/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(2, 4 + 2 * Arg_1) + nat(1 + Arg_0) + nat(1 + Arg_1)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 339 ms]
(2) BOUNDS(1, max(2, 4 + 2 * Arg_1) + nat(1 + Arg_0) + nat(1 + Arg_1))
(3) Loat Proof [FINISHED, 397 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B, C) -> Com_1(f2(-(1) + A, B, C)) :|: A >= 31
f2(A, B, C) -> Com_1(f300(A, -(1) + B, C)) :|: 30 >= A
f300(A, B, C) -> Com_1(f2(A, B, C)) :|: B >= 21
f300(A, B, C) -> Com_1(f1(A, B, D)) :|: 20 >= B
f3(A, B, C) -> Com_1(f300(A, B, C)) :|: TRUE

The start-symbols are:[f3_3]


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(1) Koat2 Proof (FINISHED)
YES( ?, 2+max([0, 2*(1+Arg_1)])+max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)})



Initial Complexity Problem:

  Start:  f3  

  Program_Vars:  Arg_0, Arg_1, Arg_2  

  Temp_Vars:  D  

  Locations:  f1, f2, f3, f300  

  Transitions:

  0: f2->f2

  1: f2->f300

  4: f3->f300

  3: f300->f1

  2: f300->f2  



Timebounds: 

  Overall timebound: 2+max([0, 2*(1+Arg_1)])+max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)}

  0: f2->f2: max([0, 1+Arg_0]) {O(n)}

  1: f2->f300: max([0, 2*(1+Arg_1)]) {O(n)}

  4: f3->f300: 1 {O(1)}

  2: f300->f2: max([0, 1+Arg_1]) {O(n)}

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



Costbounds:

  Overall costbound: 2+max([0, 2*(1+Arg_1)])+max([0, 1+Arg_0])+max([0, 1+Arg_1]) {O(n)}

  0: f2->f2: max([0, 1+Arg_0]) {O(n)}

  1: f2->f300: max([0, 2*(1+Arg_1)]) {O(n)}

  4: f3->f300: 1 {O(1)}

  2: f300->f2: max([0, 1+Arg_1]) {O(n)}

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



Sizebounds:

`Lower:

  0: f2->f2, Arg_0: 30 {O(1)}

  0: f2->f2, Arg_1: 21 {O(1)}

  0: f2->f2, Arg_2: Arg_2 {O(n)}

  1: f2->f300, Arg_0: min([30, Arg_0]) {O(n)}

  1: f2->f300, Arg_1: 20 {O(1)}

  1: f2->f300, Arg_2: Arg_2 {O(n)}

  4: f3->f300, Arg_0: Arg_0 {O(n)}

  4: f3->f300, Arg_1: Arg_1 {O(n)}

  4: f3->f300, Arg_2: Arg_2 {O(n)}

  2: f300->f2, Arg_0: min([30, Arg_0]) {O(n)}

  2: f300->f2, Arg_1: 21 {O(1)}

  2: f300->f2, Arg_2: Arg_2 {O(n)}

  3: f300->f1, Arg_0: min([30, Arg_0]) {O(n)}

  3: f300->f1, Arg_1: min([20, Arg_1]) {O(n)}

`Upper:

  0: f2->f2, Arg_0: max([30, Arg_0]) {O(n)}

  0: f2->f2, Arg_1: Arg_1 {O(n)}

  0: f2->f2, Arg_2: Arg_2 {O(n)}

  1: f2->f300, Arg_0: 30 {O(1)}

  1: f2->f300, Arg_1: Arg_1 {O(n)}

  1: f2->f300, Arg_2: Arg_2 {O(n)}

  4: f3->f300, Arg_0: Arg_0 {O(n)}

  4: f3->f300, Arg_1: Arg_1 {O(n)}

  4: f3->f300, Arg_2: Arg_2 {O(n)}

  2: f300->f2, Arg_0: max([30, Arg_0]) {O(n)}

  2: f300->f2, Arg_1: Arg_1 {O(n)}

  2: f300->f2, Arg_2: Arg_2 {O(n)}

  3: f300->f1, Arg_0: max([30, Arg_0]) {O(n)}

  3: f300->f1, Arg_1: 20 {O(1)}


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(2)
BOUNDS(1, max(2, 4 + 2 * Arg_1) + nat(1 + Arg_0) + nat(1 + Arg_1))

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f2 -> f2 : A'=-1+A, [ A>=31 ], cost: 1

      1: f2 -> f300 : B'=-1+B, [ 30>=A ], cost: 1

      2: f300 -> f2 : [ B>=21 ], cost: 1

      3: f300 -> f1 : C'=free, [ 20>=B ], cost: 1

      4: f3 -> f300 : [], cost: 1



Checking for constant complexity:

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

      4: f3 -> f300 : [], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

      0: f2 -> f2 : A'=-1+A, [ A>=31 ], cost: 1

      1: f2 -> f300 : B'=-1+B, [ 30>=A ], cost: 1

      2: f300 -> f2 : [ B>=21 ], cost: 1

      4: f3 -> f300 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f2 -> f2 : A'=-1+A, [ A>=31 ], cost: 1



   Accelerated rule 0 with metering function -30+A, yielding the new rule 5.

   Removing the simple loops: 0.



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

   Start location: f3

      1: f2 -> f300 : B'=-1+B, [ 30>=A ], cost: 1

      5: f2 -> f2 : A'=30, [ A>=31 ], cost: -30+A

      2: f300 -> f2 : [ B>=21 ], cost: 1

      4: f3 -> f300 : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      1: f2 -> f300 : B'=-1+B, [ 30>=A ], cost: 1

      2: f300 -> f2 : [ B>=21 ], cost: 1

      6: f300 -> f2 : A'=30, [ B>=21 && A>=31 ], cost: -29+A

      4: f3 -> f300 : [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: f3

      7: f300 -> f300 : B'=-1+B, [ B>=21 && 30>=A ], cost: 2

      8: f300 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -28+A

      4: f3 -> f300 : [], cost: 1



Accelerating simple loops of location 1.

   Accelerating the following rules:

      7: f300 -> f300 : B'=-1+B, [ B>=21 && 30>=A ], cost: 2

      8: f300 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -28+A



   Accelerated rule 7 with metering function -20+B, yielding the new rule 9.

   Found no metering function for rule 8.

   Removing the simple loops: 7.



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

   Start location: f3

      8: f300 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -28+A

      9: f300 -> f300 : B'=20, [ B>=21 && 30>=A ], cost: -40+2*B

      4: f3 -> f300 : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      4: f3 -> f300 : [], cost: 1

     10: f3 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -27+A

     11: f3 -> f300 : B'=20, [ B>=21 && 30>=A ], cost: -39+2*B



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

   Start location: f3

     10: f3 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -27+A

     11: f3 -> f300 : B'=20, [ B>=21 && 30>=A ], cost: -39+2*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

     10: f3 -> f300 : A'=30, B'=-1+B, [ B>=21 && A>=31 ], cost: -27+A

     11: f3 -> f300 : B'=20, [ B>=21 && 30>=A ], cost: -39+2*B



Computing asymptotic complexity for rule 10

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -27+A (+), -30+A (+/+!), -20+B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

      B / n

   Resulting cost -27+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: -27+n

   Rule cost:   -27+A

   Rule guard:  [ B>=21 && A>=31 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)