/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 23 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f18(A, B, C, D, E, F) -> Com_1(f18(A, B + 1, C, D, E, F)) :|: A >= B + 1 f24(A, B, C, D, E, F) -> Com_1(f24(A, B + 1, C, D, E, F)) :|: A >= B + 1 f31(A, B, C, D, E, F) -> Com_1(f31(A, B + 1, C, D, E, F)) :|: A >= B + 1 f31(A, B, C, D, E, F) -> Com_1(f39(A, B, C, D, E, F)) :|: B >= A f24(A, B, C, D, E, F) -> Com_1(f31(A, 0, C, D, E, F)) :|: B >= A f18(A, B, C, D, E, F) -> Com_1(f24(A, 0, C, D, E, F)) :|: B >= A f0(A, B, C, D, E, F) -> Com_1(f18(10, 0, 10, G, 10, H)) :|: TRUE The start-symbols are:[f0_6] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 100) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f18(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f24(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f31(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f39(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f31(Ar_0, 0, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f24(Ar_0, 0, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f18(10, 0, 10, Fresh_0, 10, Fresh_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [Ar_0, Ar_1]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f18(10, 0)) (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f18(10, 0)) (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 3 Pol(f0) = 3 Pol(f18) = 3 Pol(f24) = 2 Pol(f31) = 1 Pol(f39) = 0 orients all transitions weakly and the transitions f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f18(10, 0)) (Comp: 3, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 10 Pol(f0) = 10 Pol(f18) = V_1 Pol(f24) = V_1 Pol(f31) = V_1 - V_2 Pol(f39) = V_1 - V_2 orients all transitions weakly and the transition f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f18(10, 0)) (Comp: 3, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: 10, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f24) = V_1 - V_2 Pol(f18) = V_1 - V_2 and size complexities S("f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 10 S("f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ? S("f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 10 S("f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ? S("f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 10 S("f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 10 S("f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ]", 0-0) = 10 S("f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ]", 0-1) = 10 S("f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ]", 0-0) = 10 S("f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ]", 0-1) = 0 S("f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ]", 0-0) = 10 S("f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ]", 0-1) = 0 S("f0(Ar_0, Ar_1) -> Com_1(f18(10, 0))", 0-0) = 10 S("f0(Ar_0, Ar_1) -> Com_1(f18(10, 0))", 0-1) = 0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 orients the transitions f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] weakly and the transitions f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f18(10, 0)) (Comp: 3, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f24(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f31(Ar_0, 0)) [ Ar_1 >= Ar_0 ] (Comp: 3, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f39(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: 10, Cost: 1) f31(Ar_0, Ar_1) -> Com_1(f31(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 40, Cost: 1) f24(Ar_0, Ar_1) -> Com_1(f24(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 40, Cost: 1) f18(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 Complexity upper bound 100 Time: 0.058 sec (SMT: 0.048 sec) ---------------------------------------- (2) BOUNDS(1, 1)