/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 226 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 522 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v_i_0, v_m, v_n) -> Com_1(eval_start_bb0_in(v_i_0, v_m, v_n)) :|: TRUE eval_start_bb0_in(v_i_0, v_m, v_n) -> Com_1(eval_start_0(v_i_0, v_m, v_n)) :|: TRUE eval_start_0(v_i_0, v_m, v_n) -> Com_1(eval_start_1(v_i_0, v_m, v_n)) :|: TRUE eval_start_1(v_i_0, v_m, v_n) -> Com_1(eval_start_2(v_i_0, v_m, v_n)) :|: TRUE eval_start_2(v_i_0, v_m, v_n) -> Com_1(eval_start_3(v_i_0, v_m, v_n)) :|: TRUE eval_start_3(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_n, v_m, v_n)) :|: 0 < v_m eval_start_3(v_i_0, v_m, v_n) -> Com_1(eval_start_bb4_in(v_i_0, v_m, v_n)) :|: 0 >= v_m eval_start_bb1_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb2_in(v_i_0, v_m, v_n)) :|: v_i_0 > 0 eval_start_bb1_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb3_in(v_i_0, v_m, v_n)) :|: v_i_0 <= 0 eval_start_bb2_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_i_0 - 1, v_m, v_n)) :|: v_i_0 < v_m eval_start_bb2_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_i_0 - v_m, v_m, v_n)) :|: v_i_0 >= v_m eval_start_bb3_in(v_i_0, v_m, v_n) -> Com_1(eval_start_stop(v_i_0, v_m, v_n)) :|: TRUE eval_start_bb4_in(v_i_0, v_m, v_n) -> Com_1(eval_start_8(v_i_0, v_m, v_n)) :|: TRUE eval_start_8(v_i_0, v_m, v_n) -> Com_1(eval_start_9(v_i_0, v_m, v_n)) :|: TRUE eval_start_9(v_i_0, v_m, v_n) -> Com_1(eval_start_stop(v_i_0, v_m, v_n)) :|: TRUE The start-symbols are:[eval_start_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 6*Ar_2 + 14) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(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) evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 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(evalstartbb1in) = 2 Pol(evalstartbb4in) = 0 Pol(evalstartbb2in) = 2 Pol(evalstartbb3in) = 1 Pol(evalstartstop) = 0 Pol(evalstart8) = 0 Pol(evalstart9) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalstart8: -X_1 >= 0 For symbol evalstart9: -X_1 >= 0 For symbol evalstartbb1in: -X_2 + X_3 >= 0 /\ X_1 - 1 >= 0 For symbol evalstartbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0 For symbol evalstartbb3in: -X_2 + X_3 >= 0 /\ -X_2 >= 0 /\ X_1 - X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalstartbb4in: -X_1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ -Ar_1 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartbb2in) = 2*V_2 - 1 Pol(evalstartbb1in) = 2*V_2 and size complexities S("evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ]", 0-0) = Ar_0 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ]", 0-1) = Ar_2 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ]", 0-2) = Ar_2 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= 1 ]", 0-2) = Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_1 ]", 0-0) = Ar_0 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_1 ]", 0-1) = Ar_2 S("evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_1 ]", 0-2) = Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-2) = Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-0) = Ar_0 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-1) = Ar_2 S("evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ -Ar_1 + Ar_2 >= 0 /\\ Ar_0 + Ar_2 - 2 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 2 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-2) = Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ -Ar_1 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-0) = Ar_0 S("evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ -Ar_1 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-1) = Ar_2 S("evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\\ -Ar_1 >= 0 /\\ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 ]", 0-2) = Ar_2 S("evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 orients the transitions evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] weakly and the transitions evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalstart9(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalstart8(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart9(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart8(Ar_0, Ar_1, Ar_2)) [ -Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ -Ar_1 >= 0 /\ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2*Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - Ar_0, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: 2*Ar_2, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_0 + Ar_2 - 2 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 ] (Comp: 2*Ar_2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2)) [ -Ar_1 + Ar_2 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb1in(Ar_0, Ar_2, Ar_2)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 6*Ar_2 + 14 Time: 0.211 sec (SMT: 0.166 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 : B'=C, [ A>=1 ], cost: 1 6: evalstart3 -> evalstartbb4in : [ 0>=A ], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 8: evalstartbb1in -> evalstartbb3in : [ 0>=B ], cost: 1 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 11: evalstartbb3in -> evalstartstop : [], cost: 1 12: evalstartbb4in -> evalstart8 : [], cost: 1 13: evalstart8 -> evalstart9 : [], cost: 1 14: evalstart9 -> 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 : B'=C, [ A>=1 ], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalstartstart 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 20: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ B>=1 && A>=1+B ], cost: 2 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 Accelerating simple loops of location 6. Accelerating the following rules: 20: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ B>=1 && A>=1+B ], cost: 2 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 Accelerated rule 20 with metering function B, yielding the new rule 22. Found no metering function for rule 21. Removing the simple loops: 20. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 22: evalstartbb1in -> evalstartbb1in : B'=0, [ B>=1 && A>=1+B ], cost: 2*B Chained accelerated rules (with incoming rules): Start location: evalstartstart 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 23: evalstartstart -> evalstartbb1in : B'=C-A, [ A>=1 && C>=1 && C>=A ], cost: 8 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C Computing asymptotic complexity for rule 24 Simplified the guard: 24: evalstartstart -> evalstartbb1in : B'=0, [ C>=1 && A>=1+C ], cost: 6+2*C Solved the limit problem by the following transformations: Created initial limit problem: C (+/+!), -C+A (+/+!), 6+2*C (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n,A==1+n} resulting limit problem: [solved] Solution: C / n A / 1+n Resulting cost 6+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: 6+2*n Rule cost: 6+2*C Rule guard: [ C>=1 && A>=1+C ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)