/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 114 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 741 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_0(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_0(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_1(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_1(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_2(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_2(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_3(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_3(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_4(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_4(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_5(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_5(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_6(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_6(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_7(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_7(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_8(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_start_8(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, 0, 0)) :|: TRUE eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 < v_n eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 >= v_n eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0 + 1)) :|: TRUE eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb4_in(v_m, v_n, v_x_0, v_y_0)) :|: v_y_0 < v_m eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb5_in(v_m, v_n, v_x_0, v_y_0)) :|: v_y_0 >= v_m eval_start_bb4_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0 + 1)) :|: TRUE eval_start_bb5_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_stop(v_m, v_n, v_x_0, v_y_0)) :|: TRUE The start-symbols are:[eval_start_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 3*Ar_2 + 3*Ar_3 + 19) 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(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, Ar_3)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, 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_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb5in(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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, 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_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb5in(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) = 2 Pol(evalstartbb0in) = 2 Pol(evalstart0) = 2 Pol(evalstart1) = 2 Pol(evalstart2) = 2 Pol(evalstart3) = 2 Pol(evalstart4) = 2 Pol(evalstart5) = 2 Pol(evalstart6) = 2 Pol(evalstart7) = 2 Pol(evalstart8) = 2 Pol(evalstartbb1in) = 2 Pol(evalstartbb2in) = 2 Pol(evalstartbb3in) = 2 Pol(evalstartbb4in) = 2 Pol(evalstartbb5in) = 1 Pol(evalstartstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] 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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, 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_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalstartbb5in(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_3 Pol(evalstartbb0in) = V_3 Pol(evalstart0) = V_3 Pol(evalstart1) = V_3 Pol(evalstart2) = V_3 Pol(evalstart3) = V_3 Pol(evalstart4) = V_3 Pol(evalstart5) = V_3 Pol(evalstart6) = V_3 Pol(evalstart7) = V_3 Pol(evalstart8) = V_3 Pol(evalstartbb1in) = -V_1 + V_3 Pol(evalstartbb2in) = -V_1 + V_3 - 1 Pol(evalstartbb3in) = -V_1 + V_3 Pol(evalstartbb4in) = -V_1 + V_3 Pol(evalstartbb5in) = -V_1 + V_3 Pol(evalstartstop) = -V_1 + V_3 Pol(koat_start) = V_3 orients all transitions weakly and the transition evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalstartbb5in(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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalstartbb5in(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_4 + 1 Pol(evalstartbb0in) = V_4 + 1 Pol(evalstart0) = V_4 + 1 Pol(evalstart1) = V_4 + 1 Pol(evalstart2) = V_4 + 1 Pol(evalstart3) = V_4 + 1 Pol(evalstart4) = V_4 + 1 Pol(evalstart5) = V_4 + 1 Pol(evalstart6) = V_4 + 1 Pol(evalstart7) = V_4 + 1 Pol(evalstart8) = V_4 + 1 Pol(evalstartbb1in) = -V_2 + V_4 + 1 Pol(evalstartbb2in) = -V_2 + V_4 Pol(evalstartbb3in) = -V_2 + V_4 + 1 Pol(evalstartbb4in) = -V_2 + V_4 Pol(evalstartbb5in) = -V_2 + V_4 Pol(evalstartstop) = -V_2 + V_4 Pol(koat_start) = V_4 + 1 orients all transitions weakly and the transition evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] strictly and produces the following problem: 6: 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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: Ar_3 + 1, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalstartbb5in(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 6 produces the following problem: 7: 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(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart7(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(0, 0, Ar_2, Ar_3)) (Comp: Ar_2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: Ar_3 + Ar_2 + 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: Ar_3 + 1, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: Ar_3 + 1, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalstartbb5in(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 3*Ar_2 + 3*Ar_3 + 19 Time: 0.140 sec (SMT: 0.103 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 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstart5 : [], cost: 1 7: evalstart5 -> evalstart6 : [], cost: 1 8: evalstart6 -> evalstart7 : [], cost: 1 9: evalstart7 -> evalstart8 : [], cost: 1 10: evalstart8 -> evalstartbb1in : A'=0, B'=0, [], cost: 1 11: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 12: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1 13: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 14: evalstartbb3in -> evalstartbb4in : [ D>=1+B ], cost: 1 15: evalstartbb3in -> evalstartbb5in : [ B>=D ], cost: 1 16: evalstartbb4in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 17: evalstartbb5in -> 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 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstart5 : [], cost: 1 7: evalstart5 -> evalstart6 : [], cost: 1 8: evalstart6 -> evalstart7 : [], cost: 1 9: evalstart7 -> evalstart8 : [], cost: 1 10: evalstart8 -> evalstartbb1in : A'=0, B'=0, [], cost: 1 11: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 12: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1 13: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 14: evalstartbb3in -> evalstartbb4in : [ D>=1+B ], cost: 1 16: evalstartbb4in -> evalstartbb1in : A'=1+A, B'=1+B, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 27: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 11 28: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ C>=1+A ], cost: 2 30: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ A>=C && D>=1+B ], cost: 3 Accelerating simple loops of location 11. Accelerating the following rules: 28: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ C>=1+A ], cost: 2 30: evalstartbb1in -> evalstartbb1in : A'=1+A, B'=1+B, [ A>=C && D>=1+B ], cost: 3 Accelerated rule 28 with metering function C-A, yielding the new rule 31. Accelerated rule 30 with metering function D-B, yielding the new rule 32. Removing the simple loops: 28 30. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 27: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 11 31: evalstartbb1in -> evalstartbb1in : A'=C, B'=C-A+B, [ C>=1+A ], cost: 2*C-2*A 32: evalstartbb1in -> evalstartbb1in : A'=D+A-B, B'=D, [ A>=C && D>=1+B ], cost: 3*D-3*B Chained accelerated rules (with incoming rules): Start location: evalstartstart 27: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 11 33: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 11+2*C 34: evalstartstart -> evalstartbb1in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 11+3*D Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 33: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 11+2*C 34: evalstartstart -> evalstartbb1in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 11+3*D ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 33: evalstartstart -> evalstartbb1in : A'=C, B'=C, [ C>=1 ], cost: 11+2*C 34: evalstartstart -> evalstartbb1in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 11+3*D Computing asymptotic complexity for rule 33 Solved the limit problem by the following transformations: Created initial limit problem: C (+/+!), 11+2*C (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n} resulting limit problem: [solved] Solution: C / n Resulting cost 11+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: 11+2*n Rule cost: 11+2*C Rule guard: [ C>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)