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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 131 ms]
(2) BOUNDS(INF, INF)


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

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B, C) -> Com_1(f3(A, D, C)) :|: A >= 500
f1(A, B, C) -> Com_1(f1(1 + A, B, C)) :|: 499 >= A
f2(A, B, C) -> Com_1(f3(E, D, E)) :|: E >= 500
f2(A, B, C) -> Com_1(f1(1 + D, B, D)) :|: 499 >= D

The start-symbols are:[f2_3]


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

(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f2

      0: f1 -> f3 : B'=free, [ A>=500 ], cost: 1

      1: f1 -> f1 : A'=1+A, [ 499>=A ], cost: 1

      2: f2 -> f3 : A'=free_1, B'=free_2, C'=free_1, [ free_1>=500 ], cost: 1

      3: f2 -> f1 : A'=1+free_3, C'=free_3, [ 499>=free_3 ], cost: 1



Checking for constant complexity:

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

      2: f2 -> f3 : A'=free_1, B'=free_2, C'=free_1, [ free_1>=500 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f2

      1: f1 -> f1 : A'=1+A, [ 499>=A ], cost: 1

      3: f2 -> f1 : A'=1+free_3, C'=free_3, [ 499>=free_3 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      1: f1 -> f1 : A'=1+A, [ 499>=A ], cost: 1



   Accelerated rule 1 with metering function 500-A, yielding the new rule 4.

   Removing the simple loops: 1.



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

   Start location: f2

      4: f1 -> f1 : A'=500, [ 499>=A ], cost: 500-A

      3: f2 -> f1 : A'=1+free_3, C'=free_3, [ 499>=free_3 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f2

      3: f2 -> f1 : A'=1+free_3, C'=free_3, [ 499>=free_3 ], cost: 1

      5: f2 -> f1 : A'=500, C'=free_3, [ 499>=1+free_3 ], cost: 500-free_3



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

   Start location: f2

      5: f2 -> f1 : A'=500, C'=free_3, [ 499>=1+free_3 ], cost: 500-free_3



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f2

      5: f2 -> f1 : A'=500, C'=free_3, [ 499>=1+free_3 ], cost: 500-free_3



Computing asymptotic complexity for rule 5

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      500-free_3 (+), 499-free_3 (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free_3==-n}

      resulting limit problem:

      [solved]



   Solution:

      free_3 / -n

   Resulting cost 500+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: 500+n

   Rule cost:   500-free_3

   Rule guard:  [ 499>=1+free_3 ]



WORST_CASE(INF,?)


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

(2)
BOUNDS(INF, INF)