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1 |
| -# sage.doctest: needs sage.graphs |
| 1 | +# sage.doctest: needs sage.graphs sage.modules |
2 | 2 | r"""
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3 | 3 | This file contains doctests of the article ::
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4 | 4 |
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506 | 506 |
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507 | 507 | Sage example in fsm-in-sage.tex, line 1091::
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508 | 508 |
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509 |
| - sage: var('y') # optional - sage.symbolic |
| 509 | + sage: var('y') # needs sage.symbolic |
510 | 510 | y
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511 | 511 | sage: def am_entry(trans):
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512 | 512 | ....: return y^add(trans.word_out) / 2
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513 |
| - sage: A = W.adjacency_matrix(entry=am_entry) # optional - sage.symbolic |
| 513 | + sage: A = W.adjacency_matrix(entry=am_entry) # needs sage.symbolic |
514 | 514 |
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515 | 515 |
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516 | 516 | Sage example in fsm-in-sage.tex, line 1097::
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520 | 520 |
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521 | 521 | Sage example in fsm-in-sage.tex, line 1099::
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522 | 522 |
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523 |
| - sage: latex(A) # optional - sage.symbolic |
| 523 | + sage: latex(A) # needs sage.symbolic |
524 | 524 | \left(\begin{array}{ccccccccc}
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525 | 525 | \frac{1}{2} & \frac{1}{2} \, y^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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526 | 526 | 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\
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536 | 536 |
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537 | 537 | Sage example in fsm-in-sage.tex, line 1109::
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538 | 538 |
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539 |
| - sage: A1mI = (A.subs(y=1) - A.parent().identity_matrix()) # optional - sage.symbolic |
540 |
| - sage: (pi_not_normalized,) = A1mI.left_kernel().basis() # optional - sage.symbolic |
541 |
| - sage: pi = pi_not_normalized / pi_not_normalized.norm(p=1) # optional - sage.symbolic |
| 539 | + sage: A1mI = (A.subs(y=1) - A.parent().identity_matrix()) # needs sage.symbolic |
| 540 | + sage: (pi_not_normalized,) = A1mI.left_kernel().basis() # needs sage.symbolic |
| 541 | + sage: pi = pi_not_normalized / pi_not_normalized.norm(p=1) # needs sage.symbolic |
542 | 542 |
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543 | 543 |
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544 | 544 | Sage example in fsm-in-sage.tex, line 1110::
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545 | 545 |
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546 |
| - sage: str(pi) # optional - sage.symbolic |
| 546 | + sage: str(pi) # needs sage.symbolic |
547 | 547 | '(1/9, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9)'
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548 | 548 |
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549 | 549 |
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550 | 550 | Sage example in fsm-in-sage.tex, line 1117::
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551 | 551 |
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552 |
| - sage: expected_output = derivative(A, y).subs(y=1) * vector(len(W.states())*[1]) # optional - sage.symbolic |
| 552 | + sage: expected_output = derivative(A, y).subs(y=1) * vector(len(W.states())*[1]) # needs sage.symbolic |
553 | 553 |
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554 | 554 |
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555 | 555 | Sage example in fsm-in-sage.tex, line 1118::
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556 | 556 |
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557 |
| - sage: latex(expected_output) # optional - sage.symbolic |
| 557 | + sage: latex(expected_output) # needs sage.symbolic |
558 | 558 | \left(1,\,0,\,0,\,0,\,\frac{1}{2},\,1,\,1,\,\frac{1}{2},\,1\right)
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559 | 559 |
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560 | 560 |
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561 | 561 | Sage example in fsm-in-sage.tex, line 1126::
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562 | 562 |
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563 |
| - sage: pi * expected_output # optional - sage.symbolic |
| 563 | + sage: pi * expected_output # needs sage.symbolic |
564 | 564 | 5/9
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565 | 565 |
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566 | 566 |
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567 | 567 | Sage example in fsm-in-sage.tex, line 1127::
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568 | 568 |
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569 |
| - sage: latex(pi * expected_output) # optional - sage.symbolic |
| 569 | + sage: latex(pi * expected_output) # needs sage.symbolic |
570 | 570 | \frac{5}{9}
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571 | 571 |
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572 | 572 |
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573 | 573 | Sage example in fsm-in-sage.tex, line 1129::
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574 | 574 |
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575 |
| - sage: latex(pi * expected_output) # optional - sage.symbolic |
| 575 | + sage: latex(pi * expected_output) # needs sage.symbolic |
576 | 576 | \frac{5}{9}
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577 | 577 |
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578 | 578 |
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579 | 579 | Sage example in fsm-in-sage.tex, line 1145::
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580 | 580 |
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581 |
| - sage: var('k') # optional - sage.symbolic |
| 581 | + sage: var('k') # needs sage.symbolic |
582 | 582 | k
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583 |
| - sage: moments = W.asymptotic_moments(k) # optional - sage.symbolic |
| 583 | + sage: moments = W.asymptotic_moments(k) # needs sage.symbolic |
584 | 584 |
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585 | 585 |
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586 | 586 | Sage example in fsm-in-sage.tex, line 1155::
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587 | 587 |
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588 |
| - sage: latex(moments['expectation']) # optional - sage.symbolic |
| 588 | + sage: latex(moments['expectation']) # needs sage.symbolic |
589 | 589 | \frac{5}{9} \, k + \mathcal{O}\left(1\right)
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590 | 590 |
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591 | 591 |
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592 | 592 | Sage example in fsm-in-sage.tex, line 1162::
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593 | 593 |
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594 |
| - sage: latex(moments['variance']) # optional - sage.symbolic |
| 594 | + sage: latex(moments['variance']) # needs sage.symbolic |
595 | 595 | \frac{44}{243} \, k + \mathcal{O}\left(1\right)
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596 | 596 |
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597 | 597 |
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598 | 598 | Sage example in fsm-in-sage.tex, line 1192::
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599 | 599 |
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600 |
| - sage: expectation_binary = Id.asymptotic_moments(k)['expectation'] # optional - sage.symbolic |
| 600 | + sage: expectation_binary = Id.asymptotic_moments(k)['expectation'] # needs sage.symbolic |
601 | 601 |
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602 | 602 |
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603 | 603 | Sage example in fsm-in-sage.tex, line 1195::
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604 | 604 |
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605 |
| - sage: latex(expectation_binary) # optional - sage.symbolic |
| 605 | + sage: latex(expectation_binary) # needs sage.symbolic |
606 | 606 | \frac{1}{2} \, k + \mathcal{O}\left(1\right)
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607 | 607 |
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608 | 608 |
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609 | 609 | Sage example in fsm-in-sage.tex, line 1202::
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610 | 610 |
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611 |
| - sage: expectation_NAF = Weight(NAF).asymptotic_moments(k)['expectation'] # optional - sage.symbolic |
| 611 | + sage: expectation_NAF = Weight(NAF).asymptotic_moments(k)['expectation'] # needs sage.symbolic |
612 | 612 |
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613 | 613 |
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614 | 614 | Sage example in fsm-in-sage.tex, line 1205::
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615 | 615 |
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616 |
| - sage: latex(expectation_NAF) # optional - sage.symbolic |
| 616 | + sage: latex(expectation_NAF) # needs sage.symbolic |
617 | 617 | \frac{1}{3} \, k + \mathcal{O}\left(1\right)
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618 | 618 |
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619 | 619 |
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620 | 620 | Sage example in fsm-in-sage.tex, line 1211::
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621 | 621 |
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622 |
| - sage: Abs = transducers.abs([-1, 0, 1]) # optional - sage.symbolic |
| 622 | + sage: Abs = transducers.abs([-1, 0, 1]) # needs sage.symbolic |
623 | 623 |
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624 | 624 |
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625 | 625 | Sage example in fsm-in-sage.tex, line 1216::
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626 | 626 |
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627 |
| - sage: latex(moments['expectation']) # optional - sage.symbolic |
| 627 | + sage: latex(moments['expectation']) # needs sage.symbolic |
628 | 628 | \frac{5}{9} \, k + \mathcal{O}\left(1\right)
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629 | 629 |
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630 | 630 | """
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