Resolution Experiment 1 Unsatisfiable E2 = or [ "A", - - PDF document

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Resolution Experiment 1 Unsatisfiable E2 = or [ "A", - - PDF document

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Resolution Experiment 1 Unsatisfiable E2 = or [ "A", "B" ] , or "A", "C", "D" , or "C", "D" , or "B", "C" , or "A",

slide-1
SLIDE 1

Resolution Experiment 1

Unsatisfiable

E2 = or["A", "B"], or"A", "C", "D", or"C", "D",

  • r"B", "C", or"A", "B", "C", or"A", "B", "C", "D"

Resolution

Block[{$RecursionLimit = 5000}, isSatisfiable[E2, show → {proof, statistics}]]

Proof 14 steps level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 15 15 1 14 1 13 2 156 26 130 26 104 25 53 26 3 819 52 767 303 464 102 342 20 4 425 14 411 135 276 82 193 1 total 1415 92 1323 465 858 210 588 60 Proof 14 steps: 9 {3, 6}

  • rA, B, C

{3, 6} {} 11 {1, 5}

  • rB, C

{1, 5} {} 13 {3, 5}

  • rA, B, D

{3, 5} {} 14 {4, 5}

  • rA, C

{4, 5} {} 15 {1, 4}

  • rA, C

{1, 4} {} 18 {2, 3}

  • rA, C

{2, 3} {} 19 {1, 2}

  • r[B, C, D]

{1, 2} {} 22 {11, 19}

  • r[B, D]

{1, 2, 5} {1} 27 {9, 18}

  • rB, C

{2, 3, 6} {3} 42 {14, 15}

  • r[C]

{1, 4, 5} {4} 43 {1, 13}

  • rB, D

{1, 3, 5} {} 56 {22, 43}

  • r[B]

{1, 2, 3, 5} {1, 5} 59 {27, 42}

  • r[B]

{1, 2, 3, 4, 5, 6} {} 66 {56, 59}  {1, 2, 3, 4, 5, 6} {1, 2, 3, 5}

Resolution with forward subsumption

In[ ]:= Block[{$RecursionLimit = 5000},

isSatisfiable[E2, use → {forwardSubsumption}, show → {proof, statistics}]]

slide-2
SLIDE 2

Proof 14 steps level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 15 15 1 14 1 13 2 156 26 130 26 104 25 43 21 15 3 390 30 360 172 188 30 124 28 6 4 38 2 36 25 11 8 2 1 total 599 58 541 224 317 56 175 51 35 Proof 14 steps: 9 {3, 6}

  • rA, B, C

11 {1, 5}

  • rB, C

13 {3, 5}

  • rA, B, D

14 {4, 5}

  • rA, C

15 {1, 4}

  • rA, C

18 {2, 3}

  • rA, C

19 {1, 2}

  • r[B, C, D]

22 {11, 19}

  • r[B, D]

25 {9, 18}

  • rB, C

32 {14, 15}

  • r[C]

33 {1, 13}

  • rB, D

38 {22, 33}

  • r[B]

40 {25, 32}

  • r[B]

41 {38, 40} 

Resolution with backward subsumption

In[ ]:= Block[{$RecursionLimit = 5000},

isSatisfiable[E2, use → {subsumption}, show → {proof, statistics}]]

Proof 11 steps level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 9 9 1 8 8 3 2 11 3 8 3 5 5 7 3 12 2 10 6 4 4 7 4 8 3 5 3 2 2 2 5 1 1 1 1 total 41 8 33 13 20 20 19 Proof 11 steps: 7 {1, 6}

  • r[A, C, D]

10 {4, 5}

  • rA, C

11 {1, 4}

  • rA, C

14 {2, 3}

  • rA, C

17 {7, 14}

  • r[C, D]

18 {10, 14}

  • r[A]

19 {3, 11}

  • rA, D

20 {17, 19}

  • r[A, C]

23 {11, 18}

  • r[C]

25 {20, 23}

  • r[A]

26 {18, 25} 

2 Resolution-Experiments-presentation.nb

slide-3
SLIDE 3

Resolution Experiment 2

Unsatisfiable

In[ ]:=

F1 = or"A", "B", "C", or"A", "C", "D", or["B", "C"], or"B", "C", "D",

  • r"A", "D", or"A", "C", "D", or"B", "C", or"A", "B", "C", "D", or"A", "B", "C"

Resolution

Block[{$RecursionLimit = 5000}, isSatisfiable[F1, show → {initialSet, levels, clauses, proof, statistics}]]

Set 0 : 1 {}

  • rA, B, C

2 {}

  • rA, C, D

3 {}

  • r[B, C]

4 {}

  • rB, C, D

5 {}

  • rA, D

6 {}

  • rA, C, D

7 {}

  • rB, C

8 {} orA, B, C, D 9 {}

  • rA, B, C

Resolution with forward subsumption

Block[{$RecursionLimit = 5000}, isSatisfiable[F1, use → {forwardSubsumption}, show → {subsumed, initialSet, levels, clauses, proof, statistics}]]

Set initial: 1 {}

  • rA, B, C

2 {}

  • rA, C, D

3 {}

  • r[B, C]

4 {}

  • rB, C, D

5 {}

  • rA, D

6 {}

  • rA, C, D

7 {}

  • rB, C

8 {} orA, B, C, D 9 {}

  • rA, B, C

Level 1 resolution 2 {2, 9} resolvent orB, C, D added as clause 10 resolution 3 {3, 9} resolvent or[A, C] added as clause 11 resolution 5 {5, 9} resolvent orB, C, D added as clause 12

Resolution-Experiments-presentation.nb 3

slide-4
SLIDE 4

resolution 6 {6, 9} resolvent orA, B, D added as clause 13 resolution 7 {7, 9} resolvent orA, B added as clause 14 resolution 11 {3, 8} resolvent orA, B, D added as clause 15 resolution 16 {1, 7} resolvent orA, C added as clause 16 resolution 17 {2, 7} resolvent orA, B, D added as clause 17 resolution 19 {4, 7} resolvent orC, D added as clause 18 resolution 27 {1, 5} resolvent orB, C, D added as clause 19 resolution 28 {2, 5} resolvent orA, C added as clause 20 resolution 30 {4, 5} resolvent orA, B, C added as clause 21 resolution 32 {2, 4} resolvent orA, B, D added as clause 22 resolution 33 {3, 4} resolvent orB, D added as clause 23 resolution 34 {1, 3} resolvent or[A, B] added as clause 24 Level 2 resolution 37 {2, 24} resolvent orB, C, D subsumed by 3 resolution 39 {5, 24} resolvent or[B, D] added as clause 25 resolution 44 {10, 24} resolvent orA, C, D subsumed by 11 resolution 46 {12, 24} resolvent or[A, C, D] subsumed by 11 resolution 47 {13, 24} resolvent orA, D added as clause 26 resolution 48 {14, 24} resolvent or[A] added as clause 27 resolution 55 {21, 24} resolvent orB, C added as clause 28 resolution 60 {5, 23} resolvent orA, B added as clause 29 resolution 64 {9, 23} resolvent orA, C, D subsumed by 11 resolution 65 {10, 23} resolvent orC, D added as clause 30 resolution 72 {17, 23} resolvent orA, D added as clause 31 resolution 82 {7, 22} resolvent orA, C, D subsumed by 18 resolution 86 {11, 22} resolvent orB, C, D subsumed by 3 resolution 101 {7, 21} resolvent orA, C added as clause 32 resolution 111 {17, 21} resolvent orA, C, D subsumed by 18 resolution 119 {7, 20} resolvent orA, B added as clause 33 resolution 121 {9, 20} resolvent orB, C added as clause 34 resolution 123 {11, 20} resolvent or[C] added as clause 35 resolution 127 {15, 20} resolvent orB, C, D subsumed by 3 resolution 131 {19, 20} resolvent orA, B, D subsumed by 5 resolution 136 {7, 19} resolvent orC, D added as clause 36 resolution 140 {11, 19} resolvent or[A, B, D] subsumed by 24 resolution 143 {14, 19} resolvent orA, C, D subsumed by 16

4 Resolution-Experiments-presentation.nb

slide-5
SLIDE 5

resolution 155 {10, 18} resolvent orB, D added as clause 37 resolution 165 {4, 17} resolvent orA, C, D subsumed by 18 resolution 167 {6, 17} resolvent orB, C, D subsumed by 7 resolution 172 {12, 17} resolvent orA, B, C subsumed by 20 resolution 176 {16, 17} resolvent orB, C, D subsumed by 7 resolution 186 {12, 16} resolvent orA, B, D subsumed by 14 resolution 191 {2, 15} resolvent orB, C, D subsumed by 3 resolution 196 {9, 15} resolvent orA, C, D subsumed by 11 resolution 197 {10, 15} resolvent orA, C, D subsumed by 11 resolution 206 {5, 14} resolvent orB, D added as clause 38 resolution 215 {3, 13} resolvent orA, C, D subsumed by 11 resolution 224 {2, 12} resolvent orA, B, C subsumed by 20 resolution 225 {3, 12} resolvent or[C, D] subsumed by 35 resolution 235 {5, 11} resolvent or[C, D] subsumed by 35 resolution 243 {5, 10} resolvent orA, B, C subsumed by 20 Level 3 resolution 247 {1, 38} resolvent orA, C, D subsumed by 16 resolution 248 {2, 38} resolvent orA, B, C subsumed by 20 resolution 249 {3, 38} resolvent or[C, D] subsumed by 35 resolution 251 {6, 38} resolvent orA, B, C subsumed by 7 resolution 265 {21, 38} resolvent orA, C, D subsumed by 5 resolution 268 {24, 38} resolvent or[A, D] subsumed by 27 resolution 269 {25, 38} resolvent or[D] added as clause 39 resolution 281 {37, 38} resolvent or[B] added as clause 40 resolution 300 {21, 37} resolvent orA, C, D subsumed by 18 resolution 302 {23, 37} resolvent or[D] added as clause 41 resolution 323 {9, 36} resolvent orA, B, D subsumed by 14 resolution 325 {11, 36} resolvent or[A, D] subsumed by 27 resolution 327 {13, 36} resolvent orA, B, C subsumed by 7 resolution 331 {17, 36} resolvent orA, B, C subsumed by 7 resolution 332 {18, 36} resolvent or[C] added as clause 42 resolution 374 {28, 35} resolvent or[B] added as clause 43 resolution 378 {32, 35} resolvent or[A] added as clause 44 resolution 394 {15, 34} resolvent orA, C, D subsumed by 11 resolution 403 {25, 34} resolvent or[C, D] subsumed by 35 resolution 415 {4, 33} resolvent orA, C, D subsumed by 18

Resolution-Experiments-presentation.nb 5

slide-6
SLIDE 6

resolution 417 {6, 33} resolvent orB, C, D subsumed by 7 resolution 429 {19, 33} resolvent orA, C, D subsumed by 5 resolution 452 {12, 32} resolvent orA, B, D subsumed by 5 resolution 453 {13, 32} resolvent orB, C, D subsumed by 7 resolution 482 {12, 31} resolvent orA, B, C subsumed by 20 resolution 536 {12, 29} resolvent orA, C, D subsumed by 5 resolution 568 {17, 28} resolvent orA, C, D subsumed by 18 resolution 625 {2, 25} resolvent orA, B, C subsumed by 3 resolution 631 {9, 25} resolvent or[A, C, D] subsumed by 11 resolution 634 {12, 25} resolvent or[C, D] subsumed by 35 resolution 636 {14, 25} resolvent or[A, D] subsumed by 27 Level 4 resolution 672 {27, 44} resolvent  added as clause 45 Proof 10 steps level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 36 36 10 26 9 2 15 2 240 30 210 60 150 37 75 24 14 3 427 28 399 185 214 33 150 25 6 4 27 27 15 12 11 1 total 730 58 672 270 402 79 238 49 36 Proof 10 steps: 11 {3, 9}

  • r[A, C]

14 {7, 9}

  • rA, B

20 {2, 5}

  • rA, C

21 {4, 5}

  • rA, B, C

24 {1, 3}

  • r[A, B]

27 {14, 24}

  • r[A]

32 {7, 21}

  • rA, C

35 {11, 20}

  • r[C]

44 {32, 35}

  • r[A]

45 {27, 44} 

The previous proof:

Proof: 11 {3, 9}

  • r[A, C]

14 {7, 9}

  • rA, B

20 {2, 5}

  • rC, A

21 {4, 5}

  • rB, A, C

24 {1, 3}

  • r[A, B]

30 {14, 24}

  • r[A]

37 {7, 21}

  • rA, C

40 {11, 20}

  • r[C]

68 {37, 40}

  • r[A]

69 {30, 68} 

6 Resolution-Experiments-presentation.nb

slide-7
SLIDE 7

Resolution with backward subsumption

Block[{$RecursionLimit = 5000}, isSatisfiable[F1, use → {subsumption}, show → {subsumed, initialSet, levels, clauses, proof, statistics}]]

Resolution-Experiments-presentation.nb 7

slide-8
SLIDE 8

Set initial: 1 {}

  • rA, B, C

2 {}

  • rA, C, D

3 {}

  • r[B, C]

4 {}

  • rB, C, D

5 {}

  • rA, D

6 {}

  • rA, C, D

7 {}

  • rB, C

8 {} orA, B, C, D 9 {}

  • rA, B, C

Level 1 resolution 2 {2, 9} resolvent orB, C, D added as clause 10 resolution 3 {3, 9} resolvent or[A, C] added as clause 11 back-subsumes {9} resolution 8 {1, 7} resolvent orA, C added as clause 12 back-subsumes {1, 6, 8} resolution 9 {2, 7} resolvent orA, B, D added as clause 13 resolution 11 {4, 7} resolvent orC, D added as clause 14 back-subsumes {4} resolution 13 {2, 5} resolvent orA, C added as clause 15 back-subsumes {2} Level 2 resolution 16 {7, 15} resolvent orA, B added as clause 16 back-subsumes {13} resolution 18 {11, 15} resolvent or[C] added as clause 17 back-subsumes {3, 10, 11, 15} resolution 20 {5, 12} resolvent orC, D added as clause 18 Level 3 resolution 22 {14, 18} resolvent or[C] added as clause 19 back-subsumes {7, 12, 14, 18} Level 4 resolution 27 {17, 19} resolvent  added as clause 20 level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 14 14 4 10 3 1 6 6 2 8 2 6 3 3 3 5 3 6 2 4 3 1 1 4 4 3 3 2 1 1 total 31 4 27 12 15 3 1 11 15 Proof 8 steps: 11 {3, 9}

  • r[A, C]

12 {1, 7}

  • rA, C

14 {4, 7}

  • rC, D

15 {2, 5}

  • rA, C

17 {11, 15}

  • r[C]

18 {5, 12}

  • rC, D

19 {14, 18}

  • r[C]

20 {17, 19} 

8 Resolution-Experiments-presentation.nb

slide-9
SLIDE 9

Satisfiable

In[ ]:=

F2 = or"A", "B", "C", or"A", "C", "D", or["B", "C"], or"B", "C", "D",

  • r"A", "D", or"A", "C", "D", or"B", "C", or"A", "B", "C", "D"

This the set F1 without the last clause.

Block[{$RecursionLimit = 5000}, isSatisfiable[F2, show → {initialSet, statistics}]]

level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 28 28 10 18 6 2 10 2 125 20 105 40 65 12 42 11 3 253 22 231 97 134 18 110 6 4 189 12 177 104 73 18 55 total 595 54 541 251 290 54 209 27 Set initial: 1 {}

  • rA, B, C

2 {}

  • rA, C, D

3 {}

  • r[B, C]

4 {}

  • rB, C, D

5 {}

  • rA, D

6 {}

  • rA, C, D

7 {}

  • rB, C

8 {} orA, B, C, D level pairs s.ancest-

  • rs

resoluti-

  • ns

failed resolven- ts true prezent f.subsum- ed clauses b.subsum- ed 1 28 28 10 18 6 2 10 2 125 20 105 40 65 12 42 11 3 253 22 231 97 134 18 110 6 4 189 12 177 104 73 18 55 total 595 54 541 251 290 54 209 27

Resolution-Experiments-presentation.nb 9

slide-10
SLIDE 10

Set final: Level 1 {}

  • rA, B, C

2 {}

  • rA, C, D

3 {}

  • r[B, C]

4 {}

  • rB, C, D

5 {}

  • rA, D

6 {}

  • rA, C, D

7 {}

  • rB, C

8 {}

  • rA, B, C, D

Level 1 9 {3, 8}

  • rA, B, D

10 {1, 7}

  • rA, C

11 {2, 7}

  • rA, B, D

12 {4, 7}

  • rC, D

13 {1, 5}

  • rB, C, D

14 {2, 5}

  • rA, C

15 {4, 5}

  • rA, B, C

16 {2, 4}

  • rA, B, D

17 {3, 4}

  • rB, D

18 {1, 3}

  • r[A, B]

Level 2 19 {2, 18}

  • rB, C, D

20 {5, 18}

  • r[B, D]

21 {15, 18}

  • rB, C

22 {5, 17}

  • rA, B

23 {11, 17}

  • rA, D

24 {7, 16}

  • rA, C, D

25 {7, 15}

  • rA, C

26 {7, 14}

  • rA, B

27 {13, 14}

  • rA, B, D

28 {7, 13}

  • rC, D

29 {6, 11}

  • rB, C, D

Level 3 30 {5, 29}

  • rA, B, C

31 {12, 28}

  • r[C]

32 {2, 27}

  • rA, B, C

33 {7, 27}

  • rA, C, D

34 {22, 26}

  • r[A]

35 {18, 22}

  • r[B]

Remark : The unit clauses or[A], or[B], or[C] subsume all other clauses, thus they describe a satisfying interpretation. This is exactly the interpretation which falsifies the clause which was removed from F1.

10 Resolution-Experiments-presentation.nb