/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 23 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 388 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_easy2_start(v__0, v_z) -> Com_1(eval_easy2_bb0_in(v__0, v_z)) :|: TRUE eval_easy2_bb0_in(v__0, v_z) -> Com_1(eval_easy2_0(v__0, v_z)) :|: TRUE eval_easy2_0(v__0, v_z) -> Com_1(eval_easy2_1(v__0, v_z)) :|: TRUE eval_easy2_1(v__0, v_z) -> Com_1(eval_easy2_2(v__0, v_z)) :|: TRUE eval_easy2_2(v__0, v_z) -> Com_1(eval_easy2_3(v__0, v_z)) :|: TRUE eval_easy2_3(v__0, v_z) -> Com_1(eval_easy2_4(v__0, v_z)) :|: TRUE eval_easy2_4(v__0, v_z) -> Com_1(eval_easy2_5(v__0, v_z)) :|: TRUE eval_easy2_5(v__0, v_z) -> Com_1(eval_easy2_6(v__0, v_z)) :|: TRUE eval_easy2_6(v__0, v_z) -> Com_1(eval_easy2_bb1_in(v_z, v_z)) :|: TRUE eval_easy2_bb1_in(v__0, v_z) -> Com_1(eval_easy2_bb2_in(v__0, v_z)) :|: v__0 > 0 eval_easy2_bb1_in(v__0, v_z) -> Com_1(eval_easy2_bb3_in(v__0, v_z)) :|: v__0 <= 0 eval_easy2_bb2_in(v__0, v_z) -> Com_1(eval_easy2_bb1_in(v__0 - 1, v_z)) :|: TRUE eval_easy2_bb3_in(v__0, v_z) -> Com_1(eval_easy2_stop(v__0, v_z)) :|: TRUE The start-symbols are:[eval_easy2_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_1 + 15) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaleasy2start(Ar_0, Ar_1) -> Com_1(evaleasy2bb0in(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb0in(Ar_0, Ar_1) -> Com_1(evaleasy20(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy20(Ar_0, Ar_1) -> Com_1(evaleasy21(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy21(Ar_0, Ar_1) -> Com_1(evaleasy22(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy22(Ar_0, Ar_1) -> Com_1(evaleasy23(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy23(Ar_0, Ar_1) -> Com_1(evaleasy24(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy24(Ar_0, Ar_1) -> Com_1(evaleasy25(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy25(Ar_0, Ar_1) -> Com_1(evaleasy26(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy26(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_1, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_0 - 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy2start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) evaleasy2start(Ar_0, Ar_1) -> Com_1(evaleasy2bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(Ar_0, Ar_1) -> Com_1(evaleasy20(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy20(Ar_0, Ar_1) -> Com_1(evaleasy21(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy21(Ar_0, Ar_1) -> Com_1(evaleasy22(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy22(Ar_0, Ar_1) -> Com_1(evaleasy23(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy23(Ar_0, Ar_1) -> Com_1(evaleasy24(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy24(Ar_0, Ar_1) -> Com_1(evaleasy25(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy25(Ar_0, Ar_1) -> Com_1(evaleasy26(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy26(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_1, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_0 - 1, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy2start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy2start) = 2 Pol(evaleasy2bb0in) = 2 Pol(evaleasy20) = 2 Pol(evaleasy21) = 2 Pol(evaleasy22) = 2 Pol(evaleasy23) = 2 Pol(evaleasy24) = 2 Pol(evaleasy25) = 2 Pol(evaleasy26) = 2 Pol(evaleasy2bb1in) = 2 Pol(evaleasy2bb2in) = 2 Pol(evaleasy2bb3in) = 1 Pol(evaleasy2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evaleasy2start(Ar_0, Ar_1) -> Com_1(evaleasy2bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(Ar_0, Ar_1) -> Com_1(evaleasy20(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy20(Ar_0, Ar_1) -> Com_1(evaleasy21(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy21(Ar_0, Ar_1) -> Com_1(evaleasy22(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy22(Ar_0, Ar_1) -> Com_1(evaleasy23(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy23(Ar_0, Ar_1) -> Com_1(evaleasy24(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy24(Ar_0, Ar_1) -> Com_1(evaleasy25(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy25(Ar_0, Ar_1) -> Com_1(evaleasy26(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy26(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_1, Ar_1)) (Comp: ?, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy2start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy2start) = V_2 + 1 Pol(evaleasy2bb0in) = V_2 + 1 Pol(evaleasy20) = V_2 + 1 Pol(evaleasy21) = V_2 + 1 Pol(evaleasy22) = V_2 + 1 Pol(evaleasy23) = V_2 + 1 Pol(evaleasy24) = V_2 + 1 Pol(evaleasy25) = V_2 + 1 Pol(evaleasy26) = V_2 + 1 Pol(evaleasy2bb1in) = V_1 + 1 Pol(evaleasy2bb2in) = V_1 Pol(evaleasy2bb3in) = V_1 Pol(evaleasy2stop) = V_1 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaleasy2start(Ar_0, Ar_1) -> Com_1(evaleasy2bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(Ar_0, Ar_1) -> Com_1(evaleasy20(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy20(Ar_0, Ar_1) -> Com_1(evaleasy21(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy21(Ar_0, Ar_1) -> Com_1(evaleasy22(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy22(Ar_0, Ar_1) -> Com_1(evaleasy23(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy23(Ar_0, Ar_1) -> Com_1(evaleasy24(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy24(Ar_0, Ar_1) -> Com_1(evaleasy25(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy25(Ar_0, Ar_1) -> Com_1(evaleasy26(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy26(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_1, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evaleasy2bb2in(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy2start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evaleasy2start(Ar_0, Ar_1) -> Com_1(evaleasy2bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy2bb0in(Ar_0, Ar_1) -> Com_1(evaleasy20(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy20(Ar_0, Ar_1) -> Com_1(evaleasy21(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy21(Ar_0, Ar_1) -> Com_1(evaleasy22(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy22(Ar_0, Ar_1) -> Com_1(evaleasy23(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy23(Ar_0, Ar_1) -> Com_1(evaleasy24(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy24(Ar_0, Ar_1) -> Com_1(evaleasy25(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy25(Ar_0, Ar_1) -> Com_1(evaleasy26(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy26(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_1, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb2in(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy2bb1in(Ar_0, Ar_1) -> Com_1(evaleasy2bb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: Ar_1 + 1, Cost: 1) evaleasy2bb2in(Ar_0, Ar_1) -> Com_1(evaleasy2bb1in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evaleasy2bb3in(Ar_0, Ar_1) -> Com_1(evaleasy2stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy2start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 15 Time: 0.079 sec (SMT: 0.069 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evaleasy2start 0: evaleasy2start -> evaleasy2bb0in : [], cost: 1 1: evaleasy2bb0in -> evaleasy20 : [], cost: 1 2: evaleasy20 -> evaleasy21 : [], cost: 1 3: evaleasy21 -> evaleasy22 : [], cost: 1 4: evaleasy22 -> evaleasy23 : [], cost: 1 5: evaleasy23 -> evaleasy24 : [], cost: 1 6: evaleasy24 -> evaleasy25 : [], cost: 1 7: evaleasy25 -> evaleasy26 : [], cost: 1 8: evaleasy26 -> evaleasy2bb1in : A'=B, [], cost: 1 9: evaleasy2bb1in -> evaleasy2bb2in : [ A>=1 ], cost: 1 10: evaleasy2bb1in -> evaleasy2bb3in : [ 0>=A ], cost: 1 11: evaleasy2bb2in -> evaleasy2bb1in : A'=-1+A, [], cost: 1 12: evaleasy2bb3in -> evaleasy2stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evaleasy2start -> evaleasy2bb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evaleasy2start 0: evaleasy2start -> evaleasy2bb0in : [], cost: 1 1: evaleasy2bb0in -> evaleasy20 : [], cost: 1 2: evaleasy20 -> evaleasy21 : [], cost: 1 3: evaleasy21 -> evaleasy22 : [], cost: 1 4: evaleasy22 -> evaleasy23 : [], cost: 1 5: evaleasy23 -> evaleasy24 : [], cost: 1 6: evaleasy24 -> evaleasy25 : [], cost: 1 7: evaleasy25 -> evaleasy26 : [], cost: 1 8: evaleasy26 -> evaleasy2bb1in : A'=B, [], cost: 1 9: evaleasy2bb1in -> evaleasy2bb2in : [ A>=1 ], cost: 1 11: evaleasy2bb2in -> evaleasy2bb1in : A'=-1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evaleasy2start 20: evaleasy2start -> evaleasy2bb1in : A'=B, [], cost: 9 21: evaleasy2bb1in -> evaleasy2bb1in : A'=-1+A, [ A>=1 ], cost: 2 Accelerating simple loops of location 9. Accelerating the following rules: 21: evaleasy2bb1in -> evaleasy2bb1in : A'=-1+A, [ A>=1 ], cost: 2 Accelerated rule 21 with metering function A, yielding the new rule 22. Removing the simple loops: 21. Accelerated all simple loops using metering functions (where possible): Start location: evaleasy2start 20: evaleasy2start -> evaleasy2bb1in : A'=B, [], cost: 9 22: evaleasy2bb1in -> evaleasy2bb1in : A'=0, [ A>=1 ], cost: 2*A Chained accelerated rules (with incoming rules): Start location: evaleasy2start 20: evaleasy2start -> evaleasy2bb1in : A'=B, [], cost: 9 23: evaleasy2start -> evaleasy2bb1in : A'=0, [ B>=1 ], cost: 9+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evaleasy2start 23: evaleasy2start -> evaleasy2bb1in : A'=0, [ B>=1 ], cost: 9+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evaleasy2start 23: evaleasy2start -> evaleasy2bb1in : A'=0, [ B>=1 ], cost: 9+2*B Computing asymptotic complexity for rule 23 Solved the limit problem by the following transformations: Created initial limit problem: 9+2*B (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 9+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 9+2*n Rule cost: 9+2*B Rule guard: [ B>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)