Worked on sep seqs.
This commit is contained in:
Joshua Moerman
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5 changed files with 221 additions and 1 deletions
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content/esm.table
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content/esm.table
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states Hopcroft Moore
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546 0.05548176 0.05162218
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547 0.065968448 0.374567635
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571 0.151784094 2.562025297
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610 0.161709531 2.923059564
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612 0.162847423 2.949833339
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616 0.16393522 2.979295117
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634 0.169730862 3.136451995
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644 0.171697908 3.205603966
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646 0.173519053 3.247711454
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664 0.174829094 3.303879469
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674 0.176219619 3.404398614
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704 0.18026932 3.772591464
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723 0.176796323 4.025279554
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724 0.176144222 4.041852225
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727 0.177380196 4.078740205
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750 0.184511916 4.376803632
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752 0.184925295 4.39675801
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755 0.18608039 4.427337413
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779 0.191979151 4.696313345
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796 0.19801254 4.906009044
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808 0.199559131 4.969958634
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816 0.202306127 5.06486171
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820 0.203567904 5.11139433
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826 0.206071634 5.194849765
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849 0.217511686 5.553222482
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851 0.220003201 5.5797187
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881 0.226420016 5.94774453
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951 0.235091864 6.33370116
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952 0.234470623 6.357592105
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976 0.243413677 6.785026456
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998 0.247111407 7.115376566
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1022 0.254264636 7.475588529
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1028 0.253825238 7.676966117
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1031 0.253552779 7.655207879
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1032 0.253959583 7.687817395
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1036 0.256927602 7.753280811
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1038 0.256454243 7.785150879
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1044 0.2676729 7.914658705
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1048 0.268592945 7.993362548
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1090 0.280256629 8.571307839
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1096 0.283057389 8.72110485
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1104 0.285299766 8.860082549
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1172 0.30334844 9.682438227
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1182 0.308970132 9.785819737
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1222 0.323826492 10.508031711
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1230 0.325670362 10.670810609
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1242 0.331428484 10.796530463
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1248 0.32975737 10.897072507
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1262 0.33353854 11.303006909
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1268 0.338205563 11.351776371
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1460 0.371807782 13.490904423
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1466 0.372692439 13.528562317
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1486 0.380183351 14.004736754
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1492 0.382841945 14.188868672
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1495 0.382159405 14.265695125
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1513 0.388669725 14.900034072
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1516 0.390405231 14.81499996
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1518 0.390463274 14.944426519
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1522 0.392151819 14.915203734
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1524 0.393186661 14.961747551
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1526 0.394158427 15.037075056
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1573 0.408168072 16.133197125
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1598 0.416766067 16.678992988
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1660 0.44328243 18.142114009
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1665 0.444965405 18.25355816
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1666 0.444720035 18.28403416
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1667 0.446062015 18.284690403
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1668 0.446125714 18.329058714
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1670 0.446133206 18.572377324
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1672 0.446952057 18.695455536
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1708 0.456918651 18.790907562
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1712 0.458244796 18.905278887
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1720 0.461813708 19.17002062
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1728 0.465456384 19.392921399
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1736 0.46909665 19.581211756
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1738 0.469009293 19.63986836
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1739 0.470088086 19.654523832
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1745 0.472925164 19.834706477
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1747 0.476427047 19.882721961
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1756 0.477945503 20.126510562
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1776 0.48555338 21.050697212
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1779 0.487291725 20.769860328
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1785 0.487422511 20.95968171
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1792 0.490868025 21.171034734
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1799 0.495984897 21.381192056
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1829 0.500577054 22.305283908
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1835 0.501130431 22.349754728
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1841 0.499636037 22.458577516
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1853 0.502867922 22.789015879
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1857 0.503177778 22.824136936
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1863 0.506992088 22.968349427
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1869 0.511573083 23.149349308
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1870 0.510409209 23.185619132
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1891 0.51604806 23.597859463
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1915 0.52134704 23.854239145
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1927 0.526302999 24.077963239
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1935 0.529154809 24.256248614
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1951 0.533841463 24.888246072
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1972 0.537198755 25.293137881
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1978 0.538614268 25.390662904
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2032 0.565800375 26.03466246
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2053 0.559773001 26.420268108
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2101 0.569853806 27.579091597
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2125 0.575521253 28.225772617
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2137 0.57871672 28.277690829
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2155 0.581133036 28.500376699
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2158 0.582197362 28.494612045
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2167 0.584903052 28.531839432
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2170 0.588296814 28.575386555
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2172 0.588168789 28.727190877
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2202 0.601001684 29.645737322
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2214 0.607209659 30.433077486
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2218 0.610239584 30.515088605
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2224 0.612489408 30.80970033
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2228 0.609919931 30.970134524
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2246 0.616805458 30.836588838
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2364 0.688856681 36.422097475
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2382 0.699198195 37.22853454
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2386 0.699697687 37.399438633
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2390 0.696379643 37.491589119
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2448 0.722756074 40.559368969
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2822 0.892213459 56.246352395
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2894 0.925655807 58.298946257
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2902 0.930451456 58.91838622
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2908 0.934032602 59.191770267
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2915 0.939078333 59.149010129
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3042 0.979294733 63.624004421
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3109 1.008006854 66.701448478
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3164 1.030117558 69.210002572
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3172 1.028983357 68.766214254
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3326 1.06337783 72.518414203
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3364 1.080085845 75.032993864
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3368 1.081815962 74.785882543
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3380 1.086156479 75.969527635
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3406 1.095992942 76.987838799
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3410 1.11796171 79.799157453
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15
content/hopcroft_a.table
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15
content/hopcroft_a.table
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states Hopcroft Moore
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4 0.000004394 0.000008367
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8 0.000007727 0.000020106
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16 0.000014457 0.000052275
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32 0.000030602 0.000155723
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64 0.000072037 0.000509607
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128 0.000193457 0.001816708
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256 0.000563881 0.00676876
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512 0.00145047 0.026100461
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1024 0.003533843 0.101946121
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2048 0.008529486 0.407693
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4096 0.021424756 1.64542877
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8192 0.051799602 6.570335332
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16384 0.12096594 26.285563125
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32768 0.295170838 106.794193669
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15
content/hopcroft_b.table
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content/hopcroft_b.table
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states Hopcroft Moore
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4 0.000004181 0.00000485
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8 0.000007397 0.000012002
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16 0.000013752 0.000030596
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32 0.000027895 0.000084655
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64 0.000061301 0.000258466
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128 0.000149001 0.00085492
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256 0.000390771 0.003013328
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512 0.000959998 0.011248316
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1024 0.002402636 0.043437952
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2048 0.005954948 0.1706022
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4096 0.015253576 0.684592527
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8192 0.035884835 2.758716619
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16384 0.082455912 11.050600273
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32768 0.194492715 43.719043747
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@ -647,9 +647,60 @@ This can be done in $\bigO(n^2)$ time using a trie data structure.
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\stopsection
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\startsection
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[title={Experimental Results}]
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\todo{ga verder bij \tt 732}
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We have implemented \in{Algorithms}[alg:increase-k, alg:increase-k-v3] in Go, and we have compared their running time on two sets of FSMs.
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\footnote{Available at \todo{https://gitlab.science.ru.nl/rick/partition/}.}
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The first set is from \cite{Smeenk}, where FSMs for embedded control software were automatically constructed.
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These FSMs are of increasing size, varying from 546 to 3~410 states, with 78 inputs and up to 151 outputs.
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The second set is inferred from \citet[Hopcroft1971], where two classes of finite automata, $A$ and $B$, are described that serve as a worst case for \in{Algorithms}[alg:increase-k, alg:increase-k-v3] respectively.
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The FSMs that we have constructed for these automata have 1 input, 2 outputs, and $2^2$~--~$2^{15}$ states.
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The running times in seconds on an Intel Core i5-2500 are plotted in \in{Figure}[fig:results].
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We note that different slopes imply different complexity classes, since both axes have a logarithmic scale.
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\startplacefigure
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[title={Running time in seconds of \in{Algorithm}[alg:increase-k] (grey) and \in{Algorithm}[alg:increase-k-v3] (black).},
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reference=fig:results]
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\startcombination[2*1]
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{\starttikzpicture
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\startloglogaxis[width=0.5\textwidth, xtick={500, 1000, 2000, 3000}, xticklabels={500, 1000, 2000, 3000}]
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\addplot[color=darkgray] table[x=states, y=Moore] {esm.table};
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\addplot[color=black] table[x=states, y=Hopcroft] {esm.table};
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\stoploglogaxis
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\stoptikzpicture} {(a) Embedded control software.}
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{\starttikzpicture
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\startloglogaxis[width=0.5\textwidth, xtick={4, 64, 2048, 32768}, xticklabels={$2^2$, $2^6$, $2^{11}$, $2^{15}$}]
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\addplot[color=darkgray, dashed] table[x=states, y=Moore] {hopcroft_b.table};
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\addplot[color=black, dashed] table[x=states, y=Hopcroft] {hopcroft_b.table};
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\addplot[color=darkgray] table[x=states, y=Moore] {hopcroft_a.table};
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\addplot[color=black] table[x=states, y=Hopcroft] {hopcroft_a.table};
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\stoploglogaxis
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\stoptikzpicture} {(b) Class $A$ (dashed) and class $B$ (solid).}
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\stopcombination
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\stopplacefigure
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\stopsection
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\startsection
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[title={Conclusion}]
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In this paper we have described an efficient algorithm for constructing a set of minimal-length sequences that pairwise distinguish all states of a finite state machine.
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By extending Hopcroft's minimisation algorithm, we are able to construct such sequences in $\bigO(m \log n)$ for a machine with $m$ transitions and $n$ states.
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This improves on the traditional $\bigO(mn)$ method that is based on the classic algorithm by Moore.
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As an upshot, the sequences obtained form a characterisation set and a separating family, which play a crucial in conformance testing.
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Two key observations were required for a correct adaptation of Hopcroft's algorithm.
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First, it is required to perform splits in order of the length of their associated sequences.
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This guarantees minimality of the obtained separating sequences.
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Second, it is required to consider nodes as a candidate before any one of its children are considered as a candidate.
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This order follows naturally from the construction of a splitting tree.
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Experimental results show that our algorithm outperforms the classic approach for both worst-case finite state machines and models of embedded control software.
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Applications of minimal separating sequences such as the ones described by \citet[dorofeeva2010fsm, Smeenk] therefore show that our algorithm is useful in practice.
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\stopsection
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\referencesifcomponent
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\stopchapter
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\stopcomponent
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@ -3,6 +3,8 @@
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\usemodule[tikz]
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\usetikzlibrary[automata, arrows, positioning, matrix, fit, decorations.pathreplacing]
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\usemodule[pgfplots]
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% defaults
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\tikzset{node distance=2cm} % 'shorten >=2pt' only for automata
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