WORST_CASE(?, O(EXP)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: ?, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(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) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: ?, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f) = 1 Pol(g) = 1 Pol(g1) = 1 Pol(h) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transition g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: ?, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: 1, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f) = 2*V_1 Pol(g) = 2*V_1 Pol(g1) = 2*V_1 + 1 Pol(h) = 2*V_1 Pol(koat_start) = 2*V_1 orients all transitions weakly and the transition g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: 2*Ar_0, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: ?, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: 1, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 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) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: 2*Ar_0, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: 2*Ar_0, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: 1, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(h) = V_2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ]", 0-0) = Ar_0 S("h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ]", 0-1) = pow(2, 2*Ar_0) S("h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ]", 0-2) = pow(2, 2*Ar_0) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ]", 0-0) = Ar_0 S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ]", 0-1) = pow(2, 2*Ar_0) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ]", 0-2) = pow(2, 2*Ar_0) S("g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2))", 0-0) = Ar_0 S("g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2))", 0-1) = pow(2, 2*Ar_0) S("g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2))", 0-2) = pow(2, 2*Ar_0) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ]", 0-0) = Ar_0 S("g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ]", 0-1) = pow(2, 2*Ar_0) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ]", 0-2) = pow(2, 2*Ar_0) S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0))", 0-0) = Ar_0 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0))", 0-1) = 1 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0))", 0-2) = 0 orients the transitions h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] weakly and the transition h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, 1, 0)) (Comp: 2*Ar_0, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g1(Ar_0 - 1, Ar_1, Ar_1)) [ Ar_0 > 0 ] (Comp: 2*Ar_0, Cost: 1) g1(Ar_0, Ar_1, Ar_2) -> Com_1(g(Ar_0, Ar_2 + Ar_1, Ar_2)) (Comp: 1, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ Ar_0 <= 0 ] (Comp: pow(2, 2*Ar_0), Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 > 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound pow(2, 2*Ar_0) + 4*Ar_0 + 2 Time: 1.024 sec (SMT: 0.996 sec)