/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).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, 18 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f0(A, B, C, D) -> Com_1(f1(A, B, 2, D)) :|: A >= 0 && 3 >= A && 3 >= B && B >= 0 f1(A, B, C, D) -> Com_1(f1(A, B + 1, C, B + 1)) :|: C + A >= 2 * B + 1 && 0 >= 2 f1(A, B, C, D) -> Com_1(f1(A, B + 1, C, B + 1)) :|: C + A >= 2 * B + 1 f1(A, B, C, D) -> Com_1(f1(A, B - 1, C, B - 1)) :|: 2 * B >= 2 + C + A f1(A, B, C, D) -> Com_1(f1(A, B - 1, C, B - 1)) :|: 2 * B >= 2 + C + A && 0 >= 2 f1(A, B, C, D) -> Com_1(f1(A, B, C, B)) :|: 0 >= 1 && 2 * B >= C + A && C + A + 1 >= 2 * B The start-symbols are:[f0_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 23) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\ 3 >= Ar_0 /\ 3 >= Ar_1 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 /\ 0 >= 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 /\ 0 >= 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_1)) [ 0 >= 1 /\ 2*Ar_1 >= Ar_2 + Ar_0 /\ Ar_2 + Ar_0 + 1 >= 2*Ar_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: f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 /\ 0 >= 2 ] f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 /\ 0 >= 2 ] f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_1)) [ 0 >= 1 /\ 2*Ar_1 >= Ar_2 + Ar_0 /\ Ar_2 + Ar_0 + 1 >= 2*Ar_1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\ 3 >= Ar_0 /\ 3 >= Ar_1 /\ Ar_1 >= 0 ] (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) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\ 3 >= Ar_0 /\ 3 >= Ar_1 /\ Ar_1 >= 0 ] (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(f1) = -V_1 + 2*V_2 - V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-0) = 3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-1) = 3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-2) = 2 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-3) = Ar_3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-0) = 3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-1) = ? S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-2) = 2 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-3) = ? S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-0) = 3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-1) = ? S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-2) = 2 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-3) = ? orients the transitions f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] weakly and the transition f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] strictly and produces the following problem: 4: T: (Comp: 11, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\ 3 >= Ar_0 /\ 3 >= Ar_1 /\ Ar_1 >= 0 ] (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(f1) = V_1 - 2*V_2 + V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-0) = 3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-1) = 3 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-2) = 2 S("f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\\ 3 >= Ar_0 /\\ 3 >= Ar_1 /\\ Ar_1 >= 0 ]", 0-3) = Ar_3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-0) = 3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-1) = ? S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-2) = 2 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ]", 0-3) = ? S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-0) = 3 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-1) = 14 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-2) = 2 S("f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ]", 0-3) = 15 orients the transitions f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] weakly and the transition f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] strictly and produces the following problem: 5: T: (Comp: 11, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 - 1, Ar_2, Ar_1 - 1)) [ 2*Ar_1 >= Ar_2 + Ar_0 + 2 ] (Comp: 11, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2, Ar_1 + 1)) [ Ar_2 + Ar_0 >= 2*Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, 2, Ar_3)) [ Ar_0 >= 0 /\ 3 >= Ar_0 /\ 3 >= Ar_1 /\ Ar_1 >= 0 ] (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 23 Time: 0.092 sec (SMT: 0.082 sec) ---------------------------------------- (2) BOUNDS(1, 1)