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1:- :~ navhome/'/docs/'

2 sort/'11'

3 title/'2h: Set Logic'

4 ==

6;>

8# 2h: set logic

10### `++in`

12Set operations

14Core whose arms contain a variety of functions that operate on `set`s. Its

15sample accepts the input `set` to be manipulated.

17#### Accepts

19A `set`.

21#### Source

23```

24 ++ in

25 ~/ %in

26 |_ a=(set)

27```

29#### Examples

31```

32 > ~(. in (sy "asd"))

33 <13.evb [nlr(^$1{@tD $1}) <414.fvk 101.jzo 1.ypj %164>]>

34```

36### `+-all:in`

38Logical AND

40Computes the logical AND on every element in `a` slammed with `b`, producing a

41flag.

43#### Accepts

45`a` is a `set`.

47`b` is a wet gate that accepts a noun and produces a flag.

49#### Produces

51A flag.

53#### Source

55```

56 +- all

57 ~/ %all

58 |* b=$-(* ?)

59 |- ^- ?

60 ?~ a

61 &

62 ?&((b n.a) $(a l.a) $(a r.a))

63```

65#### Examples

67```

68 > =b (sy `(list [@t *])`[['a' 1] ['b' [2 3]] ~])

69 > (~(all in b) |=(a=* ?@(+.a & |)))

70 %.n

72 > =b (sy `(list @t)`['john' 'bonita' 'daniel' 'madeleine' ~])

73 > (~(all in b) |=(a=@t (gte a 100)))

74 %.y

75```

77---

78### `+-any:in`

80Logical OR

82Computes the logical OR on every element of `a` slammed with `b`, producing a

83flag.

85#### Accepts

87`a` is a `set`.

89`b` is a gate that accepts a noun and produces a flag.

91#### Produces

93A flag.

95#### Source

97```

98 +- any

99 ~/ %any

100 |* b=$-(* ?)

101 |- ^- ?

102 ?~ a

103 |

104 ?|((b n.a) $(a l.a) $(a r.a))

105```

106

107#### Examples

108

109```

110 > =b (sy `(list [@t *])`[['a' 1] ['b' [2 3]] ~])

111 > (~(any in b) |=(a/* ?@(+.a & |)))

112 %.y

113

114 > =b (sy `(list @t)`['john' 'bonita' 'daniel' 'madeleine' ~])

115 > (~(any in b) |=(a=@t (lte a 100)))

116 %.n

117```

118

119---

120#### `+-apt:in`

121

122Check correctness

123

124Computes whether `a` has a correct horizontal order and a correct vertical

125order, producing a flag.

127#### Accepts

129`a` is a set.

131#### Produces

133A flag.

135#### Source

137```

138 +- apt

139 =| [l=(unit) r=(unit)]

140 |- ^- ?

141 ?~ a &

142 ?& ?~(l & (hor n.a u.l))

143 ?~(r & (hor u.r n.a))

144 ?~(l.a & ?&((vor n.a n.l.a) $(a l.a, l `n.a)))

145 ?~(r.a & ?&((vor n.a n.r.a) $(a r.a, r `n.a)))

146 ==

147```

148

149#### Examples

150

151```

152 > ~(apt in ~)

153 %.y

154

155 > =a (sy [1 2 3 4 5 6 ~])

156 > a

157 [n=6 l={5 4} r={1 3 2}]

158 > ~(apt in a)

159 %.y

160

161 > =z ?~ a ~ a(n 7)

162 > z

163 [n=7 l={5 4} r={1 3 2}]

164 > ~(apt in z)

165 %.n

166```

167

168#### Discussion

169

170See section [`2f`](../2f) for more information on noun ordering.

171

172---

173### `+-bif:in`

174

175Bifurcate

176

177Splits set `a` into two sets `l` and `r`, with any noun `b` removed.

179#### Accepts

181`a` is a set.

183`b` is a noun.

185#### Produces

187A cell of two sets.

188

189#### Source

190

191```

192 +- bif

193 ~/ %bif

194 |* b=*

195 ^+ [l=a r=a]

196 =< [+< +>]

197 |- ^+ a

198 ?~ a

199 [b ~ ~]

200 ?: =(b n.a)

201 a

202 ?: (hor b n.a)

203 =+ c=$(a l.a)

204 ?> ?=(^ c)

205 [n.c l.c [n.a r.c r.a]]

206 =+ c=$(a r.a)

207 ?> ?=(^ c)

208 [n.c [n.a l.a l.c] r.c]

209```

210

211#### Examples

212

213```

214 > =a (sy `(list @)`[1 2 3 4 5 6 ~])

215 > a

216 [n=6 l={5 4} r={1 3 2}]

217

218 > (~(bif in a) 6)

219 [l=[n=4 l=[n=5 l={} r={}] r=~] r=[n=2 l=[n=1 l={} r={3}] r=~]]

220

221 > (~(bif in a) 100)

222 [l=[n=6 l=[n=4 l={5} r={}] r=[n=2 l={1 3} r={}]] r=~]

223```

224

225---

226### `+-del:in`

227

228Remove noun

229

230Removes `b` from the `set` `a`.

232#### Accepts

234`a` is a set.

236`b` is a noun.

238#### Produces

240A set.

242#### Source

244```

245 +- del

246 ~/ %del

247 |* b=*

248 |- ^+ a

249 ?~ a

250 ~

251 ?. =(b n.a)

252 ?: (hor b n.a)

253 [n.a $(a l.a) r.a]

254 [n.a l.a $(a r.a)]

255 |- ^- {$?($~ _a)}

256 ?~ l.a r.a

257 ?~ r.a l.a

258 ?: (vor n.l.a n.r.a)

259 [n.l.a l.l.a $(l.a r.l.a)]

260 [n.r.a $(r.a l.r.a) r.r.a]

261```

262

263

264#### Examples

265

266```

267 > =b (sy `(list @t)`['a' 'b' 'c' ~])

268 > (~(del in b) 'a')

269 [n='b' l=[n='c' l={} r={}] r=~]

270

271 > =b (sy `(list @t)`['john' 'bonita' 'daniel' 'madeleine' ~])

272 > (~(del in b) 'john')

273 [n='madeleine' l=[n='daniel' l={} r={'bonita'}] r=~]

274

275 > (~(del in b) 'susan')

276 [n='madeleine' l=[n='daniel' l={} r={'bonita'}] r=[n='john' l={} r={}]]

277```

278

279---

280### `+-dif:in`

281

282Difference

283

284Computes the difference between `a` and `b`, producing the set of items in `a`

285that are not in `b`.

287#### Accepts

289`a` is a set.

291`b` is a set.

293#### Produces

295A set.

297#### Source

299```

300 +- dif

301 ~/ %dif

302 |* b=_a

303 |- ^+ a

304 ?~ b

305 a

306 =+ c=(bif n.b)

307 ?> ?=(^ c)

308 =+ d=$(a l.c, b l.b)

309 =+ e=$(a r.c, b r.b)

310 |- ^- {$?($~ _a)}

311 ?~ e d

312 ?: (vor n.d n.e)

313 [n.d l.d $(d r.d)]

314 [n.e $(e l.e) r.e]

315```

316

317#### Examples

318

319```

320 > =c (sy 1 2 3 ~)

321 > =d (sy 2 3 4 ~)

322 > (~(dif in c) d)

323 [n=1 l=~ r=~]

324

325 > (~(dif in d) c)

326 [n=4 l=~ r=~]

327

328 > `(set @ud)`(~(dif in c) d)

329 {1}

330

331 > `(set @ud)`(~(dif in d) c)

332 {4}

333```

334

335---

336### `+-dig:in`

337

338Address b in a

339

340Produce the tree address of `b` within `a`.

342#### Accepts

344`a` is a set.

346`b` is a noun.

348#### Produces

350The `unit` of an atom.

351

352#### Source

353

354```

355 +- dig

356 |= b=*

357 =+ c=1

358 |- ^- (unit @)

359 ?~ a ~

360 ?: =(b n.a) [~ u=(peg c 2)]

361 ?: (hor b n.a)

362 $(a l.a, c (peg c 6))

363 $(a r.a, c (peg c 7))

364```

365

366

367#### Examples

368

369```

370 > =a (sy `(list @)`[1 2 3 4 5 6 7 ~])

371 > `(set @)`a

372 {5 4 7 6 1 3 2}

373

374 > -.a

375 n=6

376

377 > (~(dig in a) 7)

378 [~ u=12]

379

380 > (~(dig in a) 2)

381 [~ u=14]

382

383 > (~(dig in a) 6)

384 [~ u=2]

385```

386

387#### Discussion

388

389For more on the tree addressing system, see section [1b](../1b).

390

391---

392### `+-gas:in`

393

394Concatenate

395

396Insert the elements of a `list` `b` into a `set` `a`.

398#### Accepts

400`a` is a set.

402`b` is a list.

404#### Produces

406A `set`.

408#### Source

410```

411 +- gas

412 ~/ %gas

413 |= b=(list _?>(?=(^ a) n.a))

414 |- ^+ a

415 ?~ b

416 a

417 $(b t.b, a (put i.b))

418```

419

420#### Examples

421

422```

423 > =b `(set @t)`(sy `(list @t)`['bonita' 'madeleine' 'rudolf' 'john' ~])

424 > b

425 {'bonita' 'madeleine' 'rudolf' 'john'}

426

427 > (~(gas in b) `(list @t)`['14' 'things' 'number' '1.337' ~])

428 [n='number' l={'1.337' '14'} r={'things' 'bonita' 'madeleine' 'rudolf' 'john'}]

429

430 > =s (sy `(list @t)`['a' 'b' 'c' 'd' 'e' ~])

431 > (~(gas in s) `(list @t)`['1' '2' '3' ~])

432 [n='e' l=[n='1' l={} r={'3' '2'}] r=[n='b' l={'d' 'a' 'c'} r={}]]

433```

434

435---

436### `+-has:in`

437

438b in a?

439

440Checks if `b` is an element of `a`, producing a flag.

442#### Accepts

444`a` is a set.

446`b` is a noun.

448#### Produces

450A flag.

452#### Source

454```

455 +- has

456 ~/ %has

457 |* b=*

458 |- ^- ?

459 ?~ a

460 |

461 ?: =(b n.a)

462 &

463 ?: (hor b n.a)

464 $(a l.a)

465 $(a r.a)

466```

467

468#### Examples

469

470```

471 > =a (~(gas in `(set @t)`~) `(list @t)`['a' 'b' 'c' ~])

472 > (~(has in a) 'a')

473 %.y

474

475 > (~(has in a) 'z')

476 %.n

477```

478

479---

480### `+-int:in`

481

482Intersection

483

484Produces a set of the intersection between two sets of the same type,

485`a` and `b`.

487#### Accepts

489`a` is a set.

491`b` is a set.

493#### Produces

495A `set`.

497#### Source

499```

500 +- int

501 ~/ %int

502 |* b=_a

503 |- ^+ a

504 ?~ b

505 ~

506 ?~ a

507 ~

508 ?. (vor n.a n.b)

509 $(a b, b a)

510 ?: =(n.b n.a)

511 [n.a $(a l.a, b l.b) $(a r.a, b r.b)]

512 ?: (hor n.b n.a)

513 %- uni(a $(a l.a, b [n.b l.b ~])) $(b r.b)

514 %- uni(a $(a r.a, b [n.b ~ r.b])) $(b l.b)

515```

516

517

518#### Examples

519

520```

521 > (~(int in `(set @t)`(sy "acmo")) `(set @t)`(sy "ham"))

522 [n='m' l={'a'} r={}]

523

524 > (~(int in (sy "acmo")) ~)

525 ~

526

527 > (~(int in `(set @t)`(sy "ac")) `(set @t)`(sy "ha"))

528 [n='a' l={} r={}]

529```

530

531---

532### `+-put:in`

533

534Put b in a

535

536Add an element `b` to the set `a`, producing a set.

538#### Accepts

540`a` is a set.

542`b` is a noun.

544#### Produces

546A `set`.

548#### Source

550```

551 +- put

552 ~/ %put

553 |* b=*

554 |- ^+ a

555 ?~ a

556 [b ~ ~]

557 ?: =(b n.a)

558 a

559 ?: (hor b n.a)

560 =+ c=$(a l.a)

561 ?> ?=(^ c)

562 ?: (vor n.a n.c)

563 [n.a c r.a]

564 [n.c l.c [n.a r.c r.a]]

565 =+ c=$(a r.a)

566 ?> ?=(^ c)

567 ?: (vor n.a n.c)

568 [n.a l.a c]

569 [n.c [n.a l.a l.c] r.c]

570```

571

572#### Examples

573

574```

575 > =a (~(gas in `(set @t)`~) `(list @t)`['a' 'b' 'c' ~])

576 > =b (~(put in a) 'd')

577 > `(set @t)`b

578 {'d' 'a' 'c' 'b'}

579

580 > -.l.+.b

581 n='d'

582```

583

584---

585### `+-rep:in`

586

587Accumulate

588

589Accumulate the elements of `a` using a gate `b`.

591#### Accepts

593`a` is a `set`.

595`b` is a gate.

597#### Produces

599A noun.

601#### Source

603```

604 +- rep :: replace by product

605 |* b=_|=([* *] +<+)

606 |-

607 ?~ a +<+.b

608 $(a r.a, +<+.b $(a l.a, +<+.b (b n.a +<+.b)))

609```

610

611#### Examples

612

613```

614 > =a (~(gas in *(set @)) [1 2 3 ~])

615 > `(set @)`a

616 {1 3 2}

617

618 > (~(rep in a) |=([a=@ b=@] (add a b)))

619 b=6

620```

621

622---

623### `+-run:in`

624

625Apply gate to set

626

627Produce a `set` containing the products of gate `b` applied to each element

628in `a`.

630#### Accepts

632`a` is a `set`.

634`b` is a gate.

636#### Produces

638A set.

640#### Source

642```

643 +- run

644 ~/ %run

645 |* b=gate

646 =| c=(set _?>(?=(^ a) (b n.a)))

647 |- ?~ a c

648 =. c (~(put in c) (b n.a))

649 =. c $(a l.a, c c)

650 $(a r.a, c c)

651```

652

653#### Examples

654

655```

656 > =s (sy ~["a" "A" "b" "c"])

657 > `(set tape)`s

658 {"A" "a" "c" "b"}

659

660 > (~(run in s) cuss)

661 {"A" "C" "B"}

662```

663

664---

665### `+-tap:in`

666

667Set to list

668

669Flattens the `set` `a` into a `list`.

671#### Accepts

673`a` is an set.

675#### Produces

677A list.

679#### Source

681```

682 +- tap

683 ~/ %tap

684 |= b=(list _?>(?=(^ a) n.a))

685 ^+ b

686 ?~ a

687 b

688 $(a r.a b [n.a $(a l.a)])

689```

690

691

692#### Examples

693

694```

695 > =s (sy `(list @t)`['a' 'b' 'c' 'd' 'e' ~])

696 > s

697 [n='e' l={} r={'d' 'a' 'c' 'b'}]

698

699 > ~(tap in s)

700 ~['b' 'c' 'a' 'd' 'e']

701```

702

703---

704### `+-uni:in`

705

706Union

707

708Produces a set of the union between two sets of the same type, `a` and

709`b`.

711#### Accepts

713`a` is a set.

715`b` is a set.

717#### Produces

719A set.

721#### Source

723```

724 +- uni

725 ~/ %uni

726 |* b=_a

727 ?: =(a b) a

728 |- ^+ a

729 ?~ b

730 a

731 ?~ a

732 b

733 ?: (vor n.a n.b)

734 ?: =(n.b n.a)

735 [n.b $(a l.a, b l.b) $(a r.a, b r.b)]

736 ?: (hor n.b n.a)

737 $(a [n.a $(a l.a, b [n.b l.b ~]) r.a], b r.b)

738 $(a [n.a l.a $(a r.a, b [n.b ~ r.b])], b l.b)

739 ?: =(n.a n.b)

740 [n.b $(b l.b, a l.a) $(b r.b, a r.a)]

741 ?: (hor n.a n.b)

742 $(b [n.b $(b l.b, a [n.a l.a ~]) r.b], a r.a)

743 $(b [n.b l.b $(b r.b, a [n.a ~ r.a])], a l.a)

744 ::

745```

746

747

748#### Examples

749

750```

751 > (~(uni in `(set @t)`(sy "ac")) `(set @)`(sy "ha"))

752 [n='c' l={'a'} r={'h'}]

753

754 > (~(uni in `(set @t)`(sy "acmo")) ~)

755 [n='c' l={'a'} r={'m' 'o'}]

756

757 > (~(uni in `(set @t)`(sy "acmo")) `(set @t)`(sy "ham"))

758 [n='c' l={'a'} r={'m' 'o' 'h'}]

759

760 > (~(uni in `(set @t)`(sy "acmo")) `(set @t)`(sy "lep"))

761 [n='e' l={} r={'a' 'c' 'm' 'l' 'o' 'p'}]

762```

763

764---

765### `+-wyt:in`

766

767Set size

768

769Produces the number of elements in set `a` as an atom.

771#### Accepts

773`a` is an set.

775#### Produces

777An atom.

779#### Source

781```

782 +- wyt :: size of set

783 |- ^- @

784 ?~(a 0 +((add $(a l.a) $(a r.a))))

785 --

786```

787

788

789#### Examples

790

791```

792 > =a (~(put in (~(put in (sy)) 'a')) 'b')

793 > ~(wyt in a)

794 2

795

796 > =b (sy `(list @t)`['john' 'bonita' 'daniel' 'madeleine' ~])

797 > ~(wyt in b)

798 4

799```

800

801---