/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 2 + Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 54 ms]
(2) BOUNDS(1, max(2, 2 + Arg_1))
(3) Loat Proof [FINISHED, 145 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
l0(A, B) -> Com_1(l1(0, B)) :|: TRUE
l1(A, B) -> Com_1(l1(A + 1, B - 1)) :|: B >= 1
l1(A, B) -> Com_1(l2(A, B)) :|: 0 >= B

The start-symbols are:[l0_2]


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(1) Koat2 Proof (FINISHED)
YES( ?, max([2, 2+Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  l0  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:    

  Locations:  l0, l1, l2  

  Transitions:

  l0(Arg_0,Arg_1) -> l1(0,Arg_1):|:

  l1(Arg_0,Arg_1) -> l1(Arg_0+1,Arg_1-1):|:0 <= Arg_0 && 1 <= Arg_1

  l1(Arg_0,Arg_1) -> l2(Arg_0,Arg_1):|:0 <= Arg_0 && Arg_1 <= 0  



Timebounds: 

  Overall timebound: max([2, 2+Arg_1]) {O(n)}

  0: l0->l1: 1 {O(1)}

  1: l1->l1: max([0, Arg_1]) {O(n)}

  2: l1->l2: 1 {O(1)}



Costbounds:

  Overall costbound: max([2, 2+Arg_1]) {O(n)}

  0: l0->l1: 1 {O(1)}

  1: l1->l1: max([0, Arg_1]) {O(n)}

  2: l1->l2: 1 {O(1)}



Sizebounds:

`Lower:

  0: l0->l1, Arg_0: 0 {O(1)}

  0: l0->l1, Arg_1: Arg_1 {O(n)}

  1: l1->l1, Arg_0: 1 {O(1)}

  1: l1->l1, Arg_1: 0 {O(1)}

  2: l1->l2, Arg_0: 0 {O(1)}

  2: l1->l2, Arg_1: min([0, Arg_1]) {O(n)}

`Upper:

  0: l0->l1, Arg_0: 0 {O(1)}

  0: l0->l1, Arg_1: Arg_1 {O(n)}

  1: l1->l1, Arg_0: max([0, Arg_1]) {O(n)}

  1: l1->l1, Arg_1: Arg_1 {O(n)}

  2: l1->l2, Arg_0: max([0, Arg_1]) {O(n)}

  2: l1->l2, Arg_1: 0 {O(1)}


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(2)
BOUNDS(1, max(2, 2 + Arg_1))

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1

      2: l1 -> l2 : [ 0>=B ], cost: 1



Removed unreachable and leaf rules:

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1



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

   Removing the simple loops: 1.



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

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      3: l1 -> l1 : A'=A+B, B'=0, [ B>=1 ], cost: B



Chained accelerated rules (with incoming rules):

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



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

   Start location: l0

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: l0

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



Computing asymptotic complexity for rule 4

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      B / n

   Resulting cost 1+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: 1+n

   Rule cost:   1+B

   Rule guard:  [ B>=1 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)