/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).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, 38 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f8(A, B, C, D) -> Com_1(f8(A - 1, B, C, D)) :|: A >= 0 f19(A, B, C, D) -> Com_1(f19(A, B - 1, C, D)) :|: B >= 0 f28(A, B, C, D) -> Com_1(f28(A, B, C - 1, D)) :|: C >= 0 f28(A, B, C, D) -> Com_1(f36(A, B, C, D)) :|: 0 >= C + 1 f19(A, B, C, D) -> Com_1(f28(A, B, 999, D)) :|: 0 >= B + 1 f0(A, B, C, D) -> Com_1(f19(A, 999, C, 1)) :|: TRUE f8(A, B, C, D) -> Com_1(f19(A, 999, C, D)) :|: 0 >= A + 1 The start-symbols are:[f0_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2005) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 - 1, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: ?, Cost: 1) f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, ar_3)) [ 0 >= ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 - 1, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ] f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, ar_3)) [ 0 >= ar_0 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f28) = 1 Pol(f36) = 0 Pol(f19) = 2 Pol(f0) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: ?, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: 2, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f28) = V_3 + 1 Pol(f36) = V_3 Pol(f19) = 1000 Pol(f0) = 1000 Pol(koat_start) = 1000 orients all transitions weakly and the transition f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] strictly and produces the following problem: 5: T: (Comp: 2, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: 1000, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: 2, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: ?, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f28) = V_2 Pol(f36) = V_2 Pol(f19) = V_2 + 1 Pol(f0) = 1000 Pol(koat_start) = 1000 orients all transitions weakly and the transition f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] strictly and produces the following problem: 6: T: (Comp: 2, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ] (Comp: 1000, Cost: 1) f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ] (Comp: 2, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ] (Comp: 1000, Cost: 1) f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2005 Time: 0.087 sec (SMT: 0.079 sec) ---------------------------------------- (2) BOUNDS(1, 1)