/export/starexec/sandbox2/solver/bin/starexec_run_c_complexity /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 377 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_nd_loop_start(v_0, v_x.0) -> Com_1(eval_nd_loop_bb0_in(v_0, v_x.0)) :|: TRUE eval_nd_loop_bb0_in(v_0, v_x.0) -> Com_1(eval_nd_loop_bb1_in(v_0, 0)) :|: TRUE eval_nd_loop_bb1_in(v_0, v_x.0) -> Com_1(eval_nd_loop_0(v_0, v_x.0)) :|: TRUE eval_nd_loop_0(v_0, v_x.0) -> Com_2(eval_nondet_start(v_0, v_x.0), eval_nd_loop_1(nondef.0, v_x.0)) :|: TRUE eval_nd_loop_1(v_0, v_x.0) -> Com_1(eval_nd_loop_bb1_in(v_0, v_0)) :|: v_0 - v_x.0 <= 2 && v_0 - v_x.0 >= 1 && v_0 < 10 eval_nd_loop_1(v_0, v_x.0) -> Com_1(eval_nd_loop_bb2_in(v_0, v_x.0)) :|: v_0 - v_x.0 > 2 eval_nd_loop_1(v_0, v_x.0) -> Com_1(eval_nd_loop_bb2_in(v_0, v_x.0)) :|: v_0 - v_x.0 < 1 eval_nd_loop_1(v_0, v_x.0) -> Com_1(eval_nd_loop_bb2_in(v_0, v_x.0)) :|: v_0 >= 10 eval_nd_loop_bb2_in(v_0, v_x.0) -> Com_1(eval_nd_loop_stop(v_0, v_x.0)) :|: TRUE The start-symbols are:[eval_nd_loop_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 108) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 3 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopstart(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) evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 3 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalndloopstart) = 2 Pol(evalndloopbb0in) = 2 Pol(evalndloopbb1in) = 2 Pol(evalndloop0) = 2 Pol(evalndloop00) = 0 Pol(evalnondetstart) = 0 Pol(evalndloop01) = 2 Pol(evalndloop1) = 2 Pol(evalndloopbb2in) = 1 Pol(evalndloopstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 10 ] evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 3 ] evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 10 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopstart(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 evalndloop0: X_1 >= 0 For symbol evalndloop00: X_1 >= 0 For symbol evalndloop01: X_1 >= 0 For symbol evalndloop1: X_2 - X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 >= 0 For symbol evalndloopbb1in: X_1 >= 0 For symbol evalndloopbb2in: X_2 - X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 By chaining the transition koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] with all transitions in problem 4, the following new transition is obtained: koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] We thus obtain the following problem: 5: T: (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 5: evalndloopstart(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) We thus obtain the following problem: 6: T: (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 ] with all transitions in problem 6, the following new transition is obtained: evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] We thus obtain the following problem: 7: T: (Comp: ?, Cost: 2) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: ?, Cost: 2) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_1, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] with all transitions in problem 8, the following new transition is obtained: evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] We thus obtain the following problem: 9: T: (Comp: ?, Cost: 3) evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 9 produces the following problem: 10: T: (Comp: ?, Cost: 3) evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: ?, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalndloop1) = -2*V_1 + 18 Pol(evalndloop00) = 1 Pol(evalndloop01) = -2*V_1 + 18 Pol(evalndloopbb2in) = -2*V_1 Pol(evalndloopstop) = -2*V_1 Pol(evalnondetstart) = 0 Pol(evalndloop0) = 19 Pol(evalndloopbb1in) = 19 Pol(evalndloopbb0in) = 19 Pol(koat_start) = 19 orients all transitions weakly and the transitions evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] strictly and produces the following problem: 11: T: (Comp: 19, Cost: 3) evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: 19, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 11 produces the following problem: 12: T: (Comp: 19, Cost: 3) evalndloop1(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_1, ar_1, d), evalndloop01(ar_1, ar_1, d)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 + 2 >= ar_1 /\ ar_1 >= ar_0 + 1 /\ 9 >= ar_1 /\ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalndloopbb2in(ar_0, ar_1, ar_2) -> Com_1(evalndloopstop(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_0 + 3 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_1 ] (Comp: 2, Cost: 1) evalndloop1(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 >= 0 /\ ar_1 >= 10 ] (Comp: 19, Cost: 1) evalndloop00(ar_0, ar_1, ar_2) -> Com_1(evalnondetstart(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 20, Cost: 1) evalndloop01(ar_0, ar_1, ar_2) -> Com_1(evalndloop1(ar_0, ar_2, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloop0(ar_0, ar_1, ar_2) -> Com_2(evalndloop00(ar_0, ar_1, d), evalndloop01(ar_0, ar_1, d)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb1in(ar_0, ar_1, ar_2) -> Com_1(evalndloop0(ar_0, ar_1, ar_2)) [ ar_0 >= 0 ] (Comp: 1, Cost: 1) evalndloopbb0in(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb1in(0, ar_1, ar_2)) (Comp: 1, Cost: 1) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalndloopbb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 108 Time: 0.370 sec (SMT: 0.299 sec) ---------------------------------------- (2) BOUNDS(1, 1)