/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).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, 527 ms] (2) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f9(A, B, C, D) -> Com_1(f14(A, 0, E, D)) :|: 0 >= A f14(A, B, C, D) -> Com_1(f14(A, B, C - 1, D)) :|: C >= 1 f22(A, B, C, D) -> Com_1(f22(A, B, C, D)) :|: TRUE f24(A, B, C, D) -> Com_1(f27(A, B, C, D)) :|: TRUE f14(A, B, C, D) -> Com_1(f9(E, B, C, 0)) :|: 0 >= C f9(A, B, C, D) -> Com_1(f22(A, B, C, D)) :|: A >= 1 f0(A, B, C, D) -> Com_1(f9(E, 0, C, 0)) :|: TRUE The start-symbols are:[f0_4] ---------------------------------------- (1) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f0 0: f9 -> f14 : B'=0, C'=free, [ 0>=A ], cost: 1 5: f9 -> f22 : [ A>=1 ], cost: 1 1: f14 -> f14 : C'=-1+C, [ C>=1 ], cost: 1 4: f14 -> f9 : A'=free_1, D'=0, [ 0>=C ], cost: 1 2: f22 -> f22 : [], cost: 1 3: f24 -> f27 : [], cost: 1 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Removed unreachable and leaf rules: Start location: f0 0: f9 -> f14 : B'=0, C'=free, [ 0>=A ], cost: 1 5: f9 -> f22 : [ A>=1 ], cost: 1 1: f14 -> f14 : C'=-1+C, [ C>=1 ], cost: 1 4: f14 -> f9 : A'=free_1, D'=0, [ 0>=C ], cost: 1 2: f22 -> f22 : [], cost: 1 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 1: f14 -> f14 : C'=-1+C, [ C>=1 ], cost: 1 Accelerated rule 1 with metering function C, yielding the new rule 7. Removing the simple loops: 1. Accelerating simple loops of location 2. Accelerating the following rules: 2: f22 -> f22 : [], cost: 1 Accelerated rule 2 with NONTERM, yielding the new rule 8. Removing the simple loops: 2. Accelerated all simple loops using metering functions (where possible): Start location: f0 0: f9 -> f14 : B'=0, C'=free, [ 0>=A ], cost: 1 5: f9 -> f22 : [ A>=1 ], cost: 1 4: f14 -> f9 : A'=free_1, D'=0, [ 0>=C ], cost: 1 7: f14 -> f14 : C'=0, [ C>=1 ], cost: C 8: f22 -> [7] : [], cost: INF 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: f0 0: f9 -> f14 : B'=0, C'=free, [ 0>=A ], cost: 1 5: f9 -> f22 : [ A>=1 ], cost: 1 9: f9 -> f14 : B'=0, C'=0, [ 0>=A && free>=1 ], cost: 1+free 10: f9 -> [7] : [ A>=1 ], cost: INF 4: f14 -> f9 : A'=free_1, D'=0, [ 0>=C ], cost: 1 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: f0 0: f9 -> f14 : B'=0, C'=free, [ 0>=A ], cost: 1 9: f9 -> f14 : B'=0, C'=0, [ 0>=A && free>=1 ], cost: 1+free 10: f9 -> [7] : [ A>=1 ], cost: INF 4: f14 -> f9 : A'=free_1, D'=0, [ 0>=C ], cost: 1 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: f0 10: f9 -> [7] : [ A>=1 ], cost: INF 11: f9 -> f9 : A'=free_1, B'=0, C'=free, D'=0, [ 0>=A && 0>=free ], cost: 2 12: f9 -> f9 : A'=free_1, B'=0, C'=0, D'=0, [ 0>=A && free>=1 ], cost: 2+free 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Accelerating simple loops of location 0. Accelerating the following rules: 11: f9 -> f9 : A'=free_1, B'=0, C'=free, D'=0, [ 0>=A && 0>=free ], cost: 2 12: f9 -> f9 : A'=free_1, B'=0, C'=0, D'=0, [ 0>=A && free>=1 ], cost: 2+free Accelerated rule 11 with NONTERM (after strengthening guard), yielding the new rule 13. Accelerated rule 12 with NONTERM (after strengthening guard), yielding the new rule 14. Removing the simple loops:. Accelerated all simple loops using metering functions (where possible): Start location: f0 10: f9 -> [7] : [ A>=1 ], cost: INF 11: f9 -> f9 : A'=free_1, B'=0, C'=free, D'=0, [ 0>=A && 0>=free ], cost: 2 12: f9 -> f9 : A'=free_1, B'=0, C'=0, D'=0, [ 0>=A && free>=1 ], cost: 2+free 13: f9 -> [8] : [ 0>=A && 0>=free && 0>=free_1 ], cost: INF 14: f9 -> [8] : [ 0>=A && free>=1 && 0>=free_1 ], cost: INF 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: f0 10: f9 -> [7] : [ A>=1 ], cost: INF 6: f0 -> f9 : A'=free_2, B'=0, D'=0, [], cost: 1 15: f0 -> f9 : A'=free_1, B'=0, C'=free, D'=0, [ 0>=free ], cost: 3 16: f0 -> f9 : A'=free_1, B'=0, C'=0, D'=0, [ free>=1 ], cost: 3+free 17: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: INF 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: INF Eliminated locations (on tree-shaped paths): Start location: f0 17: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: INF 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: INF 19: f0 -> [7] : A'=free_2, B'=0, D'=0, [ free_2>=1 ], cost: INF 20: f0 -> [7] : A'=free_1, B'=0, C'=free, D'=0, [ 0>=free && free_1>=1 ], cost: INF 21: f0 -> [7] : A'=free_1, B'=0, C'=0, D'=0, [ free>=1 && free_1>=1 ], cost: INF ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: INF 19: f0 -> [7] : A'=free_2, B'=0, D'=0, [ free_2>=1 ], cost: INF 20: f0 -> [7] : A'=free_1, B'=0, C'=free, D'=0, [ 0>=free && free_1>=1 ], cost: INF 21: f0 -> [7] : A'=free_1, B'=0, C'=0, D'=0, [ free>=1 && free_1>=1 ], cost: INF Computing asymptotic complexity for rule 18 Resulting cost INF 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: INF Rule cost: INF Rule guard: [ 0>=free_2 ] NO ---------------------------------------- (2) BOUNDS(INF, INF)