/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).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, 31 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f0(A, B, C) -> Com_1(f1(A, B, 2)) :|: A >= 0 && 3 >= A && 3 >= B && B >= 0 f1(A, B, C) -> Com_1(f1(A, B + 1, C)) :|: C + A >= 2 * B + 1 f1(A, B, C) -> Com_1(f1(A, B - 1, C)) :|: 2 * B >= 2 + C + A The start-symbols are:[f0_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 23) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ] (Comp: ?, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] (Comp: ?, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(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) f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ] (Comp: ?, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] (Comp: ?, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = -V_1 + 2*V_2 - V_3 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = ? S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ? S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2 orients the transitions f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] weakly and the transition f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ] (Comp: ?, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] (Comp: 11, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = V_1 - 2*V_2 + V_3 and size complexities S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = 14 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3 S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ? S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3 S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2 orients the transitions f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] weakly and the transition f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ] (Comp: 11, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ] (Comp: 11, Cost: 1) f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 23 Time: 0.068 sec (SMT: 0.062 sec) ---------------------------------------- (2) BOUNDS(1, 1)