/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 542 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B) -> Com_1(f2(A - 1, B)) :|: A >= 1
f0(A, B) -> Com_1(f2(C, B - 1)) :|: B >= 1
f2(A, B) -> Com_1(f2(C, B - 1)) :|: B >= 1 && 0 >= A

The start-symbols are:[f0_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1

      2: f2 -> f2 : A'=free_1, B'=-1+B, [ B>=1 && 0>=A ], cost: 1

      1: f0 -> f2 : A'=free, B'=-1+B, [ B>=1 ], cost: 1



Checking for constant complexity:

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

      1: f0 -> f2 : A'=free, B'=-1+B, [ B>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f2 -> f2 : A'=-1+A, [ A>=1 ], cost: 1

      2: f2 -> f2 : A'=free_1, B'=-1+B, [ B>=1 && 0>=A ], cost: 1



   Accelerated rule 0 with metering function A, yielding the new rule 3.

   Accelerated rule 2 with metering function B (after strengthening guard), yielding the new rule 4.

   Nested simple loops 2 (outer loop) and 3 (inner loop) with metering function B, resulting in the new rules: 5, 6.

      During metering: Instantiating temporary variables by {free_1==0}

   Removing the simple loops: 0 2.



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

   Start location: f0

      3: f2 -> f2 : A'=0, [ A>=1 ], cost: A

      4: f2 -> f2 : A'=free_1, B'=0, [ B>=1 && 0>=A && 0>=free_1 ], cost: B

      5: f2 -> f2 : A'=0, B'=0, [ B>=1 && 0>=A && free_1>=1 ], cost: free_1*B+B

      6: f2 -> f2 : A'=0, B'=0, [ A>=1 && B>=1 && free_1>=1 ], cost: free_1*B+A+B

      1: f0 -> f2 : A'=free, B'=-1+B, [ B>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f0

      1: f0 -> f2 : A'=free, B'=-1+B, [ B>=1 ], cost: 1

      7: f0 -> f2 : A'=0, B'=-1+B, [ B>=1 && free>=1 ], cost: 1+free

      8: f0 -> f2 : A'=free_1, B'=0, [ -1+B>=1 && 0>=free_1 ], cost: B

      9: f0 -> f2 : A'=0, B'=0, [ -1+B>=1 && free_1>=1 ], cost: (-1+B)*free_1+B

     10: f0 -> f2 : A'=0, B'=0, [ free>=1 && -1+B>=1 && free_1>=1 ], cost: free+(-1+B)*free_1+B



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

   Start location: f0

      7: f0 -> f2 : A'=0, B'=-1+B, [ B>=1 && free>=1 ], cost: 1+free

      8: f0 -> f2 : A'=free_1, B'=0, [ -1+B>=1 && 0>=free_1 ], cost: B

      9: f0 -> f2 : A'=0, B'=0, [ -1+B>=1 && free_1>=1 ], cost: (-1+B)*free_1+B

     10: f0 -> f2 : A'=0, B'=0, [ free>=1 && -1+B>=1 && free_1>=1 ], cost: free+(-1+B)*free_1+B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

      7: f0 -> f2 : A'=0, B'=-1+B, [ B>=1 && free>=1 ], cost: 1+free

      8: f0 -> f2 : A'=free_1, B'=0, [ -1+B>=1 && 0>=free_1 ], cost: B

      9: f0 -> f2 : A'=0, B'=0, [ -1+B>=1 && free_1>=1 ], cost: (-1+B)*free_1+B

     10: f0 -> f2 : A'=0, B'=0, [ free>=1 && -1+B>=1 && free_1>=1 ], cost: free+(-1+B)*free_1+B



Computing asymptotic complexity for rule 7

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      free (+/+!), B (+/+!), 1+free (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      free / n

      B / 1

   Resulting cost 1+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: 1+n

   Rule cost:   1+free

   Rule guard:  [ B>=1 && free>=1 ]



WORST_CASE(INF,?)


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(2)
BOUNDS(INF, INF)