/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 132 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 608 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb0_in(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_bb0_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_0(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_0(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_1(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_1(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_2(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_2(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_3(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_3(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb1_in(v_i, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_bb1_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb2_in(v__0, v__1, v_3, v_i, v_k)) :|: v__0 > 100 eval_start_bb1_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb3_in(v__0, v__1, v_3, v_i, v_k)) :|: v__0 <= 100 eval_start_bb2_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb1_in(v__0 - 1, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_bb3_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_8(v__0, v__1, v__0 + v_k + 50, v_i, v_k)) :|: TRUE eval_start_8(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_9(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_9(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_10(v__0, v__1, v_3, v_i, v_k)) :|: TRUE eval_start_10(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb4_in(v__0, v_3, v_3, v_i, v_k)) :|: TRUE eval_start_bb4_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb5_in(v__0, v__1, v_3, v_i, v_k)) :|: v__1 >= 0 eval_start_bb4_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb6_in(v__0, v__1, v_3, v_i, v_k)) :|: v__1 < 0 eval_start_bb5_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_bb4_in(v__0, v__1 - 1, v_3, v_i, v_k)) :|: TRUE eval_start_bb6_in(v__0, v__1, v_3, v_i, v_k) -> Com_1(eval_start_stop(v__0, v__1, v_3, v_i, v_k)) :|: TRUE The start-symbols are:[eval_start_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_1 + 2*Ar_3 + 357) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: ?, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: ?, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = 7 Pol(evalstartbb0in) = 7 Pol(evalstart0) = 7 Pol(evalstart1) = 7 Pol(evalstart2) = 7 Pol(evalstart3) = 7 Pol(evalstartbb1in) = 7 Pol(evalstartbb2in) = 7 Pol(evalstartbb3in) = 6 Pol(evalstart8) = 5 Pol(evalstart9) = 4 Pol(evalstart10) = 3 Pol(evalstartbb4in) = 2 Pol(evalstartbb5in) = 2 Pol(evalstartbb6in) = 1 Pol(evalstartstop) = 0 Pol(koat_start) = 7 orients all transitions weakly and the transitions evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: 7, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: 7, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: 7, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = V_2 Pol(evalstartbb0in) = V_2 Pol(evalstart0) = V_2 Pol(evalstart1) = V_2 Pol(evalstart2) = V_2 Pol(evalstart3) = V_2 Pol(evalstartbb1in) = V_1 Pol(evalstartbb2in) = V_1 - 1 Pol(evalstartbb3in) = V_1 Pol(evalstart8) = V_1 Pol(evalstart9) = V_1 Pol(evalstart10) = V_1 Pol(evalstartbb4in) = V_1 - V_3 + V_5 Pol(evalstartbb5in) = V_1 - V_3 + V_5 Pol(evalstartbb6in) = V_1 - V_3 + V_5 Pol(evalstartstop) = V_1 - V_3 + V_5 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: 7, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: 7, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: 7, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: 7, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: 7, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: 7, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = V_4 + 151 Pol(evalstartbb0in) = V_4 + 151 Pol(evalstart0) = V_4 + 151 Pol(evalstart1) = V_4 + 151 Pol(evalstart2) = V_4 + 151 Pol(evalstart3) = V_4 + 151 Pol(evalstartbb1in) = V_4 + 151 Pol(evalstartbb2in) = V_4 + 151 Pol(evalstartbb3in) = V_1 + V_4 + 51 Pol(evalstart8) = V_3 + 1 Pol(evalstart9) = V_3 + 1 Pol(evalstart10) = V_3 + 1 Pol(evalstartbb4in) = V_5 + 1 Pol(evalstartbb5in) = V_5 Pol(evalstartbb6in) = V_5 Pol(evalstartstop) = V_5 Pol(koat_start) = V_4 + 151 orients all transitions weakly and the transition evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: 7, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: Ar_3 + 151, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: 7, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: ?, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: 7, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 101 ] (Comp: 7, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_0 + Ar_3 + 50, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 7, Cost: 1) evalstart10(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_2)) (Comp: Ar_3 + 151, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_4 >= 0 ] (Comp: 7, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 + 1 ] (Comp: Ar_3 + 151, Cost: 1) evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4 - 1)) (Comp: 7, Cost: 1) evalstartbb6in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 2*Ar_3 + 357 Time: 0.180 sec (SMT: 0.128 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'=B, [], cost: 1 6: evalstartbb1in -> evalstartbb2in : [ A>=101 ], cost: 1 7: evalstartbb1in -> evalstartbb3in : [ 100>=A ], cost: 1 8: evalstartbb2in -> evalstartbb1in : A'=-1+A, [], cost: 1 9: evalstartbb3in -> evalstart8 : C'=50+D+A, [], cost: 1 10: evalstart8 -> evalstart9 : [], cost: 1 11: evalstart9 -> evalstart10 : [], cost: 1 12: evalstart10 -> evalstartbb4in : E'=C, [], cost: 1 13: evalstartbb4in -> evalstartbb5in : [ E>=0 ], cost: 1 14: evalstartbb4in -> evalstartbb6in : [ 0>=1+E ], cost: 1 15: evalstartbb5in -> evalstartbb4in : E'=-1+E, [], 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'=B, [], cost: 1 6: evalstartbb1in -> evalstartbb2in : [ A>=101 ], cost: 1 7: evalstartbb1in -> evalstartbb3in : [ 100>=A ], cost: 1 8: evalstartbb2in -> evalstartbb1in : A'=-1+A, [], cost: 1 9: evalstartbb3in -> evalstart8 : C'=50+D+A, [], cost: 1 10: evalstart8 -> evalstart9 : [], cost: 1 11: evalstart9 -> evalstart10 : [], cost: 1 12: evalstart10 -> evalstartbb4in : E'=C, [], cost: 1 13: evalstartbb4in -> evalstartbb5in : [ E>=0 ], cost: 1 15: evalstartbb5in -> evalstartbb4in : E'=-1+E, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 22: evalstartbb1in -> evalstartbb1in : A'=-1+A, [ A>=101 ], cost: 2 26: evalstartbb1in -> evalstartbb4in : C'=50+D+A, E'=50+D+A, [ 100>=A ], cost: 5 27: evalstartbb4in -> evalstartbb4in : E'=-1+E, [ E>=0 ], cost: 2 Accelerating simple loops of location 6. Accelerating the following rules: 22: evalstartbb1in -> evalstartbb1in : A'=-1+A, [ A>=101 ], cost: 2 Accelerated rule 22 with metering function -100+A, yielding the new rule 28. Removing the simple loops: 22. Accelerating simple loops of location 12. Accelerating the following rules: 27: evalstartbb4in -> evalstartbb4in : E'=-1+E, [ E>=0 ], cost: 2 Accelerated rule 27 with metering function 1+E, yielding the new rule 29. Removing the simple loops: 27. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 26: evalstartbb1in -> evalstartbb4in : C'=50+D+A, E'=50+D+A, [ 100>=A ], cost: 5 28: evalstartbb1in -> evalstartbb1in : A'=100, [ A>=101 ], cost: -200+2*A 29: evalstartbb4in -> evalstartbb4in : E'=-1, [ E>=0 ], cost: 2+2*E Chained accelerated rules (with incoming rules): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 30: evalstartstart -> evalstartbb1in : A'=100, [ B>=101 ], cost: -194+2*B 26: evalstartbb1in -> evalstartbb4in : C'=50+D+A, E'=50+D+A, [ 100>=A ], cost: 5 31: evalstartbb1in -> evalstartbb4in : C'=50+D+A, E'=-1, [ 100>=A && 50+D+A>=0 ], cost: 107+2*D+2*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 30: evalstartstart -> evalstartbb1in : A'=100, [ B>=101 ], cost: -194+2*B 31: evalstartbb1in -> evalstartbb4in : C'=50+D+A, E'=-1, [ 100>=A && 50+D+A>=0 ], cost: 107+2*D+2*A Eliminated locations (on tree-shaped paths): Start location: evalstartstart 32: evalstartstart -> evalstartbb4in : A'=B, C'=50+D+B, E'=-1, [ 100>=B && 50+D+B>=0 ], cost: 113+2*D+2*B 33: evalstartstart -> evalstartbb4in : A'=100, C'=150+D, E'=-1, [ B>=101 && 150+D>=0 ], cost: 113+2*D+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 32: evalstartstart -> evalstartbb4in : A'=B, C'=50+D+B, E'=-1, [ 100>=B && 50+D+B>=0 ], cost: 113+2*D+2*B 33: evalstartstart -> evalstartbb4in : A'=100, C'=150+D, E'=-1, [ B>=101 && 150+D>=0 ], cost: 113+2*D+2*B Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: 101-B (+/+!), 51+D+B (+/+!), 113+2*D+2*B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {D==2*n,B==-n} resulting limit problem: [solved] Solution: D / 2*n B / -n Resulting cost 113+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: 113+2*n Rule cost: 113+2*D+2*B Rule guard: [ 100>=B && 50+D+B>=0 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)