/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 121 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 505 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: start(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: 0 >= A && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F start(A, B, C, D, E, F) -> Com_1(lbl101(A, 2, C, D, 1, F)) :|: A >= 1 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F start(A, B, C, D, E, F) -> Com_1(lbl101(A, 2, C, D, -(1), F)) :|: A >= 1 && B >= C && B <= C && D >= A && D <= A && E >= F && E <= F lbl101(A, B, C, D, E, F) -> Com_1(stop(A, B, C, D, E, F)) :|: E + A >= 0 && A >= 1 && A >= E && B >= A + 1 && B <= A + 1 && D >= A && D <= A lbl101(A, B, C, D, E, F) -> Com_1(lbl101(A, 1 + B, C, D, 1 + E, F)) :|: A >= B && E + B >= 1 && A + 1 >= B && B >= 2 && B >= E + 1 && D >= A && D <= A lbl101(A, B, C, D, E, F) -> Com_1(lbl101(A, 1 + B, C, D, E - 1, F)) :|: A >= B && E + B >= 1 && A + 1 >= B && B >= 2 && B >= E + 1 && D >= A && D <= A start0(A, B, C, D, E, F) -> Com_1(start(A, C, C, A, F, F)) :|: TRUE The start-symbols are:[start0_6] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 4*Ar_0 + 17) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_4 /\ Ar_1 = Ar_0 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 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) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_4 /\ Ar_1 = Ar_0 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(start) = 1 Pol(stop) = 0 Pol(lbl101) = 1 Pol(start0) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_4 /\ Ar_1 = Ar_0 + 1 /\ Ar_3 = Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_4 /\ Ar_1 = Ar_0 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: ?, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(lbl101) = V_1 - V_2 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-4) = Ar_4 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ]", 0-5) = Ar_5 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-0) = Ar_0 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-1) = Ar_2 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-2) = Ar_2 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-3) = Ar_0 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-4) = Ar_5 S("start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5))", 0-5) = Ar_5 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-0) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-1) = ? S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-2) = Ar_2 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-3) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-4) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-5) = Ar_5 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-0) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-1) = ? S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-2) = Ar_2 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-3) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-4) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\\ Ar_4 + Ar_1 >= 1 /\\ Ar_0 + 1 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_1 >= Ar_4 + 1 /\\ Ar_3 = Ar_0 ]", 0-5) = Ar_5 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-0) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-1) = ? S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-2) = Ar_2 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-3) = Ar_0 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-4) = Ar_0 + 2 S("lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\\ Ar_0 >= 1 /\\ Ar_0 >= Ar_4 /\\ Ar_1 = Ar_0 + 1 /\\ Ar_3 = Ar_0 ]", 0-5) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-1) = 2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = 1 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-1) = 2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = 1 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-1) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-3) = Ar_0 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-4) = Ar_5 S("start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\\ Ar_1 = Ar_2 /\\ Ar_3 = Ar_0 /\\ Ar_4 = Ar_5 ]", 0-5) = Ar_5 orients the transitions lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] weakly and the transitions lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 >= Ar_0 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, 1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, 2, Ar_2, Ar_3, -1, Ar_5)) [ Ar_0 >= 1 /\ Ar_1 = Ar_2 /\ Ar_3 = Ar_0 /\ Ar_4 = Ar_5 ] (Comp: 1, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(stop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ Ar_4 + Ar_0 >= 0 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_4 /\ Ar_1 = Ar_0 + 1 /\ Ar_3 = Ar_0 ] (Comp: 2*Ar_0 + 6, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 + 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: 2*Ar_0 + 6, Cost: 1) lbl101(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(lbl101(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4 - 1, Ar_5)) [ Ar_0 >= Ar_1 /\ Ar_4 + Ar_1 >= 1 /\ Ar_0 + 1 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_1 >= Ar_4 + 1 /\ Ar_3 = Ar_0 ] (Comp: 1, Cost: 1) start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start(Ar_0, Ar_2, Ar_2, Ar_0, Ar_5, Ar_5)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5) -> Com_1(start0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 4*Ar_0 + 17 Time: 0.140 sec (SMT: 0.115 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: start0 0: start -> stop : [ 0>=A && B==C && D==A && E==F ], cost: 1 1: start -> lbl101 : B'=2, E'=1, [ A>=1 && B==C && D==A && E==F ], cost: 1 2: start -> lbl101 : B'=2, E'=-1, [ A>=1 && B==C && D==A && E==F ], cost: 1 3: lbl101 -> stop : [ A+E>=0 && A>=1 && A>=E && B==1+A && D==A ], cost: 1 4: lbl101 -> lbl101 : B'=1+B, E'=1+E, [ A>=B && E+B>=1 && 1+A>=B && B>=2 && B>=1+E && D==A ], cost: 1 5: lbl101 -> lbl101 : B'=1+B, E'=-1+E, [ A>=B && E+B>=1 && 1+A>=B && B>=2 && B>=1+E && D==A ], cost: 1 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Removed unreachable and leaf rules: Start location: start0 1: start -> lbl101 : B'=2, E'=1, [ A>=1 && B==C && D==A && E==F ], cost: 1 2: start -> lbl101 : B'=2, E'=-1, [ A>=1 && B==C && D==A && E==F ], cost: 1 4: lbl101 -> lbl101 : B'=1+B, E'=1+E, [ A>=B && E+B>=1 && 1+A>=B && B>=2 && B>=1+E && D==A ], cost: 1 5: lbl101 -> lbl101 : B'=1+B, E'=-1+E, [ A>=B && E+B>=1 && 1+A>=B && B>=2 && B>=1+E && D==A ], cost: 1 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Simplified all rules, resulting in: Start location: start0 1: start -> lbl101 : B'=2, E'=1, [ A>=1 && B==C && D==A && E==F ], cost: 1 2: start -> lbl101 : B'=2, E'=-1, [ A>=1 && B==C && D==A && E==F ], cost: 1 4: lbl101 -> lbl101 : B'=1+B, E'=1+E, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1 5: lbl101 -> lbl101 : B'=1+B, E'=-1+E, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 4: lbl101 -> lbl101 : B'=1+B, E'=1+E, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1 5: lbl101 -> lbl101 : B'=1+B, E'=-1+E, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1 Accelerated rule 4 with metering function 1+A-B, yielding the new rule 7. Accelerated rule 5 with metering function 1+A-B, yielding the new rule 8. Removing the simple loops: 4 5. Accelerated all simple loops using metering functions (where possible): Start location: start0 1: start -> lbl101 : B'=2, E'=1, [ A>=1 && B==C && D==A && E==F ], cost: 1 2: start -> lbl101 : B'=2, E'=-1, [ A>=1 && B==C && D==A && E==F ], cost: 1 7: lbl101 -> lbl101 : B'=1+A, E'=1+A+E-B, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1+A-B 8: lbl101 -> lbl101 : B'=1+A, E'=-1-A+E+B, [ A>=B && E+B>=1 && B>=2 && B>=1+E && D==A ], cost: 1+A-B 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: start0 1: start -> lbl101 : B'=2, E'=1, [ A>=1 && B==C && D==A && E==F ], cost: 1 2: start -> lbl101 : B'=2, E'=-1, [ A>=1 && B==C && D==A && E==F ], cost: 1 9: start -> lbl101 : B'=1+A, E'=A, [ B==C && D==A && E==F && A>=2 ], cost: A 10: start -> lbl101 : B'=1+A, E'=-2+A, [ B==C && D==A && E==F && A>=2 ], cost: A 11: start -> lbl101 : B'=1+A, E'=2-A, [ B==C && D==A && E==F && A>=2 ], cost: A 12: start -> lbl101 : B'=1+A, E'=-A, [ B==C && D==A && E==F && A>=2 ], cost: A 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: start0 9: start -> lbl101 : B'=1+A, E'=A, [ B==C && D==A && E==F && A>=2 ], cost: A 10: start -> lbl101 : B'=1+A, E'=-2+A, [ B==C && D==A && E==F && A>=2 ], cost: A 11: start -> lbl101 : B'=1+A, E'=2-A, [ B==C && D==A && E==F && A>=2 ], cost: A 12: start -> lbl101 : B'=1+A, E'=-A, [ B==C && D==A && E==F && A>=2 ], cost: A 6: start0 -> start : B'=C, D'=A, E'=F, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: start0 13: start0 -> lbl101 : B'=1+A, D'=A, E'=A, [ A>=2 ], cost: 1+A 14: start0 -> lbl101 : B'=1+A, D'=A, E'=-2+A, [ A>=2 ], cost: 1+A 15: start0 -> lbl101 : B'=1+A, D'=A, E'=2-A, [ A>=2 ], cost: 1+A 16: start0 -> lbl101 : B'=1+A, D'=A, E'=-A, [ A>=2 ], cost: 1+A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: start0 16: start0 -> lbl101 : B'=1+A, D'=A, E'=-A, [ A>=2 ], cost: 1+A Computing asymptotic complexity for rule 16 Solved the limit problem by the following transformations: Created initial limit problem: -1+A (+/+!), 1+A (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n} resulting limit problem: [solved] Solution: A / n Resulting cost 1+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: 1+n Rule cost: 1+A Rule guard: [ A>=2 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)