/export/starexec/sandbox2/solver/bin/starexec_run_c_complexity /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/output/output_files/bench.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 77 ms] (2) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_t47_start(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb0_in(v_.0, v_flag.0, v_n)) :|: TRUE eval_t47_bb0_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_n, 1, v_n)) :|: TRUE eval_t47_bb1_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb2_in(v_.0, v_flag.0, v_n)) :|: v_flag.0 > 0 eval_t47_bb1_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb3_in(v_.0, v_flag.0, v_n)) :|: v_flag.0 <= 0 eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0 - 1, 1, v_n)) :|: v_.0 > 0 eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0, 1, v_n)) :|: v_.0 > 0 && v_.0 <= 0 eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0 - 1, 0, v_n)) :|: v_.0 <= 0 && v_.0 > 0 eval_t47_bb2_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_bb1_in(v_.0, 0, v_n)) :|: v_.0 <= 0 eval_t47_bb3_in(v_.0, v_flag.0, v_n) -> Com_1(eval_t47_stop(v_.0, v_flag.0, v_n)) :|: TRUE The start-symbols are:[eval_t47_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*ar_1 + 8) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ] evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: ?, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: ?, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 1, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: 1, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalt47bb3in) = 1 Pol(evalt47stop) = 0 Pol(evalt47bb1in) = 2 Pol(evalt47bb2in) = 2 Pol(evalt47bb0in) = 2 Pol(evalt47start) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: 2, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 1, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: 1, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalt47bb3in) = V_3 Pol(evalt47stop) = V_3 Pol(evalt47bb1in) = V_3 Pol(evalt47bb2in) = 1 Pol(evalt47bb0in) = 1 Pol(evalt47start) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] strictly and produces the following problem: 5: T: (Comp: 2, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: 2, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: 1, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ?, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 1, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: 1, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalt47bb3in) = V_1 Pol(evalt47stop) = V_1 Pol(evalt47bb1in) = V_1 Pol(evalt47bb2in) = V_1 Pol(evalt47bb0in) = V_2 Pol(evalt47start) = V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] strictly and produces the following problem: 6: T: (Comp: 2, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: 2, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: 1, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ar_1, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ?, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 1, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: 1, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 2, Cost: 1) evalt47bb3in(ar_0, ar_1, ar_2) -> Com_1(evalt47stop(ar_0, ar_1, ar_2)) (Comp: 2, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] (Comp: 1, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ] (Comp: ar_1, Cost: 1) evalt47bb2in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ] (Comp: ar_1 + 1, Cost: 1) evalt47bb1in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ] (Comp: 1, Cost: 1) evalt47bb0in(ar_0, ar_1, ar_2) -> Com_1(evalt47bb1in(ar_1, ar_1, 1)) (Comp: 1, Cost: 1) evalt47start(ar_0, ar_1, ar_2) -> Com_1(evalt47bb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalt47start(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*ar_1 + 8 Time: 0.073 sec (SMT: 0.066 sec) ---------------------------------------- (2) BOUNDS(1, n^1)