/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 101 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 628 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalSimpleSingle2start(A, B, C, D) -> Com_1(evalSimpleSingle2entryin(A, B, C, D)) :|: TRUE evalSimpleSingle2entryin(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(0, 0, C, D)) :|: TRUE evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2bbin(A, B, C, D)) :|: 0 >= E + 1 evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2bbin(A, B, C, D)) :|: E >= 1 evalSimpleSingle2bb4in(A, B, C, D) -> Com_1(evalSimpleSingle2returnin(A, B, C, D)) :|: TRUE evalSimpleSingle2bbin(A, B, C, D) -> Com_1(evalSimpleSingle2bb1in(A, B, C, D)) :|: C >= B + 1 evalSimpleSingle2bbin(A, B, C, D) -> Com_1(evalSimpleSingle2bb2in(A, B, C, D)) :|: B >= C evalSimpleSingle2bb1in(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(A + 1, B + 1, C, D)) :|: TRUE evalSimpleSingle2bb2in(A, B, C, D) -> Com_1(evalSimpleSingle2bb3in(A, B, C, D)) :|: D >= A + 1 evalSimpleSingle2bb2in(A, B, C, D) -> Com_1(evalSimpleSingle2returnin(A, B, C, D)) :|: A >= D evalSimpleSingle2bb3in(A, B, C, D) -> Com_1(evalSimpleSingle2bb4in(A + 1, B + 1, C, D)) :|: TRUE evalSimpleSingle2returnin(A, B, C, D) -> Com_1(evalSimpleSingle2stop(A, B, C, D)) :|: TRUE The start-symbols are:[evalSimpleSingle2start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 6*Ar_3 + 6*Ar_2 + 24) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(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) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalSimpleSingle2start) = 2 Pol(evalSimpleSingle2entryin) = 2 Pol(evalSimpleSingle2bb4in) = 2 Pol(evalSimpleSingle2bbin) = 2 Pol(evalSimpleSingle2returnin) = 1 Pol(evalSimpleSingle2bb1in) = 2 Pol(evalSimpleSingle2bb2in) = 2 Pol(evalSimpleSingle2bb3in) = 2 Pol(evalSimpleSingle2stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalSimpleSingle2start) = V_3 + 1 Pol(evalSimpleSingle2entryin) = V_3 + 1 Pol(evalSimpleSingle2bb4in) = -V_2 + V_3 + 1 Pol(evalSimpleSingle2bbin) = -V_2 + V_3 + 1 Pol(evalSimpleSingle2returnin) = -V_2 + V_3 Pol(evalSimpleSingle2bb1in) = -V_2 + V_3 Pol(evalSimpleSingle2bb2in) = -V_2 + V_3 + 1 Pol(evalSimpleSingle2bb3in) = -V_2 + V_3 Pol(evalSimpleSingle2stop) = -V_2 + V_3 Pol(koat_start) = V_3 + 1 orients all transitions weakly and the transition evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalSimpleSingle2start) = V_4 + 1 Pol(evalSimpleSingle2entryin) = V_4 + 1 Pol(evalSimpleSingle2bb4in) = -V_1 + V_4 + 1 Pol(evalSimpleSingle2bbin) = -V_1 + V_4 + 1 Pol(evalSimpleSingle2returnin) = -V_1 + V_4 Pol(evalSimpleSingle2bb1in) = -V_1 + V_4 Pol(evalSimpleSingle2bb2in) = -V_1 + V_4 + 1 Pol(evalSimpleSingle2bb3in) = -V_1 + V_4 Pol(evalSimpleSingle2stop) = -V_1 + V_4 Pol(koat_start) = V_4 + 1 orients all transitions weakly and the transition evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: Ar_3 + 1, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: ?, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(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) evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalSimpleSingle2entryin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(0, 0, Ar_2, Ar_3)) (Comp: Ar_3 + Ar_2 + 3, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= E + 1 ] (Comp: Ar_3 + Ar_2 + 3, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3)) [ E >= 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 2*Ar_3 + 2*Ar_2 + 6, Cost: 1) evalSimpleSingle2bbin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 ] (Comp: Ar_2 + 1, Cost: 1) evalSimpleSingle2bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: Ar_3 + 1, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSingle2bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_3 ] (Comp: Ar_3 + 1, Cost: 1) evalSimpleSingle2bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2bb4in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalSimpleSingle2returnin(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalSimpleSingle2start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 6*Ar_3 + 6*Ar_2 + 24 Time: 0.113 sec (SMT: 0.087 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalSimpleSingle2start 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 2: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ 0>=1+free ], cost: 1 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ free_1>=1 ], cost: 1 4: evalSimpleSingle2bb4in -> evalSimpleSingle2returnin : [], cost: 1 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 9: evalSimpleSingle2bb2in -> evalSimpleSingle2returnin : [ A>=D ], cost: 1 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 11: evalSimpleSingle2returnin -> evalSimpleSingle2stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalSimpleSingle2start 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 2: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ 0>=1+free ], cost: 1 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [ free_1>=1 ], cost: 1 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 Simplified all rules, resulting in: Start location: evalSimpleSingle2start 0: evalSimpleSingle2start -> evalSimpleSingle2entryin : [], cost: 1 1: evalSimpleSingle2entryin -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 1 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [], cost: 1 5: evalSimpleSingle2bbin -> evalSimpleSingle2bb1in : [ C>=1+B ], cost: 1 6: evalSimpleSingle2bbin -> evalSimpleSingle2bb2in : [ B>=C ], cost: 1 7: evalSimpleSingle2bb1in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 8: evalSimpleSingle2bb2in -> evalSimpleSingle2bb3in : [ D>=1+A ], cost: 1 10: evalSimpleSingle2bb3in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalSimpleSingle2start 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 3: evalSimpleSingle2bb4in -> evalSimpleSingle2bbin : [], cost: 1 13: evalSimpleSingle2bbin -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 2 15: evalSimpleSingle2bbin -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 3 Eliminated locations (on tree-shaped paths): Start location: evalSimpleSingle2start 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 16: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 3 17: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 4 Accelerating simple loops of location 2. Accelerating the following rules: 16: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ C>=1+B ], cost: 3 17: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=1+A, B'=1+B, [ B>=C && D>=1+A ], cost: 4 Accelerated rule 16 with metering function C-B, yielding the new rule 18. Accelerated rule 17 with metering function D-A, yielding the new rule 19. Removing the simple loops: 16 17. Accelerated all simple loops using metering functions (where possible): Start location: evalSimpleSingle2start 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 18: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=C+A-B, B'=C, [ C>=1+B ], cost: 3*C-3*B 19: evalSimpleSingle2bb4in -> evalSimpleSingle2bb4in : A'=D, B'=D-A+B, [ B>=C && D>=1+A ], cost: 4*D-4*A Chained accelerated rules (with incoming rules): Start location: evalSimpleSingle2start 12: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=0, B'=0, [], cost: 2 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D Removed unreachable locations (and leaf rules with constant cost): Start location: evalSimpleSingle2start 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalSimpleSingle2start 20: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=C, B'=C, [ C>=1 ], cost: 2+3*C 21: evalSimpleSingle2start -> evalSimpleSingle2bb4in : A'=D, B'=D, [ 0>=C && D>=1 ], cost: 2+4*D Computing asymptotic complexity for rule 20 Solved the limit problem by the following transformations: Created initial limit problem: C (+/+!), 2+3*C (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n} resulting limit problem: [solved] Solution: C / n Resulting cost 2+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 2+3*n Rule cost: 2+3*C Rule guard: [ C>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)