/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 84 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 527 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb0_in(v__0, v__01, v_x, v_y)) :|: TRUE eval_start_bb0_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_0(v__0, v__01, v_x, v_y)) :|: TRUE eval_start_0(v__0, v__01, v_x, v_y) -> Com_1(eval_start_1(v__0, v__01, v_x, v_y)) :|: TRUE eval_start_1(v__0, v__01, v_x, v_y) -> Com_1(eval_start_2(v__0, v__01, v_x, v_y)) :|: TRUE eval_start_2(v__0, v__01, v_x, v_y) -> Com_1(eval_start_3(v__0, v__01, v_x, v_y)) :|: TRUE eval_start_3(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb1_in(v_x, v__01, v_x, v_y)) :|: TRUE eval_start_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb2_in(v__0, v__01, v_x, v_y)) :|: v__0 < v_y eval_start_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v_y, v_x, v_y)) :|: v__0 >= v_y eval_start_bb2_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01, v_x, v_y)) :|: TRUE eval_start_bb3_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb4_in(v__0, v__01, v_x, v_y)) :|: v__01 < v__0 eval_start_bb3_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb5_in(v__0, v__01, v_x, v_y)) :|: v__01 >= v__0 eval_start_bb4_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v__01 + 1, v_x, v_y)) :|: TRUE eval_start_bb5_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_stop(v__0, v__01, v_x, v_y)) :|: TRUE The start-symbols are:[eval_start_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 14*Ar_1 + 20*Ar_2 + 21) 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(Ar_1, 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_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_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_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_0 >= Ar_3 + 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_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (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(evalstartbb1in(Ar_1, 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_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_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_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_0 >= Ar_3 + 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_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (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) = 3 Pol(evalstartbb0in) = 3 Pol(evalstart0) = 3 Pol(evalstart1) = 3 Pol(evalstart2) = 3 Pol(evalstart3) = 3 Pol(evalstartbb1in) = 3 Pol(evalstartbb2in) = 3 Pol(evalstartbb3in) = 2 Pol(evalstartbb4in) = 2 Pol(evalstartbb5in) = 1 Pol(evalstartstop) = 0 Pol(koat_start) = 3 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_3 >= Ar_0 ] evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] 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(Ar_1, 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_2 >= Ar_0 + 1 ] (Comp: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_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_0 >= Ar_3 + 1 ] (Comp: 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (Comp: 3, 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_2 + V_3 Pol(evalstartbb0in) = -V_2 + V_3 Pol(evalstart0) = -V_2 + V_3 Pol(evalstart1) = -V_2 + V_3 Pol(evalstart2) = -V_2 + V_3 Pol(evalstart3) = -V_2 + 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_2 + 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(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 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: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_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_0 >= Ar_3 + 1 ] (Comp: 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (Comp: 3, 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(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 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: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: Ar_1 + Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_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_0 >= Ar_3 + 1 ] (Comp: 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (Comp: 3, 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(evalstartbb4in) = V_1 - V_4 Pol(evalstartbb3in) = V_1 - V_4 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1))", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1))", 0-1) = Ar_1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1))", 0-2) = Ar_2 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1))", 0-3) = ? S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-1) = Ar_1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-2) = Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ]", 0-3) = ? S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ]", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ]", 0-2) = Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ]", 0-3) = ? S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-0) = 2*Ar_1 + 2*Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-1) = Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-2) = Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-3) = Ar_2 S("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 ]", 0-0) = 2*Ar_1 + 2*Ar_2 S("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 ]", 0-1) = Ar_1 S("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 ]", 0-2) = Ar_2 S("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 ]", 0-3) = Ar_3 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ] 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_0 >= Ar_3 + 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(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 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: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: Ar_1 + Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 6*Ar_1 + 9*Ar_2 + 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ] (Comp: 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (Comp: 3, 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(evalstartbb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + 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: 3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: Ar_1 + Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 6*Ar_1 + 9*Ar_2 + 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 + 1 ] (Comp: 3, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 ] (Comp: 6*Ar_1 + 9*Ar_2 + 3, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) (Comp: 3, 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 14*Ar_1 + 20*Ar_2 + 21 Time: 0.163 sec (SMT: 0.125 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 : [ C>=1+A ], cost: 1 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 8: evalstartbb2in -> evalstartbb1in : A'=1+A, [], cost: 1 9: evalstartbb3in -> evalstartbb4in : [ A>=1+D ], cost: 1 10: evalstartbb3in -> evalstartbb5in : [ D>=A ], cost: 1 11: evalstartbb4in -> evalstartbb3in : D'=1+D, [], cost: 1 12: 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 -> evalstartbb1in : A'=B, [], cost: 1 6: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 8: evalstartbb2in -> evalstartbb1in : A'=1+A, [], cost: 1 9: evalstartbb3in -> evalstartbb4in : [ A>=1+D ], cost: 1 11: evalstartbb4in -> evalstartbb3in : D'=1+D, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 18: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 2 19: evalstartbb3in -> evalstartbb3in : D'=1+D, [ A>=1+D ], cost: 2 Accelerating simple loops of location 6. Accelerating the following rules: 18: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 2 Accelerated rule 18 with metering function C-A, yielding the new rule 20. Removing the simple loops: 18. Accelerating simple loops of location 8. Accelerating the following rules: 19: evalstartbb3in -> evalstartbb3in : D'=1+D, [ A>=1+D ], cost: 2 Accelerated rule 19 with metering function -D+A, yielding the new rule 21. Removing the simple loops: 19. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 20: evalstartbb1in -> evalstartbb1in : A'=C, [ C>=1+A ], cost: 2*C-2*A 21: evalstartbb3in -> evalstartbb3in : D'=A, [ A>=1+D ], cost: -2*D+2*A Chained accelerated rules (with incoming rules): Start location: evalstartstart 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 22: evalstartstart -> evalstartbb1in : A'=C, [ C>=1+B ], cost: 6+2*C-2*B 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 23: evalstartbb1in -> evalstartbb3in : D'=A, [ A>=1+C ], cost: 1-2*C+2*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 22: evalstartstart -> evalstartbb1in : A'=C, [ C>=1+B ], cost: 6+2*C-2*B 23: evalstartbb1in -> evalstartbb3in : D'=A, [ A>=1+C ], cost: 1-2*C+2*A Eliminated locations (on tree-shaped paths): Start location: evalstartstart 24: evalstartstart -> evalstartbb3in : A'=B, D'=B, [ B>=1+C ], cost: 7-2*C+2*B 25: evalstartstart -> [14] : [ C>=1+B ], cost: 6+2*C-2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 24: evalstartstart -> evalstartbb3in : A'=B, D'=B, [ B>=1+C ], cost: 7-2*C+2*B 25: evalstartstart -> [14] : [ C>=1+B ], cost: 6+2*C-2*B Computing asymptotic complexity for rule 24 Solved the limit problem by the following transformations: Created initial limit problem: -C+B (+/+!), 7-2*C+2*B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==0,B==n} resulting limit problem: [solved] Solution: C / 0 B / n Resulting cost 7+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: 7+2*n Rule cost: 7-2*C+2*B Rule guard: [ B>=1+C ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)