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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 223 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f3(A, B) -> Com_1(f0(0, B)) :|: TRUE
f0(A, B) -> Com_1(f0(A, B - 1)) :|: B >= 1
f4(A, B) -> Com_1(f4(A, B)) :|: TRUE
f0(A, B) -> Com_1(f4(-(1), B)) :|: 0 >= B

The start-symbols are:[f3_2]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f3 -> f0 : A'=0, [], cost: 1

      1: f0 -> f0 : B'=-1+B, [ B>=1 ], cost: 1

      3: f0 -> f4 : A'=-1, [ 0>=B ], cost: 1

      2: f4 -> f4 : [], cost: 1



Checking for constant complexity:

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

      0: f3 -> f0 : A'=0, [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f0 -> f0 : B'=-1+B, [ B>=1 ], cost: 1



   Accelerated rule 1 with metering function B, yielding the new rule 4.

   Removing the simple loops: 1.



Accelerating simple loops of location 2.

   Accelerating the following rules:

      2: f4 -> f4 : [], cost: 1



   Accelerated rule 2 with NONTERM, yielding the new rule 5.

   Removing the simple loops: 2.



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

   Start location: f3

      0: f3 -> f0 : A'=0, [], cost: 1

      3: f0 -> f4 : A'=-1, [ 0>=B ], cost: 1

      4: f0 -> f0 : B'=0, [ B>=1 ], cost: B

      5: f4 -> [4] : [], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f3

      0: f3 -> f0 : A'=0, [], cost: 1

      6: f3 -> f0 : A'=0, B'=0, [ B>=1 ], cost: 1+B

      3: f0 -> f4 : A'=-1, [ 0>=B ], cost: 1

      7: f0 -> [4] : A'=-1, [ 0>=B ], cost: NONTERM



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

   Start location: f3

      0: f3 -> f0 : A'=0, [], cost: 1

      6: f3 -> f0 : A'=0, B'=0, [ B>=1 ], cost: 1+B

      7: f0 -> [4] : A'=-1, [ 0>=B ], cost: NONTERM



Eliminated locations (on tree-shaped paths):

   Start location: f3

      8: f3 -> [4] : A'=-1, [ 0>=B ], cost: NONTERM

      9: f3 -> [4] : A'=-1, B'=0, [ B>=1 ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

      8: f3 -> [4] : A'=-1, [ 0>=B ], cost: NONTERM

      9: f3 -> [4] : A'=-1, B'=0, [ B>=1 ], cost: NONTERM



Computing asymptotic complexity for rule 8

   Guard is satisfiable, yielding nontermination

   Resulting cost NONTERM has complexity: Nonterm



Found new complexity Nonterm.



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

   Complexity:  Nonterm

   Cpx degree:  Nonterm

   Solved cost: NONTERM

   Rule cost:   NONTERM

   Rule guard:  [ 0>=B ]



NO


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(2)
BOUNDS(INF, INF)