/ | ||

1156 | |||

1157 | output = (A & B & C) | (output & (A | B | C)); | ||

1158 | |||

1159 | /* End of code. */ | ||

1160 | \end{verbatim} | ||

1161 | \caption{C-Language Implementation Of 3/3 Debouncing} | ||

1162 | \label{fig:cbal0:svct0:sttd1:02} | ||

1163 | \end{figure} | ||

1164 | |||

1165 | \subsection{N/N Debouncing Vertical Counter} | ||

1166 | %Subsection Tag: NNC0. | ||

1167 | \label{cbal0:svct0:snnc0} | ||

1168 | |||

1169 | \index{debouncing}\index{debouncing!N/N}It is clear from | ||

1170 | the design of the 3/3 debouncing | ||

1171 | vertical counter (Section \ref{cbal0:svct0:sttd1}, immediately | ||

1172 | above) that the design of the 3/3 debouncing vertical counter | ||

1173 | can be generalized to cover any number of recent samples. | ||

1174 | We call such a debouncing vertical counter an N/N debouncing | ||

1175 | vertical counter. In order for the output of such a counter to | ||

1176 | turn 1, all $N$ most recent samples must be 1. Similarly, in order | ||

1177 | for such a counter to turn 0, all $N$ most recent samples must be 0. | ||

1178 | |||

1179 | We rely on a logical argument to design such a N/N debouncing | ||

1180 | vertical counter, rather than on Karnaugh maps. If $I_1 \ldots I_N$ | ||

1181 | are the $N$ most recent input samples and $O$ is the most recent | ||

1182 | output (assumed stored in RAM), it is clear from the form of | ||

1183 | (\ref{eq:cbal0:svct0:sttd1:01}) that the formula for the new output | ||

1184 | as a function of the inputs $I_1 \ldots I_N$ and the | ||

1185 | most recent output $O_{k-1}$ must be: | ||

1186 | |||

1187 | \begin{equation} | ||

1188 | \label{eq:cbal0:svct0:snnc0:01} | ||

1189 | O_k = I_1 I_2 \ldots I_N + (O_{k-1} (I_1 + I_2 + \ldots + I_N)) . | ||

1190 | \end{equation} | ||

1191 | |||

1192 | The form of (\ref{eq:cbal0:svct0:snnc0:01}) is intuitively | ||

1193 | plausible. If the previous output $O_{k-1}$ is 0, the only | ||

1194 | circumstance under which the next output $O_k$ will become 1 | ||

1195 | is if all $N$ inputs $I_1, I_2, \ldots , I_N$ are 1. | ||

1196 | Similarly, if the previous output $O_{k-1}$ is 1, the only | ||

1197 | circumstance under which $O_k$ will become 0 | ||

1198 | is if all $N$ inputs $I_1, I_2, \ldots , I_N$ are 0. This | ||

1199 | is the desired behavior. | ||

1200 | |||

1201 | For an N/N counter when a large number of previous samples | ||

1202 | are required to be at the same value (say, $N=10$, for example), | ||

1203 | it may seem that the efficiency of the N/N vertical counter | ||

1204 | design presented here will break down. However, it must be | ||

1205 | remembered that the vertical counter approach manipulates | ||

1206 | a large number (usually 8 or 16) inputs at a time. It can be | ||

1207 | shown easily that an N/N debouncing vertical counter requires | ||

1208 | $2N$ operations to determine the next output. Thus, for a | ||

1209 | 10/10 debouncing vertical counter, the necessary software | ||

1210 | may execute in as little as 20 machine instructions. It would | ||

1211 | be difficult to find another method which will perform 10/10 | ||

1212 | debouncing for 8 or 16 inputs in only 20 machine instructions, thus | ||

1213 | we maintain that the N/N debouncing vertical counter | ||

1214 | approach presented is quite efficient even for larger | ||

1215 | $N$. | ||

1216 | |||

1217 | |||

1218 | \section{Authors And Acknowledgements} | ||

1219 | %Section tag: ACK0 | ||

1220 | This chapter was primarily written by David T. Ashley | ||

1221 | \cite{bibref:i:daveashley}. | ||

1222 | |||

1223 | We are very grateful to \index{Dattalo, Scott}Scott Dattalo | ||

1224 | \cite{bibref:i:scottdattalo}, | ||

1225 | whose web pages about vertical counters provided much of the | ||

1226 | material for the Section \ref{cbal0:svct0}. | ||

1227 | Special thanks to \index{Virgil}Virgil \cite{bibref:i:virgil}, | ||

1228 | \index{Dresner, Norm}Norm Dresner \cite{bibref:i:normdresner}, | ||

1229 | and \index{Kaskelma, Heikki}Heikki Kaskelma \cite{bibref:i:heikkikaskelma} | ||

1230 | for mathematical assistance provided via the | ||

1231 | \index{sci.math newsgroup@\texttt{sci.math} newsgroup}% | ||

1232 | \texttt{sci.math} \cite{bibref:n:scimathnewsgroup} newsgroup. | ||

1233 | |||

1234 | |||

1235 | \section{Exercises} | ||

1236 | |||

1237 | |||

1238 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||

1239 | |||

1240 | \noindent\begin{figure}[!b] | ||

1241 | \noindent\rule[-0.25in]{\textwidth}{1pt} | ||

1242 | \begin{tiny} | ||

1243 | \begin{verbatim} | ||

1244 | dashley | 275 | $HeadURL$ |

1245 | $Revision$ | ||

1246 | $Date$ | ||

1247 | $Author$ | ||

1248 | dashley | 140 | \end{verbatim} |

1249 | \end{tiny} | ||

1250 | \noindent\rule[0.25in]{\textwidth}{1pt} | ||

1251 | \end{figure} | ||

1252 | |||

1253 | dashley | 275 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

1254 | dashley | 140 | % |

1255 | %End of file C_BAL0.TEX |

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