/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 2129 ms]
(2) BOUNDS(INF, INF)


----------------------------------------

(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]


----------------------------------------

(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



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



   Accelerated rule 25 with NONTERM (after adding A<=B), yielding the new rule 30.

   Accelerated rule 25 with backward acceleration, yielding the new rule 31.

   Accelerated rule 25 with backward acceleration, yielding the new rule 32.

   Accelerated rule 26 with metering function A, yielding the new rule 33.

   Accelerated rule 27 with metering function -2+A, yielding the new rule 34.

   Removing the simple loops: 25 26 27.



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

   Start location: f0

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

     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 -> [8] : [ A>=1 && 0>=1+B && A<=B ], cost: INF

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

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

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

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



Chained accelerated rules (with incoming rules):

   Start location: f0

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

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

     36: 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



Removed unreachable locations (and leaf rules with constant cost):

   Start location: f0

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

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

     36: 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

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

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

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

     40: 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

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

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

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

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



Computing asymptotic complexity for rule 37

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -free_1 (+/+!), 2+free (+), -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,?)


----------------------------------------

(2)
BOUNDS(INF, INF)