/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(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2 + Arg_0, 3)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 140 ms]
(2) BOUNDS(1, max(2 + Arg_0, 3))
(3) Loat Proof [FINISHED, 146 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f5(A, B) -> Com_1(f5(-(1) + A, B)) :|: A >= 1
f5(A, B) -> Com_1(f1(A, C)) :|: 0 >= A
f300(A, B) -> Com_1(f5(-(1) + A, B)) :|: A >= 1
f300(A, B) -> Com_1(f1(A, C)) :|: 0 >= A

The start-symbols are:[f300_2]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([3, 2+Arg_0]) {O(n)})



Initial Complexity Problem:

  Start:  f300  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:  C  

  Locations:  f1, f300, f5  

  Transitions:

  f300(Arg_0,Arg_1) -> f1(Arg_0,C):|:Arg_0 <= 0

  f300(Arg_0,Arg_1) -> f5(-1+Arg_0,Arg_1):|:1 <= Arg_0

  f5(Arg_0,Arg_1) -> f1(Arg_0,C):|:0 <= Arg_0 && Arg_0 <= 0

  f5(Arg_0,Arg_1) -> f5(-1+Arg_0,Arg_1):|:0 <= Arg_0 && 1 <= Arg_0  



Timebounds: 

  Overall timebound: max([3, 2+Arg_0]) {O(n)}

  2: f300->f5: 1 {O(1)}

  3: f300->f1: 1 {O(1)}

  0: f5->f5: max([0, -1+Arg_0]) {O(n)}

  1: f5->f1: 1 {O(1)}



Costbounds:

  Overall costbound: max([3, 2+Arg_0]) {O(n)}

  2: f300->f5: 1 {O(1)}

  3: f300->f1: 1 {O(1)}

  0: f5->f5: max([0, -1+Arg_0]) {O(n)}

  1: f5->f1: 1 {O(1)}



Sizebounds:

`Lower:

  2: f300->f5, Arg_0: 0 {O(1)}

  2: f300->f5, Arg_1: Arg_1 {O(n)}

  3: f300->f1, Arg_0: Arg_0 {O(n)}

  0: f5->f5, Arg_0: 0 {O(1)}

  0: f5->f5, Arg_1: Arg_1 {O(n)}

  1: f5->f1, Arg_0: 0 {O(1)}

`Upper:

  2: f300->f5, Arg_0: -1+Arg_0 {O(n)}

  2: f300->f5, Arg_1: Arg_1 {O(n)}

  3: f300->f1, Arg_0: 0 {O(1)}

  0: f5->f5, Arg_0: -1+Arg_0 {O(n)}

  0: f5->f5, Arg_1: Arg_1 {O(n)}

  1: f5->f1, Arg_0: 0 {O(1)}


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(2)
BOUNDS(1, max(2 + Arg_0, 3))

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f300

      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1

      1: f5 -> f1 : B'=free, [ 0>=A ], cost: 1

      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1

      3: f300 -> f1 : B'=free_1, [ 0>=A ], cost: 1



Removed unreachable and leaf rules:

   Start location: f300

      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1

      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f5 -> f5 : A'=-1+A, [ A>=1 ], cost: 1



   Accelerated rule 0 with metering function A, yielding the new rule 4.

   Removing the simple loops: 0.



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

   Start location: f300

      4: f5 -> f5 : A'=0, [ A>=1 ], cost: A

      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f300

      2: f300 -> f5 : A'=-1+A, [ A>=1 ], cost: 1

      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A



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

   Start location: f300

      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f300

      5: f300 -> f5 : A'=0, [ -1+A>=1 ], cost: A



Computing asymptotic complexity for rule 5

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+A (+/+!), A (+) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



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

   Complexity:  Poly(n^1)

   Cpx degree:  1

   Solved cost: n

   Rule cost:   A

   Rule guard:  [ -1+A>=1 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)