/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).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, 625 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 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 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: NONTERM 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: NONTERM 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: NONTERM 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: NONTERM 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: NONTERM 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: NONTERM 14: f9 -> [8] : [ 0>=A && free>=1 && 0>=free_1 ], cost: NONTERM 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: NONTERM 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: NONTERM 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: NONTERM Eliminated locations (on tree-shaped paths): Start location: f0 17: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: NONTERM 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: NONTERM 19: f0 -> [7] : A'=free_2, B'=0, D'=0, [ free_2>=1 ], cost: NONTERM 20: f0 -> [7] : A'=free_1, B'=0, C'=free, D'=0, [ 0>=free && free_1>=1 ], cost: NONTERM 21: f0 -> [7] : A'=free_1, B'=0, C'=0, D'=0, [ free>=1 && free_1>=1 ], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 18: f0 -> [8] : A'=free_2, B'=0, D'=0, [ 0>=free_2 ], cost: NONTERM 19: f0 -> [7] : A'=free_2, B'=0, D'=0, [ free_2>=1 ], cost: NONTERM 20: f0 -> [7] : A'=free_1, B'=0, C'=free, D'=0, [ 0>=free && free_1>=1 ], cost: NONTERM 21: f0 -> [7] : A'=free_1, B'=0, C'=0, D'=0, [ free>=1 && free_1>=1 ], cost: NONTERM Computing asymptotic complexity for rule 18 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>=free_2 ] NO ---------------------------------------- (2) BOUNDS(INF, INF)