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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 169 ms]
(2) BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_0^2) + nat(2 * Arg_0))
(3) Loat Proof [FINISHED, 336 ms]
(4) BOUNDS(n^2, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B) -> Com_1(f3(A, A)) :|: A >= 1
f3(A, B) -> Com_1(f3(A, B - 1)) :|: B >= 1
f3(A, B) -> Com_1(f3(A - 1, A - 1)) :|: 0 >= B && A >= 2

The start-symbols are:[f1_2]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)})



Initial Complexity Problem:

  Start:  f1  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:    

  Locations:  f1, f3  

  Transitions:

  f1(Arg_0,Arg_1) -> f3(Arg_0,Arg_0):|:1 <= Arg_0

  f3(Arg_0,Arg_1) -> f3(Arg_0,Arg_1-1):|:Arg_1 <= Arg_0 && 1 <= Arg_0 && 1 <= Arg_1

  f3(Arg_0,Arg_1) -> f3(Arg_0-1,Arg_0-1):|:Arg_1 <= Arg_0 && 1 <= Arg_0 && Arg_1 <= 0 && 2 <= Arg_0  



Timebounds: 

  Overall timebound: max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)}

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

  1: f3->f3: max([0, Arg_0*(-1+Arg_0)])+max([0, Arg_0]) {O(n^2)}

  2: f3->f3: max([0, Arg_0]) {O(n)}



Costbounds:

  Overall costbound: max([1, 1+Arg_0*(-1+Arg_0)])+max([0, Arg_0])+max([0, Arg_0]) {O(n^2)}

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

  1: f3->f3: max([0, Arg_0*(-1+Arg_0)])+max([0, Arg_0]) {O(n^2)}

  2: f3->f3: max([0, Arg_0]) {O(n)}



Sizebounds:

`Lower:

  0: f1->f3, Arg_0: 1 {O(1)}

  0: f1->f3, Arg_1: 1 {O(1)}

  1: f3->f3, Arg_0: 1 {O(1)}

  1: f3->f3, Arg_1: 0 {O(1)}

  2: f3->f3, Arg_0: 1 {O(1)}

  2: f3->f3, Arg_1: 1 {O(1)}

`Upper:

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

  0: f1->f3, Arg_1: Arg_0 {O(n)}

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

  1: f3->f3, Arg_1: max([Arg_0, -1+Arg_0]) {O(n)}

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

  2: f3->f3, Arg_1: -1+Arg_0 {O(n)}


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(2)
BOUNDS(1, max(1, 1 + -1 * Arg_0 + Arg_0^2) + nat(2 * Arg_0))

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f1

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

      1: f3 -> f3 : B'=-1+B, [ B>=1 ], cost: 1

      2: f3 -> f3 : A'=-1+A, B'=-1+A, [ 0>=B && A>=2 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f3 -> f3 : B'=-1+B, [ B>=1 ], cost: 1

      2: f3 -> f3 : A'=-1+A, B'=-1+A, [ 0>=B && A>=2 ], cost: 1



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

   Found no metering function for rule 2.

   Nested simple loops 2 (outer loop) and 3 (inner loop) with metering function -1+A, resulting in the new rules: 4, 5.

   Removing the simple loops: 1 2.



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

   Start location: f1

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

      3: f3 -> f3 : B'=0, [ B>=1 ], cost: B

      4: f3 -> f3 : A'=1, B'=1, [ B>=1 && A>=2 ], cost: -3/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2

      5: f3 -> f3 : A'=1, B'=1, [ 0>=B && -1+A>=2 ], cost: -2-1/2*(-2+A)^2+3/2*A+(-1+A)*(-2+A)



Chained accelerated rules (with incoming rules):

   Start location: f1

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

      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A

      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2



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

   Start location: f1

      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A

      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f1

      6: f1 -> f3 : B'=0, [ A>=1 ], cost: 1+A

      7: f1 -> f3 : A'=1, B'=1, [ A>=2 ], cost: -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2



Computing asymptotic complexity for rule 6

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      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 1+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 7

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+A (+/+!), -1+3/2*A+1/2*A^2 (+) [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 -1+3/2*n+1/2*n^2 has complexity: Poly(n^2)



Found new complexity Poly(n^2).



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

   Complexity:  Poly(n^2)

   Cpx degree:  2

   Solved cost: -1+3/2*n+1/2*n^2

   Rule cost:   -1/2+(-1+A)*A+3/2*A-1/2*(-1+A)^2

   Rule guard:  [ A>=2 ]



WORST_CASE(Omega(n^2),?)


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(4)
BOUNDS(n^2, INF)