/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 2052 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B, C, D) -> Com_1(f6(E, 0, C, D)) :|: TRUE
f6(A, B, C, D) -> Com_1(f10(A - 1, B + 1, C, D)) :|: A >= 1
f10(A, B, C, D) -> Com_1(f14(A, B - 1, A - 1, D)) :|: B >= 1
f14(A, B, C, D) -> Com_1(f14(A, B, C - 1, 0)) :|: C >= 1
f14(A, B, C, D) -> Com_1(f14(A - 1, B + 1, C - 1, E)) :|: C >= 1 && 0 >= E + 1
f14(A, B, C, D) -> Com_1(f14(A - 1, B + 1, C - 1, E)) :|: C >= 1 && E >= 1
f14(A, B, C, D) -> Com_1(f10(A, B, C, D)) :|: 0 >= C
f10(A, B, C, D) -> Com_1(f6(A, B, C, D)) :|: 0 >= B
f6(A, B, C, D) -> Com_1(f25(A, B, C, D)) :|: 0 >= A

The start-symbols are:[f0_4]


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(1) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      8: f6 -> f25 : [ 0>=A ], cost: 1

      2: f10 -> f14 : B'=-1+B, C'=-1+A, [ B>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

      3: f14 -> f14 : C'=-1+C, D'=0, [ C>=1 ], cost: 1

      4: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_1, [ C>=1 && 0>=1+free_1 ], cost: 1

      5: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_2, [ C>=1 && free_2>=1 ], cost: 1

      6: f14 -> f10 : [ 0>=C ], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      2: f10 -> f14 : B'=-1+B, C'=-1+A, [ B>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

      3: f14 -> f14 : C'=-1+C, D'=0, [ C>=1 ], cost: 1

      4: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_1, [ C>=1 && 0>=1+free_1 ], cost: 1

      5: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_2, [ C>=1 && free_2>=1 ], cost: 1

      6: f14 -> f10 : [ 0>=C ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 3.

   Accelerating the following rules:

      3: f14 -> f14 : C'=-1+C, D'=0, [ C>=1 ], cost: 1

      4: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_1, [ C>=1 && 0>=1+free_1 ], cost: 1

      5: f14 -> f14 : A'=-1+A, B'=1+B, C'=-1+C, D'=free_2, [ C>=1 && free_2>=1 ], cost: 1



   Accelerated rule 3 with metering function C, yielding the new rule 9.

   Accelerated rule 4 with metering function C, yielding the new rule 10.

   Accelerated rule 5 with metering function C, yielding the new rule 11.

   Removing the simple loops: 3 4 5.



Accelerated all simple loops using metering functions (where possible):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      2: f10 -> f14 : B'=-1+B, C'=-1+A, [ B>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

      6: f14 -> f10 : [ 0>=C ], cost: 1

      9: f14 -> f14 : C'=0, D'=0, [ C>=1 ], cost: C

     10: f14 -> f14 : A'=-C+A, B'=C+B, C'=0, D'=free_1, [ C>=1 && 0>=1+free_1 ], cost: C

     11: f14 -> f14 : A'=-C+A, B'=C+B, C'=0, D'=free_2, [ C>=1 && free_2>=1 ], cost: C



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      2: f10 -> f14 : B'=-1+B, C'=-1+A, [ B>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

     12: f10 -> f14 : B'=-1+B, C'=0, D'=0, [ B>=1 && -1+A>=1 ], cost: A

     13: f10 -> f14 : A'=1, B'=-2+A+B, C'=0, D'=free_1, [ B>=1 && -1+A>=1 && 0>=1+free_1 ], cost: A

     14: f10 -> f14 : A'=1, B'=-2+A+B, C'=0, D'=free_2, [ B>=1 && -1+A>=1 && free_2>=1 ], cost: A

      6: f14 -> f10 : [ 0>=C ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

     15: f10 -> f10 : B'=-1+B, C'=-1+A, [ B>=1 && 0>=-1+A ], cost: 2

     16: f10 -> f10 : B'=-1+B, C'=0, D'=0, [ B>=1 && -1+A>=1 ], cost: 1+A

     17: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_1, [ B>=1 && -1+A>=1 && 0>=1+free_1 ], cost: 1+A

     18: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_2, [ B>=1 && -1+A>=1 && free_2>=1 ], cost: 1+A



Accelerating simple loops of location 2.

   Accelerating the following rules:

     15: f10 -> f10 : B'=-1+B, C'=-1+A, [ B>=1 && 0>=-1+A ], cost: 2

     16: f10 -> f10 : B'=-1+B, C'=0, D'=0, [ B>=1 && -1+A>=1 ], cost: 1+A

     17: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_1, [ B>=1 && -1+A>=1 && 0>=1+free_1 ], cost: 1+A

     18: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_2, [ B>=1 && -1+A>=1 && free_2>=1 ], cost: 1+A



   Accelerated rule 15 with metering function B, yielding the new rule 19.

   Accelerated rule 16 with metering function B, yielding the new rule 20.

   Found no metering function for rule 17.

   Found no metering function for rule 18.

   Removing the simple loops: 15 16.



Accelerated all simple loops using metering functions (where possible):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

      7: f10 -> f6 : [ 0>=B ], cost: 1

     17: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_1, [ B>=1 && -1+A>=1 && 0>=1+free_1 ], cost: 1+A

     18: f10 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_2, [ B>=1 && -1+A>=1 && free_2>=1 ], cost: 1+A

     19: f10 -> f10 : B'=0, C'=-1+A, [ B>=1 && 0>=-1+A ], cost: 2*B

     20: f10 -> f10 : B'=0, C'=0, D'=0, [ B>=1 && -1+A>=1 ], cost: A*B+B



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

      1: f6 -> f10 : A'=-1+A, B'=1+B, [ A>=1 ], cost: 1

     21: f6 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_1, [ 1+B>=1 && -2+A>=1 && 0>=1+free_1 ], cost: 1+A

     22: f6 -> f10 : A'=1, B'=-2+A+B, C'=0, D'=free_2, [ 1+B>=1 && -2+A>=1 && free_2>=1 ], cost: 1+A

     23: f6 -> f10 : A'=-1+A, B'=0, C'=-2+A, [ A>=1 && 1+B>=1 && 0>=-2+A ], cost: 3+2*B

     24: f6 -> f10 : A'=-1+A, B'=0, C'=0, D'=0, [ 1+B>=1 && -2+A>=1 ], cost: 2+(-1+A)*(1+B)+B

      7: f10 -> f6 : [ 0>=B ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

     25: f6 -> f6 : A'=-1+A, B'=1+B, [ A>=1 && 0>=1+B ], cost: 2

     26: f6 -> f6 : A'=-1+A, B'=0, C'=-2+A, [ A>=1 && 1+B>=1 && 0>=-2+A ], cost: 4+2*B

     27: f6 -> f6 : A'=-1+A, B'=0, C'=0, D'=0, [ 1+B>=1 && -2+A>=1 ], cost: 3+(-1+A)*(1+B)+B

     28: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && 0>=1+free_1 ], cost: 1+A

     29: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && free_2>=1 ], cost: 1+A



Accelerating simple loops of location 1.

   Accelerating the following rules:

     25: f6 -> f6 : A'=-1+A, B'=1+B, [ A>=1 && 0>=1+B ], cost: 2

     26: f6 -> f6 : A'=-1+A, B'=0, C'=-2+A, [ A>=1 && 1+B>=1 && 0>=-2+A ], cost: 4+2*B

     27: f6 -> f6 : A'=-1+A, B'=0, C'=0, D'=0, [ 1+B>=1 && -2+A>=1 ], cost: 3+(-1+A)*(1+B)+B



   Found no metering function for rule 25.

   Accelerated rule 26 with metering function A, yielding the new rule 30.

   Accelerated rule 27 with metering function -2+A, yielding the new rule 31.

   Removing the simple loops: 26 27.



Accelerated all simple loops using metering functions (where possible):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

     25: f6 -> f6 : A'=-1+A, B'=1+B, [ A>=1 && 0>=1+B ], cost: 2

     28: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && 0>=1+free_1 ], cost: 1+A

     29: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && free_2>=1 ], cost: 1+A

     30: f6 -> f6 : A'=0, B'=0, C'=-1, [ A>=1 && 1+B>=1 && 0>=-2+A ], cost: 4*A

     31: f6 -> f6 : A'=2, B'=0, C'=0, D'=0, [ 1+B>=1 && -2+A>=1 ], cost: -5+A*(-2+A)+5/2*A-1/2*(-2+A)^2



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f6 : A'=free, B'=0, [], cost: 1

     32: f0 -> f6 : A'=0, B'=0, C'=-1, [ free>=1 && 0>=-2+free ], cost: 1+4*free

     33: f0 -> f6 : A'=2, B'=0, C'=0, D'=0, [ -2+free>=1 ], cost: -4+5/2*free-1/2*(-2+free)^2+free*(-2+free)

     28: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && 0>=1+free_1 ], cost: 1+A

     29: f6 -> [7] : [ 1+B>=1 && -2+A>=1 && free_2>=1 ], cost: 1+A



Eliminated locations (on tree-shaped paths):

   Start location: f0

     34: f0 -> [7] : A'=free, B'=0, [ -2+free>=1 && 0>=1+free_1 ], cost: 2+free

     35: f0 -> [7] : A'=free, B'=0, [ -2+free>=1 && free_2>=1 ], cost: 2+free

     36: f0 -> [9] : [ free>=1 && 0>=-2+free ], cost: 1+4*free

     37: f0 -> [9] : [ -2+free>=1 ], cost: -4+5/2*free-1/2*(-2+free)^2+free*(-2+free)



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     34: f0 -> [7] : A'=free, B'=0, [ -2+free>=1 && 0>=1+free_1 ], cost: 2+free

     35: f0 -> [7] : A'=free, B'=0, [ -2+free>=1 && free_2>=1 ], cost: 2+free

     36: f0 -> [9] : [ free>=1 && 0>=-2+free ], cost: 1+4*free

     37: f0 -> [9] : [ -2+free>=1 ], cost: -4+5/2*free-1/2*(-2+free)^2+free*(-2+free)



Computing asymptotic complexity for rule 34

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -2+free (+/+!), -free_1 (+/+!), 2+free (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free==n,free_1==-1}

      resulting limit problem:

      [solved]



   Solution:

      free / n

      free_1 / -1

   Resulting cost 2+n has complexity: Unbounded



Found new complexity Unbounded.



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Unbounded

   Cpx degree:  Unbounded

   Solved cost: 2+n

   Rule cost:   2+free

   Rule guard:  [ -2+free>=1 && 0>=1+free_1 ]



WORST_CASE(INF,?)


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(2)
BOUNDS(INF, INF)