/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, 2 + Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 32 ms]
(2) BOUNDS(1, max(1, 2 + Arg_0))
(3) Loat Proof [FINISHED, 133 ms]
(4) BOUNDS(n^1, INF)


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A) -> Com_1(f1(A)) :|: TRUE
f1(A) -> Com_1(f1(A - 1000)) :|: A >= 1001

The start-symbols are:[f0_1]


----------------------------------------

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



Initial Complexity Problem:

  Start:  f0  

  Program_Vars:  Arg_0  

  Temp_Vars:    

  Locations:  f0, f1  

  Transitions:

  0: f0->f1

  1: f1->f1  



Timebounds: 

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

  0: f0->f1: 1 {O(1)}

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



Costbounds:

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

  0: f0->f1: 1 {O(1)}

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



Sizebounds:

`Lower:

  0: f0->f1, Arg_0: Arg_0 {O(n)}

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

`Upper:

  0: f0->f1, Arg_0: Arg_0 {O(n)}

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


----------------------------------------

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

----------------------------------------

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      1: f1 -> f1 : A'=-1000+A, [ A>=1001 ], cost: 1



Checking for constant complexity:

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

      0: f0 -> f1 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : A'=-1000+A, [ A>=1001 ], cost: 1



   Accelerated rule 1 with metering function meter (where 1000*meter==-1000+A), yielding the new rule 2.

   Removing the simple loops: 1.



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

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      2: f1 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: meter



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f1 : [], cost: 1

      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter



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

   Start location: f0

      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

      3: f0 -> f1 : A'=-1000*meter+A, [ A>=1001 && 1000*meter==-1000+A && meter>=1 ], cost: 1+meter



Computing asymptotic complexity for rule 3

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+meter (+), -999-1000*meter+A (+/+!), 1001+1000*meter-A (+/+!), -1000+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {meter==n,A==1000+1000*n}

      resulting limit problem:

      [solved]



   Solution:

      meter / n

      A / 1000+1000*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+meter

   Rule guard:  [ A>=1001 && 1000*meter==-1000+A ]



WORST_CASE(Omega(n^1),?)


----------------------------------------

(4)
BOUNDS(n^1, INF)