/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).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, 621 ms] (2) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f6(A, B, C) -> Com_1(f9(A, D, C)) :|: 0 >= A + 1 f6(A, B, C) -> Com_1(f9(A, D, C)) :|: A >= 1 f6(A, B, C) -> Com_1(f17(0, B, C)) :|: A >= 0 && A <= 0 f17(A, B, C) -> Com_1(f24(A, B, C)) :|: 0 >= C f17(A, B, C) -> Com_1(f24(A, B, C)) :|: C >= 2 f17(A, B, C) -> Com_1(f24(A, B, 0)) :|: C >= 1 && C <= 1 f9(A, B, C) -> Com_1(f17(A, 0, 1)) :|: B >= 0 && B <= 0 f9(A, B, C) -> Com_1(f6(D, B, C)) :|: 0 >= B + 1 f9(A, B, C) -> Com_1(f6(D, B, C)) :|: B >= 1 f0(A, B, C) -> Com_1(f6(D, B, 0)) :|: TRUE The start-symbols are:[f0_3] ---------------------------------------- (1) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f0 0: f6 -> f9 : B'=free, [ 0>=1+A ], cost: 1 1: f6 -> f9 : B'=free_1, [ A>=1 ], cost: 1 2: f6 -> f17 : A'=0, [ A==0 ], cost: 1 3: f17 -> f24 : [ 0>=C ], cost: 1 4: f17 -> f24 : [ C>=2 ], cost: 1 5: f17 -> f24 : C'=0, [ C==1 ], cost: 1 6: f9 -> f17 : B'=0, C'=1, [ B==0 ], cost: 1 7: f9 -> f6 : A'=free_2, [ 0>=1+B ], cost: 1 8: f9 -> f6 : A'=free_3, [ B>=1 ], cost: 1 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 Removed unreachable and leaf rules: Start location: f0 0: f6 -> f9 : B'=free, [ 0>=1+A ], cost: 1 1: f6 -> f9 : B'=free_1, [ A>=1 ], cost: 1 7: f9 -> f6 : A'=free_2, [ 0>=1+B ], cost: 1 8: f9 -> f6 : A'=free_3, [ B>=1 ], cost: 1 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on tree-shaped paths): Start location: f0 10: f6 -> f6 : A'=free_2, B'=free, [ 0>=1+A && 0>=1+free ], cost: 2 11: f6 -> f6 : A'=free_3, B'=free, [ 0>=1+A && free>=1 ], cost: 2 12: f6 -> f6 : A'=free_2, B'=free_1, [ A>=1 && 0>=1+free_1 ], cost: 2 13: f6 -> f6 : A'=free_3, B'=free_1, [ A>=1 && free_1>=1 ], cost: 2 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 Accelerating simple loops of location 0. Accelerating the following rules: 10: f6 -> f6 : A'=free_2, B'=free, [ 0>=1+A && 0>=1+free ], cost: 2 11: f6 -> f6 : A'=free_3, B'=free, [ 0>=1+A && free>=1 ], cost: 2 12: f6 -> f6 : A'=free_2, B'=free_1, [ A>=1 && 0>=1+free_1 ], cost: 2 13: f6 -> f6 : A'=free_3, B'=free_1, [ A>=1 && free_1>=1 ], cost: 2 Accelerated rule 10 with NONTERM (after strengthening guard), yielding the new rule 14. Accelerated rule 11 with NONTERM (after strengthening guard), yielding the new rule 15. Accelerated rule 12 with NONTERM (after strengthening guard), yielding the new rule 16. Accelerated rule 13 with NONTERM (after strengthening guard), yielding the new rule 17. Removing the simple loops:. Accelerated all simple loops using metering functions (where possible): Start location: f0 10: f6 -> f6 : A'=free_2, B'=free, [ 0>=1+A && 0>=1+free ], cost: 2 11: f6 -> f6 : A'=free_3, B'=free, [ 0>=1+A && free>=1 ], cost: 2 12: f6 -> f6 : A'=free_2, B'=free_1, [ A>=1 && 0>=1+free_1 ], cost: 2 13: f6 -> f6 : A'=free_3, B'=free_1, [ A>=1 && free_1>=1 ], cost: 2 14: f6 -> [5] : [ 0>=1+A && 0>=1+free && 0>=1+free_2 ], cost: NONTERM 15: f6 -> [5] : [ 0>=1+A && free>=1 && 0>=1+free_3 ], cost: NONTERM 16: f6 -> [5] : [ A>=1 && 0>=1+free_1 && free_2>=1 ], cost: NONTERM 17: f6 -> [5] : [ A>=1 && free_1>=1 && free_3>=1 ], cost: NONTERM 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: f0 9: f0 -> f6 : A'=free_4, C'=0, [], cost: 1 18: f0 -> f6 : A'=free_2, B'=free, C'=0, [ 0>=1+free ], cost: 3 19: f0 -> f6 : A'=free_3, B'=free, C'=0, [ free>=1 ], cost: 3 20: f0 -> f6 : A'=free_2, B'=free_1, C'=0, [ 0>=1+free_1 ], cost: 3 21: f0 -> f6 : A'=free_3, B'=free_1, C'=0, [ free_1>=1 ], cost: 3 22: f0 -> [5] : A'=free_4, C'=0, [ 0>=1+free_4 ], cost: NONTERM 23: f0 -> [5] : A'=free_4, C'=0, [ 0>=1+free_4 ], cost: NONTERM 24: f0 -> [5] : A'=free_4, C'=0, [ free_4>=1 ], cost: NONTERM 25: f0 -> [5] : A'=free_4, C'=0, [ free_4>=1 ], cost: NONTERM Removed unreachable locations (and leaf rules with constant cost): Start location: f0 22: f0 -> [5] : A'=free_4, C'=0, [ 0>=1+free_4 ], cost: NONTERM 23: f0 -> [5] : A'=free_4, C'=0, [ 0>=1+free_4 ], cost: NONTERM 24: f0 -> [5] : A'=free_4, C'=0, [ free_4>=1 ], cost: NONTERM 25: f0 -> [5] : A'=free_4, C'=0, [ free_4>=1 ], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 23: f0 -> [5] : A'=free_4, C'=0, [ 0>=1+free_4 ], cost: NONTERM 25: f0 -> [5] : A'=free_4, C'=0, [ free_4>=1 ], cost: NONTERM Computing asymptotic complexity for rule 23 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>=1+free_4 ] NO ---------------------------------------- (2) BOUNDS(INF, INF)