/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 122 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f1(A, 1 + B, D, O, D, P, B, H, I, J, K, L, M, N)) :|: A >= B + 1 && B >= 0
f1(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f4(P, Q, R, U, T, F, G, O, S, V, W, C, M, N)) :|: B >= A && B >= 0 && Q >= O && O >= 2
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f4(R, S, Q, V, U, F, G, P, T, W, Y, 0, O, N)) :|: 0 >= P && 0 >= X
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f1(P, 2, R, Q, R, F, G, P, R, J, K, L, O, S)) :|: P >= 2
f3(A, B, C, D, E, F, G, H, I, J, K, L, M, N) -> Com_1(f4(P, Q, R, U, T, F, G, 1, S, V, W, D, O, N)) :|: TRUE

The start-symbols are:[f3_14]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: f3

      0: f1 -> f1 : B'=1+B, C'=D, D'=free_1, E'=D, F'=free, G'=B, [ A>=1+B && B>=0 ], cost: 1

      1: f1 -> f4 : A'=free_9, B'=free_3, C'=free_8, D'=free_2, E'=free_5, H'=free_6, Q'=free_7, J'=free_10, K'=free_4, L'=C, [ B>=A && B>=0 && free_3>=free_6 && free_6>=2 ], cost: 1

      2: f3 -> f4 : A'=free_19, B'=free_12, C'=free_18, D'=free_11, E'=free_14, H'=free_15, Q'=free_17, J'=free_21, K'=free_13, L'=0, M'=free_20, [ 0>=free_15 && 0>=free_16 ], cost: 1

      3: f3 -> f1 : A'=free_26, B'=2, C'=free_23, D'=free_25, E'=free_23, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=2 ], cost: 1

      4: f3 -> f4 : A'=free_34, B'=free_28, C'=free_33, D'=free_27, E'=free_30, H'=1, Q'=free_31, J'=free_32, K'=free_35, L'=D, M'=free_29, [], cost: 1



Checking for constant complexity:

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

      2: f3 -> f4 : A'=free_19, B'=free_12, C'=free_18, D'=free_11, E'=free_14, H'=free_15, Q'=free_17, J'=free_21, K'=free_13, L'=0, M'=free_20, [ 0>=free_15 && 0>=free_16 ], cost: 1



Removed unreachable and leaf rules:

   Start location: f3

      0: f1 -> f1 : B'=1+B, C'=D, D'=free_1, E'=D, F'=free, G'=B, [ A>=1+B && B>=0 ], cost: 1

      3: f3 -> f1 : A'=free_26, B'=2, C'=free_23, D'=free_25, E'=free_23, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=2 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: f1 -> f1 : B'=1+B, C'=D, D'=free_1, E'=D, F'=free, G'=B, [ A>=1+B && B>=0 ], cost: 1



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

   Removing the simple loops: 0.



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

   Start location: f3

      5: f1 -> f1 : B'=A, C'=free_1, D'=free_1, E'=free_1, F'=free, G'=-1+A, [ A>=1+B && B>=0 ], cost: A-B

      3: f3 -> f1 : A'=free_26, B'=2, C'=free_23, D'=free_25, E'=free_23, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=2 ], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: f3

      3: f3 -> f1 : A'=free_26, B'=2, C'=free_23, D'=free_25, E'=free_23, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=2 ], cost: 1

      6: f3 -> f1 : A'=free_26, B'=free_26, C'=free_1, D'=free_1, E'=free_1, F'=free, G'=-1+free_26, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=3 ], cost: -1+free_26



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

   Start location: f3

      6: f3 -> f1 : A'=free_26, B'=free_26, C'=free_1, D'=free_1, E'=free_1, F'=free, G'=-1+free_26, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=3 ], cost: -1+free_26



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f3

      6: f3 -> f1 : A'=free_26, B'=free_26, C'=free_1, D'=free_1, E'=free_1, F'=free, G'=-1+free_26, H'=free_26, Q'=free_23, M'=free_22, N'=free_24, [ free_26>=3 ], cost: -1+free_26



Computing asymptotic complexity for rule 6

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -1+free_26 (+), -2+free_26 (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {free_26==n}

      resulting limit problem:

      [solved]



   Solution:

      free_26 / 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+free_26

   Rule guard:  [ free_26>=3 ]



WORST_CASE(INF,?)


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(2)
BOUNDS(INF, INF)