/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 126 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 730 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb0_in(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_bb0_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_0(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_0(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_1(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_1(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_2(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_2(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_3(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_3(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb1_in(v_1, v_n, 0, v_x_1)) :|: TRUE eval_start_bb1_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb2_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_0 < v_n eval_start_bb1_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_0)) :|: v_x_0 >= v_n eval_start_bb2_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_4(v_1, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_4(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_5(nondef_0, v_n, v_x_0, v_x_1)) :|: TRUE eval_start_5(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_0)) :|: v_1 > 0 eval_start_5(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb3_in(v_1, v_n, v_x_0, v_x_1)) :|: v_1 <= 0 eval_start_bb3_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb1_in(v_1, v_n, v_x_0 + 1, v_x_1)) :|: TRUE eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb5_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_1 < v_n eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb6_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_1 >= v_n eval_start_bb5_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1 + 1)) :|: TRUE eval_start_bb6_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_stop(v_1, v_n, v_x_0, v_x_1)) :|: TRUE The start-symbols are:[eval_start_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 7*Ar_1 + 32) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Fresh_0)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Fresh_0)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = 3 Pol(evalstartbb0in) = 3 Pol(evalstart0) = 3 Pol(evalstart1) = 3 Pol(evalstart2) = 3 Pol(evalstart3) = 3 Pol(evalstartbb1in) = 3 Pol(evalstartbb2in) = 3 Pol(evalstartbb4in) = 2 Pol(evalstart4) = 3 Pol(evalstart5) = 3 Pol(evalstartbb3in) = 3 Pol(evalstartbb5in) = 2 Pol(evalstartbb6in) = 1 Pol(evalstartstop) = 0 Pol(koat_start) = 3 orients all transitions weakly and the transitions evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Fresh_0)) (Comp: 3, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 3, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 3, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = V_2 + 2 Pol(evalstartbb0in) = V_2 + 2 Pol(evalstart0) = V_2 + 2 Pol(evalstart1) = V_2 + 2 Pol(evalstart2) = V_2 + 2 Pol(evalstart3) = V_2 + 2 Pol(evalstartbb1in) = -V_1 + V_2 + 2 Pol(evalstartbb2in) = -V_1 + V_2 + 1 Pol(evalstartbb4in) = V_2 - V_3 + 1 Pol(evalstart4) = -V_1 + V_2 + 1 Pol(evalstart5) = -V_1 + V_2 + 1 Pol(evalstartbb3in) = -V_1 + V_2 + 1 Pol(evalstartbb5in) = V_2 - V_3 Pol(evalstartbb6in) = V_2 - V_3 Pol(evalstartstop) = V_2 - V_3 Pol(koat_start) = V_2 + 2 orients all transitions weakly and the transitions evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Fresh_0)) (Comp: 3, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 3, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 3, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1 + 2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 2, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Fresh_0)) (Comp: 3, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_0, Ar_3)) [ Ar_3 >= 1 ] (Comp: Ar_1 + 2, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: Ar_1 + 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 3, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: Ar_1 + 2, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 3, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 7*Ar_1 + 32 Time: 0.118 sec (SMT: 0.086 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1 6: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1 7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1 8: evalstartbb2in -> evalstart4 : [], cost: 1 9: evalstart4 -> evalstart5 : D'=free, [], cost: 1 10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1 11: evalstart5 -> evalstartbb3in : [ 0>=D ], cost: 1 12: evalstartbb3in -> evalstartbb1in : A'=1+A, [], cost: 1 13: evalstartbb4in -> evalstartbb5in : [ B>=1+C ], cost: 1 14: evalstartbb4in -> evalstartbb6in : [ C>=B ], cost: 1 15: evalstartbb5in -> evalstartbb4in : C'=1+C, [], cost: 1 16: evalstartbb6in -> evalstartstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalstartstart -> evalstartbb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1 6: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1 7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1 8: evalstartbb2in -> evalstart4 : [], cost: 1 9: evalstart4 -> evalstart5 : D'=free, [], cost: 1 10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1 11: evalstart5 -> evalstartbb3in : [ 0>=D ], cost: 1 12: evalstartbb3in -> evalstartbb1in : A'=1+A, [], cost: 1 13: evalstartbb4in -> evalstartbb5in : [ B>=1+C ], cost: 1 15: evalstartbb5in -> evalstartbb4in : C'=1+C, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1 23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3 10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1 24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2 25: evalstartbb4in -> evalstartbb4in : C'=1+C, [ B>=1+C ], cost: 2 Accelerating simple loops of location 11. Accelerating the following rules: 25: evalstartbb4in -> evalstartbb4in : C'=1+C, [ B>=1+C ], cost: 2 Accelerated rule 25 with metering function -C+B, yielding the new rule 26. Removing the simple loops: 25. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1 23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3 10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1 24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2 26: evalstartbb4in -> evalstartbb4in : C'=B, [ B>=1+C ], cost: -2*C+2*B Chained accelerated rules (with incoming rules): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1 23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3 10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1 24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2 27: evalstart5 -> evalstartbb4in : C'=B, [ D>=1 && B>=1+A ], cost: 1-2*A+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3 24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2 27: evalstart5 -> evalstartbb4in : C'=B, [ D>=1 && B>=1+A ], cost: 1-2*A+2*B Eliminated locations (on tree-shaped paths): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 28: evalstartbb1in -> evalstartbb1in : A'=1+A, D'=free, [ B>=1+A && 0>=free ], cost: 5 29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B Accelerating simple loops of location 6. Accelerating the following rules: 28: evalstartbb1in -> evalstartbb1in : A'=1+A, D'=free, [ B>=1+A && 0>=free ], cost: 5 Accelerated rule 28 with metering function -A+B, yielding the new rule 30. Removing the simple loops: 28. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B 30: evalstartbb1in -> evalstartbb1in : A'=B, D'=free, [ B>=1+A && 0>=free ], cost: -5*A+5*B Chained accelerated rules (with incoming rules): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 31: evalstartstart -> evalstartbb1in : A'=B, D'=free, [ B>=1 && 0>=free ], cost: 6+5*B 29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B Eliminated locations (on tree-shaped paths): Start location: evalstartstart 32: evalstartstart -> evalstartbb4in : A'=0, C'=B, D'=free, [ B>=1 && free>=1 ], cost: 10+2*B 33: evalstartstart -> [17] : [ B>=1 && 0>=free ], cost: 6+5*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 32: evalstartstart -> evalstartbb4in : A'=0, C'=B, D'=free, [ B>=1 && free>=1 ], cost: 10+2*B 33: evalstartstart -> [17] : [ B>=1 && 0>=free ], cost: 6+5*B Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: 10+2*B (+), free (+/+!), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {free==n,B==n} resulting limit problem: [solved] Solution: free / n B / n Resulting cost 10+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: 10+2*n Rule cost: 10+2*B Rule guard: [ B>=1 && free>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)