/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(Omega(n^1), O(n^1))
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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(n^1, max(2, 3 + Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 106 ms]
(2) BOUNDS(1, max(2, 3 + Arg_0))
(3) Loat Proof [FINISHED, 91 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f3(A, B) -> Com_1(f1(A, B)) :|: TRUE
f1(A, B) -> Com_1(f1(-(1) + A, B)) :|: A >= 0
f1(A, B) -> Com_1(f300(A, C)) :|: 0 >= A + 1

The start-symbols are:[f3_2]


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



Initial Complexity Problem:

  Start:  f3  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:  C  

  Locations:  f1, f3, f300  

  Transitions:

  f1(Arg_0,Arg_1) -> f1(-1+Arg_0,Arg_1):|:0 <= Arg_0

  f1(Arg_0,Arg_1) -> f300(Arg_0,C):|:Arg_0+1 <= 0

  f3(Arg_0,Arg_1) -> f1(Arg_0,Arg_1):|:  



Timebounds: 

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

  1: f1->f1: max([0, 1+Arg_0]) {O(n)}

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

  0: f3->f1: 1 {O(1)}



Costbounds:

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

  1: f1->f1: max([0, 1+Arg_0]) {O(n)}

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

  0: f3->f1: 1 {O(1)}



Sizebounds:

`Lower:

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

  1: f1->f1, Arg_1: Arg_1 {O(n)}

  2: f1->f300, Arg_0: min([-1, Arg_0]) {O(n)}

  0: f3->f1, Arg_0: Arg_0 {O(n)}

  0: f3->f1, Arg_1: Arg_1 {O(n)}

`Upper:

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

  1: f1->f1, Arg_1: Arg_1 {O(n)}

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

  0: f3->f1, Arg_0: Arg_0 {O(n)}

  0: f3->f1, Arg_1: Arg_1 {O(n)}


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

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f3 -> f1 : [], cost: 1

      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1

      2: f1 -> f300 : B'=free, [ 0>=1+A ], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

      0: f3 -> f1 : [], cost: 1

      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : A'=-1+A, [ A>=0 ], cost: 1



   Accelerated rule 1 with metering function 1+A, yielding the new rule 3.

   Removing the simple loops: 1.



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

   Start location: f3

      0: f3 -> f1 : [], cost: 1

      3: f1 -> f1 : A'=-1, [ A>=0 ], cost: 1+A



Chained accelerated rules (with incoming rules):

   Start location: f3

      0: f3 -> f1 : [], cost: 1

      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A



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

   Start location: f3

      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

      4: f3 -> f1 : A'=-1, [ A>=0 ], cost: 2+A



Computing asymptotic complexity for rule 4

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      2+A (+), 1+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 2+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: 2+n

   Rule cost:   2+A

   Rule guard:  [ A>=0 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)