5.11/2.37 WORST_CASE(NON_POLY, ?) 5.28/2.78 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.28/2.78 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.28/2.78 5.28/2.78 5.28/2.78 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF). 5.28/2.78 5.28/2.78 (0) CpxIntTrs 5.28/2.78 (1) Loat Proof [FINISHED, 635 ms] 5.28/2.78 (2) BOUNDS(INF, INF) 5.28/2.78 5.28/2.78 5.28/2.78 ---------------------------------------- 5.28/2.78 5.28/2.78 (0) 5.28/2.78 Obligation: 5.28/2.78 Complexity Int TRS consisting of the following rules: 5.28/2.78 f10(A, B, C, D, E) -> Com_1(f10(A - 1, B - 1, C + 1, F, E)) :|: A >= 1 && F >= 1 5.28/2.78 f10(A, B, C, D, E) -> Com_1(f10(A - 1, B, C, F, E)) :|: 0 >= F && A >= 1 && A >= B + 1 5.28/2.78 f20(A, B, C, D, E) -> Com_1(f20(A, B, C, D, E)) :|: TRUE 5.28/2.78 f22(A, B, C, D, E) -> Com_1(f25(A, B, C, D, E)) :|: TRUE 5.28/2.78 f10(A, B, C, D, E) -> Com_1(f20(A, B, C, D, E)) :|: 0 >= A 5.28/2.78 f0(A, B, C, D, E) -> Com_1(f10(8, F, 0, D, 8)) :|: F >= 1 5.28/2.78 5.28/2.78 The start-symbols are:[f0_5] 5.28/2.78 5.28/2.78 5.28/2.78 ---------------------------------------- 5.28/2.78 5.28/2.78 (1) Loat Proof (FINISHED) 5.28/2.78 5.28/2.78 5.28/2.78 ### Pre-processing the ITS problem ### 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Initial linear ITS problem 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 0: f10 -> f10 : A'=-1+A, B'=-1+B, C'=1+C, D'=free, [ A>=1 && free>=1 ], cost: 1 5.28/2.78 5.28/2.78 1: f10 -> f10 : A'=-1+A, D'=free_1, [ 0>=free_1 && A>=1 && A>=1+B ], cost: 1 5.28/2.78 5.28/2.78 4: f10 -> f20 : [ 0>=A ], cost: 1 5.28/2.78 5.28/2.78 2: f20 -> f20 : [], cost: 1 5.28/2.78 5.28/2.78 3: f22 -> f25 : [], cost: 1 5.28/2.78 5.28/2.78 5: f0 -> f10 : A'=8, B'=free_2, C'=0, E'=8, [ free_2>=1 ], cost: 1 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Removed unreachable and leaf rules: 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 0: f10 -> f10 : A'=-1+A, B'=-1+B, C'=1+C, D'=free, [ A>=1 && free>=1 ], cost: 1 5.28/2.78 5.28/2.78 1: f10 -> f10 : A'=-1+A, D'=free_1, [ 0>=free_1 && A>=1 && A>=1+B ], cost: 1 5.28/2.78 5.28/2.78 4: f10 -> f20 : [ 0>=A ], cost: 1 5.28/2.78 5.28/2.78 2: f20 -> f20 : [], cost: 1 5.28/2.78 5.28/2.78 5: f0 -> f10 : A'=8, B'=free_2, C'=0, E'=8, [ free_2>=1 ], cost: 1 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 ### Simplification by acceleration and chaining ### 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Accelerating simple loops of location 0. 5.28/2.78 5.28/2.78 Accelerating the following rules: 5.28/2.78 5.28/2.78 0: f10 -> f10 : A'=-1+A, B'=-1+B, C'=1+C, D'=free, [ A>=1 && free>=1 ], cost: 1 5.28/2.78 5.28/2.78 1: f10 -> f10 : A'=-1+A, D'=free_1, [ 0>=free_1 && A>=1 && A>=1+B ], cost: 1 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Accelerated rule 0 with metering function A, yielding the new rule 6. 5.28/2.78 5.28/2.78 Accelerated rule 1 with backward acceleration, yielding the new rule 7. 5.28/2.78 5.28/2.78 Accelerated rule 1 with backward acceleration, yielding the new rule 8. 5.28/2.78 5.28/2.78 Removing the simple loops: 0 1. 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Accelerating simple loops of location 1. 5.28/2.78 5.28/2.78 Accelerating the following rules: 5.28/2.78 5.28/2.78 2: f20 -> f20 : [], cost: 1 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Accelerated rule 2 with NONTERM, yielding the new rule 9. 5.28/2.78 5.28/2.78 Removing the simple loops: 2. 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Accelerated all simple loops using metering functions (where possible): 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 4: f10 -> f20 : [ 0>=A ], cost: 1 5.28/2.78 5.28/2.78 6: f10 -> f10 : A'=0, B'=-A+B, C'=C+A, D'=free, [ A>=1 && free>=1 ], cost: A 5.28/2.78 5.28/2.78 7: f10 -> f10 : A'=0, D'=free_1, [ 0>=free_1 && A>=1 && A>=1+B && 1>=1+B ], cost: A 5.28/2.78 5.28/2.78 8: f10 -> f10 : A'=B, D'=free_1, [ 0>=free_1 && A>=1 && A>=1+B && 1+B>=1 ], cost: A-B 5.28/2.78 5.28/2.78 9: f20 -> [6] : [], cost: INF 5.28/2.78 5.28/2.78 5: f0 -> f10 : A'=8, B'=free_2, C'=0, E'=8, [ free_2>=1 ], cost: 1 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Chained accelerated rules (with incoming rules): 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 4: f10 -> f20 : [ 0>=A ], cost: 1 5.28/2.78 5.28/2.78 12: f10 -> [6] : [ 0>=A ], cost: INF 5.28/2.78 5.28/2.78 5: f0 -> f10 : A'=8, B'=free_2, C'=0, E'=8, [ free_2>=1 ], cost: 1 5.28/2.78 5.28/2.78 10: f0 -> f10 : A'=0, B'=-8+free_2, C'=8, D'=free, E'=8, [ free_2>=1 && free>=1 ], cost: 9 5.28/2.78 5.28/2.78 11: f0 -> f10 : A'=free_2, B'=free_2, C'=0, D'=free_1, E'=8, [ free_2>=1 && 0>=free_1 && 8>=1+free_2 ], cost: 9-free_2 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Removed unreachable locations (and leaf rules with constant cost): 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 12: f10 -> [6] : [ 0>=A ], cost: INF 5.28/2.78 5.28/2.78 5: f0 -> f10 : A'=8, B'=free_2, C'=0, E'=8, [ free_2>=1 ], cost: 1 5.28/2.78 5.28/2.78 10: f0 -> f10 : A'=0, B'=-8+free_2, C'=8, D'=free, E'=8, [ free_2>=1 && free>=1 ], cost: 9 5.28/2.78 5.28/2.78 11: f0 -> f10 : A'=free_2, B'=free_2, C'=0, D'=free_1, E'=8, [ free_2>=1 && 0>=free_1 && 8>=1+free_2 ], cost: 9-free_2 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Eliminated locations (on tree-shaped paths): 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 13: f0 -> [6] : A'=0, B'=-8+free_2, C'=8, D'=free, E'=8, [ free_2>=1 && free>=1 ], cost: INF 5.28/2.78 5.28/2.78 14: f0 -> [7] : [ free_2>=1 && 0>=free_1 && 8>=1+free_2 ], cost: 9-free_2 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 ### Computing asymptotic complexity ### 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Fully simplified ITS problem 5.28/2.78 5.28/2.78 Start location: f0 5.28/2.78 5.28/2.78 13: f0 -> [6] : A'=0, B'=-8+free_2, C'=8, D'=free, E'=8, [ free_2>=1 && free>=1 ], cost: INF 5.28/2.78 5.28/2.78 14: f0 -> [7] : [ free_2>=1 && 0>=free_1 && 8>=1+free_2 ], cost: 9-free_2 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Computing asymptotic complexity for rule 13 5.28/2.78 5.28/2.78 Resulting cost INF has complexity: Nonterm 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Found new complexity Nonterm. 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 Obtained the following overall complexity (w.r.t. the length of the input n): 5.28/2.78 5.28/2.78 Complexity: Nonterm 5.28/2.78 5.28/2.78 Cpx degree: Nonterm 5.28/2.78 5.28/2.78 Solved cost: INF 5.28/2.78 5.28/2.78 Rule cost: INF 5.28/2.78 5.28/2.78 Rule guard: [ free_2>=1 && free>=1 ] 5.28/2.78 5.28/2.78 5.28/2.78 5.28/2.78 NO 5.28/2.78 5.28/2.78 5.28/2.78 ---------------------------------------- 5.28/2.78 5.28/2.78 (2) 5.28/2.78 BOUNDS(INF, INF) 5.29/2.99 EOF