/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 14 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 552 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_random1d_start(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb0_in(v_2, v_max, v_x_0)) :|: TRUE eval_random1d_bb0_in(v_2, v_max, v_x_0) -> Com_1(eval_random1d_0(v_2, v_max, v_x_0)) :|: TRUE eval_random1d_0(v_2, v_max, v_x_0) -> Com_1(eval_random1d_1(v_2, v_max, v_x_0)) :|: TRUE eval_random1d_1(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb1_in(v_2, v_max, 1)) :|: v_max > 0 eval_random1d_1(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb3_in(v_2, v_max, v_x_0)) :|: v_max <= 0 eval_random1d_bb1_in(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb2_in(v_2, v_max, v_x_0)) :|: v_x_0 <= v_max eval_random1d_bb1_in(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb3_in(v_2, v_max, v_x_0)) :|: v_x_0 > v_max eval_random1d_bb2_in(v_2, v_max, v_x_0) -> Com_1(eval_random1d_2(v_2, v_max, v_x_0)) :|: TRUE eval_random1d_2(v_2, v_max, v_x_0) -> Com_1(eval_random1d_3(nondef_0, v_max, v_x_0)) :|: TRUE eval_random1d_3(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb1_in(v_2, v_max, v_x_0 + 1)) :|: v_2 > 0 eval_random1d_3(v_2, v_max, v_x_0) -> Com_1(eval_random1d_bb1_in(v_2, v_max, v_x_0 + 1)) :|: v_2 <= 0 eval_random1d_bb3_in(v_2, v_max, v_x_0) -> Com_1(eval_random1d_stop(v_2, v_max, v_x_0)) :|: TRUE The start-symbols are:[eval_random1d_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 5*Ar_0 + 19) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 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) evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrandom1dstart) = 2 Pol(evalrandom1dbb0in) = 2 Pol(evalrandom1d0) = 2 Pol(evalrandom1d1) = 2 Pol(evalrandom1dbb1in) = 2 Pol(evalrandom1dbb3in) = 1 Pol(evalrandom1dbb2in) = 2 Pol(evalrandom1d2) = 2 Pol(evalrandom1d3) = 2 Pol(evalrandom1dstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: 2, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] (Comp: 2, Cost: 1) evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalrandom1dbb2in) = V_1 - V_2 Pol(evalrandom1d2) = V_1 - V_2 Pol(evalrandom1dbb1in) = V_1 - V_2 + 1 Pol(evalrandom1d3) = V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ]", 0-0) = Ar_0 S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ]", 0-1) = ? S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ]", 0-2) = ? S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ]", 0-0) = Ar_0 S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ]", 0-1) = ? S("evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ]", 0-2) = ? S("evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0))", 0-0) = Ar_0 S("evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0))", 0-1) = ? S("evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0))", 0-2) = ? S("evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = ? S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = ? S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-0) = Ar_0 S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-1) = ? S("evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ]", 0-0) = Ar_0 S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ]", 0-1) = 1 S("evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ]", 0-2) = Ar_2 S("evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) weakly and the transition evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: Ar_0 + 2, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: 2, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] (Comp: 2, Cost: 1) evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 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) evalrandom1dstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1dbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d0(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, 1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalrandom1d1(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: Ar_0 + 2, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: 2, Cost: 1) evalrandom1dbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: Ar_0 + 2, Cost: 1) evalrandom1dbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d2(Ar_0, Ar_1, Ar_2)) (Comp: Ar_0 + 2, Cost: 1) evalrandom1d2(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1d3(Ar_0, Ar_1, Fresh_0)) (Comp: Ar_0 + 2, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_2 >= 1 ] (Comp: Ar_0 + 2, Cost: 1) evalrandom1d3(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dbb1in(Ar_0, Ar_1 + 1, Ar_2)) [ 0 >= Ar_2 ] (Comp: 2, Cost: 1) evalrandom1dbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalrandom1dstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 5*Ar_0 + 19 Time: 0.073 sec (SMT: 0.057 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalrandom1dstart 0: evalrandom1dstart -> evalrandom1dbb0in : [], cost: 1 1: evalrandom1dbb0in -> evalrandom1d0 : [], cost: 1 2: evalrandom1d0 -> evalrandom1d1 : [], cost: 1 3: evalrandom1d1 -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 1 4: evalrandom1d1 -> evalrandom1dbb3in : [ 0>=A ], cost: 1 5: evalrandom1dbb1in -> evalrandom1dbb2in : [ A>=B ], cost: 1 6: evalrandom1dbb1in -> evalrandom1dbb3in : [ B>=1+A ], cost: 1 7: evalrandom1dbb2in -> evalrandom1d2 : [], cost: 1 8: evalrandom1d2 -> evalrandom1d3 : C'=free, [], cost: 1 9: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ C>=1 ], cost: 1 10: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ 0>=C ], cost: 1 11: evalrandom1dbb3in -> evalrandom1dstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalrandom1dstart -> evalrandom1dbb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalrandom1dstart 0: evalrandom1dstart -> evalrandom1dbb0in : [], cost: 1 1: evalrandom1dbb0in -> evalrandom1d0 : [], cost: 1 2: evalrandom1d0 -> evalrandom1d1 : [], cost: 1 3: evalrandom1d1 -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 1 5: evalrandom1dbb1in -> evalrandom1dbb2in : [ A>=B ], cost: 1 7: evalrandom1dbb2in -> evalrandom1d2 : [], cost: 1 8: evalrandom1d2 -> evalrandom1d3 : C'=free, [], cost: 1 9: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ C>=1 ], cost: 1 10: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ 0>=C ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalrandom1dstart 14: evalrandom1dstart -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 4 16: evalrandom1dbb1in -> evalrandom1d3 : C'=free, [ A>=B ], cost: 3 9: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ C>=1 ], cost: 1 10: evalrandom1d3 -> evalrandom1dbb1in : B'=1+B, [ 0>=C ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalrandom1dstart 14: evalrandom1dstart -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 4 17: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+B, C'=free, [ A>=B && free>=1 ], cost: 4 18: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+B, C'=free, [ A>=B && 0>=free ], cost: 4 Accelerating simple loops of location 4. Accelerating the following rules: 17: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+B, C'=free, [ A>=B && free>=1 ], cost: 4 18: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+B, C'=free, [ A>=B && 0>=free ], cost: 4 Accelerated rule 17 with metering function 1+A-B, yielding the new rule 19. Accelerated rule 18 with metering function 1+A-B, yielding the new rule 20. Removing the simple loops: 17 18. Accelerated all simple loops using metering functions (where possible): Start location: evalrandom1dstart 14: evalrandom1dstart -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 4 19: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=B && free>=1 ], cost: 4+4*A-4*B 20: evalrandom1dbb1in -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=B && 0>=free ], cost: 4+4*A-4*B Chained accelerated rules (with incoming rules): Start location: evalrandom1dstart 14: evalrandom1dstart -> evalrandom1dbb1in : B'=1, [ A>=1 ], cost: 4 21: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && free>=1 ], cost: 4+4*A 22: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && 0>=free ], cost: 4+4*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalrandom1dstart 21: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && free>=1 ], cost: 4+4*A 22: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && 0>=free ], cost: 4+4*A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalrandom1dstart 21: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && free>=1 ], cost: 4+4*A 22: evalrandom1dstart -> evalrandom1dbb1in : B'=1+A, C'=free, [ A>=1 && 0>=free ], cost: 4+4*A Computing asymptotic complexity for rule 21 Solved the limit problem by the following transformations: Created initial limit problem: A (+/+!), free (+/+!), 4+4*A (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n,free==1} resulting limit problem: [solved] Solution: A / n free / 1 Resulting cost 4+4*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: 4+4*n Rule cost: 4+4*A Rule guard: [ A>=1 && free>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)