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


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 843 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B) -> Com_1(f1(C, 1)) :|: TRUE
f1(A, B) -> Com_1(f1(A - 10, B - 1)) :|: B >= 1 && A >= 101
f1(A, B) -> Com_1(f1(A + 11, B + 1)) :|: B >= 1 && 100 >= A

The start-symbols are:[f0_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f0

      0: f0 -> f1 : A'=free, B'=1, [], cost: 1

      1: f1 -> f1 : A'=-10+A, B'=-1+B, [ B>=1 && A>=101 ], cost: 1

      2: f1 -> f1 : A'=11+A, B'=1+B, [ B>=1 && 100>=A ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : A'=-10+A, B'=-1+B, [ B>=1 && A>=101 ], cost: 1

      2: f1 -> f1 : A'=11+A, B'=1+B, [ B>=1 && 100>=A ], cost: 1



   Accelerated rule 1 with metering function meter (where 10*meter==-100+A) (after adding A<=B), yielding the new rule 3.

   Accelerated rule 1 with backward acceleration, yielding the new rule 4.

   Accelerated rule 2 with metering function meter_1 (where 11*meter_1==100-A), yielding the new rule 5.

      During metering: Instantiating temporary variables by {meter==1}

   Nested simple loops 2 (outer loop) and 3 (inner loop) with metering function 89+10*meter-A, resulting in the new rules: 6.

      During metering: Instantiating temporary variables by {k==-1+B}

      During metering: Instantiating temporary variables by {k==1}

   Removing the simple loops: 1 2.



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

   Start location: f0

      0: f0 -> f1 : A'=free, B'=1, [], cost: 1

      3: f1 -> f1 : A'=-10*meter+A, B'=-meter+B, [ B>=1 && A>=101 && A<=B && 10*meter==-100+A && meter>=1 ], cost: meter

      4: f1 -> f1 : A'=A-10*k, B'=-k+B, [ B>=1 && A>=101 && k>0 && 1-k+B>=1 && 10+A-10*k>=101 ], cost: k

      5: f1 -> f1 : A'=11*meter_1+A, B'=meter_1+B, [ B>=1 && 100>=A && 11*meter_1==100-A && meter_1>=1 ], cost: meter_1

      6: f1 -> f1 : A'=979-10*(89+10*meter-A)*meter+110*meter-10*A, B'=89-(89+10*meter-A)*meter+10*meter-A+B, [ B>=1 && 100>=A && 11+A>=101 && 11+A<=1+B && 10*meter==-89+A && meter>=1 && 89+10*meter-A>=1 ], cost: 89+(89+10*meter-A)*meter+10*meter-A



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f1 : A'=free, B'=1, [], cost: 1

      7: f0 -> f1 : A'=-10+free, B'=0, [ free>=101 ], cost: 2

      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1



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

   Start location: f0

      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

      8: f0 -> f1 : A'=100, B'=1+meter_1, [ 100>=100-11*meter_1 && meter_1>=1 ], cost: 1+meter_1



Computing asymptotic complexity for rule 8

   Simplified the guard:

      8: f0 -> f1 : A'=100, B'=1+meter_1, [ meter_1>=1 ], cost: 1+meter_1

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+meter_1 (+), meter_1 (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {meter_1==n}

      resulting limit problem:

      [solved]



   Solution:

      meter_1 / n

   Resulting cost 1+n has complexity: Unbounded



Found new complexity Unbounded.



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

   Complexity:  Unbounded

   Cpx degree:  Unbounded

   Solved cost: 1+n

   Rule cost:   1+meter_1

   Rule guard:  [ meter_1>=1 ]



WORST_CASE(INF,?)


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(2)
BOUNDS(INF, INF)