/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2).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, 44 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 529 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalfstart(A, B, C, D) -> Com_1(evalfentryin(A, B, C, D)) :|: TRUE evalfentryin(A, B, C, D) -> Com_1(evalfbb4in(1, B, C, D)) :|: TRUE evalfbb4in(A, B, C, D) -> Com_1(evalfbb2in(A, B, 1, D)) :|: B >= A evalfbb4in(A, B, C, D) -> Com_1(evalfreturnin(A, B, C, D)) :|: A >= B + 1 evalfbb2in(A, B, C, D) -> Com_1(evalfbb1in(A, B, C, D)) :|: D >= C evalfbb2in(A, B, C, D) -> Com_1(evalfbb3in(A, B, C, D)) :|: C >= D + 1 evalfbb1in(A, B, C, D) -> Com_1(evalfbb2in(A, B, C + 1, D)) :|: TRUE evalfbb3in(A, B, C, D) -> Com_1(evalfbb4in(A + 1, B, C, D)) :|: TRUE evalfreturnin(A, B, C, D) -> Com_1(evalfstop(A, B, C, D)) :|: TRUE The start-symbols are:[evalfstart_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 9*Ar_1 + 2*Ar_1*Ar_3 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfstart) = V_2 Pol(evalfentryin) = V_2 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 orients all transitions weakly and the transition evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_3) -> Com_1(evalfstart(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(evalfstart(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(evalfstart(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(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-0) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-1) = Ar_1 S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-2) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-3) = Ar_3 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-3) = Ar_3 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-0) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-3) = Ar_3 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = Ar_3 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) weakly and the transitions evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 2*Ar_1, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfbb2in) = -V_3 + V_4 + 1 Pol(evalfbb1in) = -V_3 + V_4 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(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(evalfstart(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(evalfstart(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(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 20 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-0) = 2*Ar_1 + 4 S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-0) = 2*Ar_1 + 4 S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-1) = Ar_1 S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-2) = ? S("evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-3) = Ar_3 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-0) = 2*Ar_1 + 4 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-3) = Ar_3 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-0) = 2*Ar_1 + 4 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-1) = Ar_1 S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-2) = ? S("evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-3) = Ar_3 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = 2*Ar_1 + 10 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = ? S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ]", 0-3) = Ar_3 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-0) = 2*Ar_1 + 4 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-2) = 1 S("evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ]", 0-3) = Ar_3 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-0) = 1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) weakly and the transition evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalfstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: Ar_1*Ar_3 + 2*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 2*Ar_1, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_3) -> Com_1(evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalfentryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(1, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, 1, Ar_3)) [ Ar_1 >= Ar_0 ] (Comp: 2, Cost: 1) evalfbb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 + 1 ] (Comp: Ar_1*Ar_3 + 2*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) evalfbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: Ar_1*Ar_3 + 2*Ar_1, Cost: 1) evalfbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb2in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 2*Ar_1, Cost: 1) evalfbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfbb4in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfreturnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 9*Ar_1 + 2*Ar_1*Ar_3 + 6 Time: 0.086 sec (SMT: 0.064 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'=1, [], cost: 1 2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1 3: evalfbb4in -> evalfreturnin : [ A>=1+B ], cost: 1 4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1 5: evalfbb2in -> evalfbb3in : [ C>=1+D ], 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'=1, [], cost: 1 2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1 4: evalfbb2in -> evalfbb1in : [ D>=C ], cost: 1 5: evalfbb2in -> evalfbb3in : [ C>=1+D ], 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'=1, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1 10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2 Accelerating simple loops of location 3. Accelerating the following rules: 10: evalfbb2in -> evalfbb2in : C'=1+C, [ D>=C ], cost: 2 Accelerated rule 10 with metering function 1-C+D, 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'=1, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2 12: evalfbb2in -> evalfbb2in : C'=1+D, [ D>=C ], cost: 2-2*C+2*D Chained accelerated rules (with incoming rules): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=1, [], cost: 2 2: evalfbb4in -> evalfbb2in : C'=1, [ B>=A ], cost: 1 13: evalfbb4in -> evalfbb2in : C'=1+D, [ B>=A && D>=1 ], cost: 1+2*D 11: evalfbb2in -> evalfbb4in : A'=1+A, [ C>=1+D ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=1, [], cost: 2 14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3 15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D Accelerating simple loops of location 2. Accelerating the following rules: 14: evalfbb4in -> evalfbb4in : A'=1+A, C'=1, [ B>=A && 1>=1+D ], cost: 3 15: evalfbb4in -> evalfbb4in : A'=1+A, C'=1+D, [ B>=A && D>=1 ], cost: 3+2*D Accelerated rule 14 with metering function 1-A+B, yielding the new rule 16. Accelerated rule 15 with metering function 1-A+B, yielding the new rule 17. Removing the simple loops: 14 15. Accelerated all simple loops using metering functions (where possible): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=1, [], cost: 2 16: evalfbb4in -> evalfbb4in : A'=1+B, C'=1, [ B>=A && 1>=1+D ], cost: 3-3*A+3*B 17: evalfbb4in -> evalfbb4in : A'=1+B, C'=1+D, [ B>=A && D>=1 ], cost: 3-3*A+3*B-2*(-1+A-B)*D Chained accelerated rules (with incoming rules): Start location: evalfstart 9: evalfstart -> evalfbb4in : A'=1, [], cost: 2 18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B 19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalfstart 18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B 19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalfstart 18: evalfstart -> evalfbb4in : A'=1+B, C'=1, [ B>=1 && 1>=1+D ], cost: 2+3*B 19: evalfstart -> evalfbb4in : A'=1+B, C'=1+D, [ B>=1 && D>=1 ], cost: 2+2*D*B+3*B Computing asymptotic complexity for rule 18 Solved the limit problem by the following transformations: Created initial limit problem: 1-D (+/+!), 2+3*B (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {D==0,B==n} resulting limit problem: [solved] Solution: D / 0 B / n Resulting cost 2+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 19 Solved the limit problem by the following transformations: Created initial limit problem: D (+/+!), 2+2*D*B+3*B (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {D==n,B==n} resulting limit problem: [solved] Solution: D / n B / n Resulting cost 2+3*n+2*n^2 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+3*n+2*n^2 Rule cost: 2+2*D*B+3*B Rule guard: [ B>=1 && D>=1 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)