/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1299 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f2(A, B, C, D) -> Com_1(f1(A, B, C, D)) :|: TRUE
f1(A, B, C, D) -> Com_1(f1(1 + A, B, E, D)) :|: E >= 1 && B >= 1 + A
f1(A, B, C, D) -> Com_1(f1(1 + A, B, E, D)) :|: 0 >= E + 1 && B >= 1 + A
f1(A, B, C, D) -> Com_1(f1(A, B, 0, D)) :|: B >= 1 + A
f1(A, B, C, D) -> Com_1(f1(1 + A, A, E, D)) :|: E >= 1 && B >= F && A >= B && A <= B
f1(A, B, C, D) -> Com_1(f1(1 + A, A, E, D)) :|: 0 >= E + 1 && B >= F && A >= B && A <= B
f1(A, B, C, D) -> Com_1(f1(A, A, 0, D)) :|: B >= E && A >= B && A <= B
f1(A, B, C, D) -> Com_1(f300(A, B, C, E)) :|: A >= B && A >= B + 1
f1(A, B, C, D) -> Com_1(f300(A, B, C, E)) :|: A >= B && B >= A + 1

The start-symbols are:[f2_4]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f2

      0: f2 -> f1 : [], cost: 1

      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1

      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1

      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1

      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && A>=free_2 && B==A ], cost: 1

      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && A>=free_4 && B==A ], cost: 1

      6: f1 -> f1 : A'=B, C'=0, [ A>=free_6 && B==A ], cost: 1

      7: f1 -> f300 : D'=free_7, [ B>=A && B>=1+A ], cost: 1

      8: f1 -> f300 : D'=free_8, [ B>=A && A>=1+B ], cost: 1



Checking for constant complexity:

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

      0: f2 -> f1 : [], cost: 1



Removed unreachable and leaf rules:

   Start location: f2

      0: f2 -> f1 : [], cost: 1

      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1

      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1

      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1

      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && A>=free_2 && B==A ], cost: 1

      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && A>=free_4 && B==A ], cost: 1

      6: f1 -> f1 : A'=B, C'=0, [ A>=free_6 && B==A ], cost: 1



Simplified all rules, resulting in:

   Start location: f2

      0: f2 -> f1 : [], cost: 1

      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1

      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1

      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1

      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && B==A ], cost: 1

      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1

      6: f1 -> f1 : A'=B, C'=0, [ B==A ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1

      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1

      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1

      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && B==A ], cost: 1

      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1

      6: f1 -> f1 : A'=B, C'=0, [ B==A ], cost: 1



   Accelerated rule 1 with metering function A-B, yielding the new rule 9.

   Accelerated rule 2 with metering function A-B, yielding the new rule 10.

   Accelerated rule 3 with NONTERM, yielding the new rule 11.

   Accelerated rule 4 with metering function 1+A-B, yielding the new rule 12.

   Accelerated rule 5 with metering function 1+A-B, yielding the new rule 13.

   Accelerated rule 6 with NONTERM, yielding the new rule 14.

   Removing the simple loops: 1 2 3 4 5 6.



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

   Start location: f2

      0: f2 -> f1 : [], cost: 1

      9: f1 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: A-B

     10: f1 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: A-B

     11: f1 -> [3] : [ A>=1+B ], cost: NONTERM

     12: f1 -> f1 : A'=A, B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 1+A-B

     13: f1 -> f1 : A'=A, B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1+A-B

     14: f1 -> [3] : [ B==A ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: f2

      0: f2 -> f1 : [], cost: 1

     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B

     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B

     17: f2 -> [3] : [ A>=1+B ], cost: NONTERM

     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B

     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B

     20: f2 -> [3] : [ B==A ], cost: NONTERM



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

   Start location: f2

     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B

     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B

     17: f2 -> [3] : [ A>=1+B ], cost: NONTERM

     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B

     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B

     20: f2 -> [3] : [ B==A ], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f2

     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B

     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B

     17: f2 -> [3] : [ A>=1+B ], cost: NONTERM

     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B

     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B

     20: f2 -> [3] : [ B==A ], cost: NONTERM



Computing asymptotic complexity for rule 15

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free==1,A==0,B==-n}

      resulting limit problem:

      [solved]



   Solution:

      free / 1

      A / 0

      B / -n

   Resulting cost 1+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 17

   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:  [ A>=1+B ]



NO


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(2)
BOUNDS(INF, INF)