/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 79 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 518 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalfstart(A, B, C) -> Com_1(evalfentryin(A, B, C)) :|: TRUE evalfentryin(A, B, C) -> Com_1(evalfbb4in(0, B, C)) :|: TRUE evalfbb4in(A, B, C) -> Com_1(evalfbb2in(A, B, 0)) :|: B >= 1 + A evalfbb4in(A, B, C) -> Com_1(evalfreturnin(A, B, C)) :|: A >= B evalfbb2in(A, B, C) -> Com_1(evalfbb1in(A, B, C)) :|: A >= C evalfbb2in(A, B, C) -> Com_1(evalfbb3in(A, B, C)) :|: C >= A + 1 evalfbb1in(A, B, C) -> Com_1(evalfbb2in(A, B, C + 1)) :|: TRUE evalfbb3in(A, B, C) -> Com_1(evalfbb4in(A + 1, B, C)) :|: TRUE evalfreturnin(A, B, C) -> Com_1(evalfstop(A, B, C)) :|: TRUE The start-symbols are:[evalfstart_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 27*Ar_1 + 4*Ar_1^2 + 29) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(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) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfstart) = 2 Pol(evalfentryin) = 2 Pol(evalfbb4in) = 2 Pol(evalfbb2in) = 2 Pol(evalfreturnin) = 1 Pol(evalfbb1in) = 2 Pol(evalfbb3in) = 2 Pol(evalfstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfstart) = V_2 + 1 Pol(evalfentryin) = V_2 + 1 Pol(evalfbb4in) = -V_1 + V_2 + 1 Pol(evalfbb2in) = -V_1 + V_2 Pol(evalfreturnin) = -V_1 + V_2 Pol(evalfbb1in) = -V_1 + V_2 Pol(evalfbb3in) = -V_1 + V_2 Pol(evalfstop) = -V_1 + V_2 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbb3in) = 1 Pol(evalfbb4in) = 0 Pol(evalfbb2in) = 2 Pol(evalfbb1in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-0) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-0) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-1) = Ar_1 S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-0) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = 0 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) weakly and the transitions evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbb2in) = V_1 - V_3 + 1 Pol(evalfbb1in) = V_1 - V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 32 S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-0) = 2*Ar_1 + 8 S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2))", 0-2) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-0) = 2*Ar_1 + 8 S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-1) = Ar_1 S("evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1))", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-0) = 2*Ar_1 + 8 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-0) = 2*Ar_1 + 8 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-0) = 2*Ar_1 + 16 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-0) = 2*Ar_1 + 8 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ]", 0-2) = 0 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-0) = 0 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) weakly and the transition evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1^2 + 11*Ar_1 + 9, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(0, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1^2 + 11*Ar_1 + 9, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 2*Ar_1^2 + 11*Ar_1 + 9, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 27*Ar_1 + 4*Ar_1^2 + 29 Time: 0.120 sec (SMT: 0.105 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb4in : A'=0, [], cost: 1 2: evalfbb4in -> evalfbb2in : C'=0, [ B>=1+A ], cost: 1 3: evalfbb4in -> evalfreturnin : [ A>=B ], cost: 1 4: evalfbb2in -> evalfbb1in : [ A>=C ], cost: 1 5: evalfbb2in -> evalfbb3in : [ C>=1+A ], cost: 1 6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1 7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1 8: evalfreturnin -> evalfstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalfstart -> evalfentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalfstart 0: evalfstart -> evalfentryin : [], cost: 1 1: evalfentryin -> evalfbb4in : A'=0, [], cost: 1 2: evalfbb4in -> evalfbb2in : C'=0, [ B>=1+A ], cost: 1 4: evalfbb2in -> evalfbb1in : [ A>=C ], cost: 1 5: evalfbb2in -> evalfbb3in : [ C>=1+A ], cost: 1 6: evalfbb1in -> evalfbb2in : C'=1+C, [], cost: 1 7: evalfbb3in -> evalfbb4in : A'=1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=0, [ B>=1+A ], cost: 1 10: evalfbb2in -> evalfbb2in : C'=1+C, [ A>=C ], cost: 2 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+A ], cost: 2 Accelerating simple loops of location 3. Accelerating the following rules: 10: evalfbb2in -> evalfbb2in : C'=1+C, [ A>=C ], cost: 2 Accelerated rule 10 with metering function 1-C+A, yielding the new rule 12. Removing the simple loops: 10. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=0, [ B>=1+A ], cost: 1 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+A ], cost: 2 12: evalfbb2in -> evalfbb2in : C'=1+A, [ A>=C ], cost: 2-2*C+2*A Chained accelerated rules (with incoming rules): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=0, [ B>=1+A ], cost: 1 13: evalfbb4in -> evalfbb2in : C'=1+A, [ B>=1+A && A>=0 ], cost: 3+2*A 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+A ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 14: evalfbb4in -> evalfbb4in : A'=1+A, C'=0, [ B>=1+A && 0>=1+A ], cost: 3 15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+A, [ B>=1+A && A>=0 ], cost: 5+2*A Accelerating simple loops of location 2. Accelerating the following rules: 14: evalfbb4in -> evalfbb4in : A'=1+A, C'=0, [ B>=1+A && 0>=1+A ], cost: 3 15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+A, [ B>=1+A && A>=0 ], cost: 5+2*A Found no metering function for rule 14. Accelerated rule 15 with metering function -A+B, yielding the new rule 16. Removing the simple loops: 15. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 14: evalfbb4in -> evalfbb4in : A'=1+A, C'=0, [ B>=1+A && 0>=1+A ], cost: 3 16: evalfbb4in -> evalfbb4in : A'=B, C'=B, [ B>=1+A && A>=0 ], cost: (A-B)^2-4*A-2*A*(A-B)+4*B Chained accelerated rules (with incoming rules): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=0, [], cost: 2 17: evalfstart -> evalfbb4in : A'=B, C'=B, [ B>=1 ], cost: 2+4*B+B^2 Removed unreachable locations (and leaf rules with constant cost): Start location: evalfstart 17: evalfstart -> evalfbb4in : A'=B, C'=B, [ B>=1 ], cost: 2+4*B+B^2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalfstart 17: evalfstart -> evalfbb4in : A'=B, C'=B, [ B>=1 ], cost: 2+4*B+B^2 Computing asymptotic complexity for rule 17 Solved the limit problem by the following transformations: Created initial limit problem: B (+/+!), 2+4*B+B^2 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 2+n^2+4*n has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 2+n^2+4*n Rule cost: 2+4*B+B^2 Rule guard: [ B>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)