/export/starexec/sandbox/solver/bin/starexec_run_c_complexity /export/starexec/sandbox/benchmark/theBenchmark.c /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).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, 173 ms] (2) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_foo_start(v_.0, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_x, v_y)) :|: TRUE eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_x, v_y)) :|: 2 * v_y >= 1 eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: 2 * v_y < 1 eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_x, v_y)) :|: v_.0 >= 0 eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_.0 < 0 eval_foo_bb2_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 - 2 * v_y + 1, v_x, v_y)) :|: TRUE eval_foo_bb3_in(v_.0, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_x, v_y)) :|: TRUE The start-symbols are:[eval_foo_start_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 4*ar_2 + 11) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(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) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb3in) = 1 Pol(evalfoobb2in) = 2 Pol(evalfoostop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) (Comp: 2, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(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 evalfoobb1in: -X_2 + X_3 >= 0 /\ X_1 - 1 >= 0 For symbol evalfoobb2in: X_3 >= 0 /\ X_2 + X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 0 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb2in) = 2*V_2 + 1 Pol(evalfoobb1in) = 2*V_2 + 2 and size complexities S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ]", 0-0) = ar_0 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ]", 0-1) = ar_2 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ]", 0-2) = ar_2 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ]", 0-0) = ar_0 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ]", 0-1) = ar_1 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 0 ]", 0-0) = ar_0 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 0 ]", 0-1) = 2*ar_0 + 2*ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 0 ]", 0-2) = ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 + 1 ]", 0-0) = ar_0 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 + 1 ]", 0-1) = 2*ar_0 + 2*ar_2 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 + 1 ]", 0-2) = ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\\ ar_1 + ar_2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\\ ar_1 + ar_2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = 2*ar_0 + 2*ar_2 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\\ ar_1 + ar_2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_2 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = ar_0 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = 2*ar_0 + 2*ar_1 + 2*ar_2 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 orients the transitions evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 0 ] weakly and the transitions evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 0 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) (Comp: 2*ar_2 + 2, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 - 2*ar_0 + 1, ar_2)) [ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ] (Comp: 2*ar_2 + 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 0 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= 2*ar_0 ] (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ 2*ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 4*ar_2 + 11 Time: 0.213 sec (SMT: 0.192 sec) ---------------------------------------- (2) BOUNDS(1, n^1)