4.97/2.34 WORST_CASE(NON_POLY, ?) 4.97/2.34 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 4.97/2.34 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.97/2.34 4.97/2.34 4.97/2.34 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(EXP, EXP). 4.97/2.34 4.97/2.34 (0) CpxIntTrs 4.97/2.34 (1) Koat Proof [FINISHED, 149 ms] 4.97/2.34 (2) BOUNDS(1, EXP) 4.97/2.34 (3) Loat Proof [FINISHED, 619 ms] 4.97/2.34 (4) BOUNDS(EXP, INF) 4.97/2.34 4.97/2.34 4.97/2.34 ---------------------------------------- 4.97/2.34 4.97/2.34 (0) 4.97/2.34 Obligation: 4.97/2.34 Complexity Int TRS consisting of the following rules: 4.97/2.34 f(A, B, C) -> Com_1(g(A, 1, 1)) :|: TRUE 4.97/2.34 g(A, B, C) -> Com_1(g(A - 1, 2 * B, C)) :|: A > 0 4.97/2.34 g(A, B, C) -> Com_1(h(A, B, C)) :|: A <= 0 4.97/2.34 h(A, B, C) -> Com_1(h(A, B - 1, 2 * C)) :|: B > 0 4.97/2.34 h(A, B, C) -> Com_1(i(A, B, C)) :|: B <= 0 4.97/2.34 i(A, B, C) -> Com_1(i(A, B, C - 1)) :|: C > 0 4.97/2.34 4.97/2.34 The start-symbols are:[f_3] 4.97/2.34 4.97/2.34 4.97/2.34 ---------------------------------------- 4.97/2.34 4.97/2.34 (1) Koat Proof (FINISHED) 4.97/2.34 YES(?, pow(2, ar_2) * 2 + pow(2, pow(2, ar_2) * 2) * 2 + ar_2 + 5) 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Initial complexity problem: 4.97/2.34 4.97/2.34 1: T: 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Repeatedly propagating knowledge in problem 1 produces the following problem: 4.97/2.34 4.97/2.34 2: T: 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 A polynomial rank function with 4.97/2.34 4.97/2.34 Pol(f) = 2 4.97/2.34 4.97/2.34 Pol(g) = 2 4.97/2.34 4.97/2.34 Pol(h) = 1 4.97/2.34 4.97/2.34 Pol(i) = 0 4.97/2.34 4.97/2.34 Pol(koat_start) = 2 4.97/2.34 4.97/2.34 orients all transitions weakly and the transitions 4.97/2.34 4.97/2.34 h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 strictly and produces the following problem: 4.97/2.34 4.97/2.34 3: T: 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 A polynomial rank function with 4.97/2.34 4.97/2.34 Pol(f) = V_3 4.97/2.34 4.97/2.34 Pol(g) = V_3 4.97/2.34 4.97/2.34 Pol(h) = V_3 4.97/2.34 4.97/2.34 Pol(i) = V_3 4.97/2.34 4.97/2.34 Pol(koat_start) = V_3 4.97/2.34 4.97/2.34 orients all transitions weakly and the transition 4.97/2.34 4.97/2.34 g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 strictly and produces the following problem: 4.97/2.34 4.97/2.34 4: T: 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ar_2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 A polynomial rank function with 4.97/2.34 4.97/2.34 Pol(h) = V_1 4.97/2.34 4.97/2.34 and size complexities 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-1) = ? 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ? 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-1) = ? 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = 1 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-1) = 1 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-0) = 1 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-1) = 1 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-2) = ar_2 4.97/2.34 4.97/2.34 orients the transitions 4.97/2.34 4.97/2.34 h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 weakly and the transition 4.97/2.34 4.97/2.34 h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 strictly and produces the following problem: 4.97/2.34 4.97/2.34 5: T: 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ar_2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: pow(2, ar_2) * 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: ?, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 A polynomial rank function with 4.97/2.34 4.97/2.34 Pol(i) = V_2 4.97/2.34 4.97/2.34 and size complexities 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 4.97/2.34 4.97/2.34 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-1) = pow(2, pow(2, ar_2) * 2) 4.97/2.34 4.97/2.34 S("i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = pow(2, pow(2, ar_2) * 2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-1) = pow(2, pow(2, ar_2) * 2) 4.97/2.34 4.97/2.34 S("h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = 1 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-0) = pow(2, ar_2) 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-1) = 1 4.97/2.34 4.97/2.34 S("g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ]", 0-2) = ar_2 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-0) = 1 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-1) = 1 4.97/2.34 4.97/2.34 S("f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2))", 0-2) = ar_2 4.97/2.34 4.97/2.34 orients the transitions 4.97/2.34 4.97/2.34 i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 weakly and the transition 4.97/2.34 4.97/2.34 i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 strictly and produces the following problem: 4.97/2.34 4.97/2.34 6: T: 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 1) f(ar_0, ar_1, ar_2) -> Com_1(g(1, 1, ar_2)) 4.97/2.34 4.97/2.34 (Comp: ar_2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(g(2*ar_0, ar_1, ar_2 - 1)) [ ar_2 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) g(ar_0, ar_1, ar_2) -> Com_1(h(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ] 4.97/2.34 4.97/2.34 (Comp: pow(2, ar_2) * 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(h(ar_0 - 1, 2*ar_1, ar_2)) [ ar_0 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 2, Cost: 1) h(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.97/2.34 4.97/2.34 (Comp: pow(2, pow(2, ar_2) * 2) * 2, Cost: 1) i(ar_0, ar_1, ar_2) -> Com_1(i(ar_0, ar_1 - 1, ar_2)) [ ar_1 >= 1 ] 4.97/2.34 4.97/2.34 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(f(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.97/2.34 4.97/2.34 start location: koat_start 4.97/2.34 4.97/2.34 leaf cost: 0 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Complexity upper bound pow(2, ar_2) * 2 + pow(2, pow(2, ar_2) * 2) * 2 + ar_2 + 5 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Time: 0.193 sec (SMT: 0.171 sec) 4.97/2.34 4.97/2.34 4.97/2.34 ---------------------------------------- 4.97/2.34 4.97/2.34 (2) 4.97/2.34 BOUNDS(1, EXP) 4.97/2.34 4.97/2.34 ---------------------------------------- 4.97/2.34 4.97/2.34 (3) Loat Proof (FINISHED) 4.97/2.34 4.97/2.34 4.97/2.34 ### Pre-processing the ITS problem ### 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Initial linear ITS problem 4.97/2.34 4.97/2.34 Start location: f 4.97/2.34 4.97/2.34 0: f -> g : A'=1, B'=1, [], cost: 1 4.97/2.34 4.97/2.34 1: g -> g : A'=2*A, C'=-1+C, [ C>=1 ], cost: 1 4.97/2.34 4.97/2.34 2: g -> h : [ 0>=C ], cost: 1 4.97/2.34 4.97/2.34 3: h -> h : A'=-1+A, B'=2*B, [ A>=1 ], cost: 1 4.97/2.34 4.97/2.34 4: h -> i : [ 0>=A ], cost: 1 4.97/2.34 4.97/2.34 5: i -> i : B'=-1+B, [ B>=1 ], cost: 1 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 ### Simplification by acceleration and chaining ### 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Accelerating simple loops of location 1. 4.97/2.34 4.97/2.34 Accelerating the following rules: 4.97/2.34 4.97/2.34 1: g -> g : A'=2*A, C'=-1+C, [ C>=1 ], cost: 1 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Accelerated rule 1 with metering function C, yielding the new rule 6. 4.97/2.34 4.97/2.34 Removing the simple loops: 1. 4.97/2.34 4.97/2.34 4.97/2.34 4.97/2.34 Accelerating simple loops of location 2. 4.97/2.34 4.97/2.34 Accelerating the following rules: 4.97/2.34 4.97/2.34 3: h -> h : A'=-1+A, B'=2*B, [ A>=1 ], cost: 1 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Accelerated rule 3 with metering function A, yielding the new rule 7. 4.97/2.35 4.97/2.35 Removing the simple loops: 3. 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Accelerating simple loops of location 3. 4.97/2.35 4.97/2.35 Accelerating the following rules: 4.97/2.35 4.97/2.35 5: i -> i : B'=-1+B, [ B>=1 ], cost: 1 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Accelerated rule 5 with metering function B, yielding the new rule 8. 4.97/2.35 4.97/2.35 Removing the simple loops: 5. 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Accelerated all simple loops using metering functions (where possible): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 0: f -> g : A'=1, B'=1, [], cost: 1 4.97/2.35 4.97/2.35 2: g -> h : [ 0>=C ], cost: 1 4.97/2.35 4.97/2.35 6: g -> g : A'=2^C*A, C'=0, [ C>=1 ], cost: C 4.97/2.35 4.97/2.35 4: h -> i : [ 0>=A ], cost: 1 4.97/2.35 4.97/2.35 7: h -> h : A'=0, B'=2^A*B, [ A>=1 ], cost: A 4.97/2.35 4.97/2.35 8: i -> i : B'=0, [ B>=1 ], cost: B 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Chained accelerated rules (with incoming rules): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 0: f -> g : A'=1, B'=1, [], cost: 1 4.97/2.35 4.97/2.35 9: f -> g : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 1+C 4.97/2.35 4.97/2.35 2: g -> h : [ 0>=C ], cost: 1 4.97/2.35 4.97/2.35 10: g -> h : A'=0, B'=2^A*B, [ 0>=C && A>=1 ], cost: 1+A 4.97/2.35 4.97/2.35 4: h -> i : [ 0>=A ], cost: 1 4.97/2.35 4.97/2.35 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Removed unreachable locations (and leaf rules with constant cost): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 0: f -> g : A'=1, B'=1, [], cost: 1 4.97/2.35 4.97/2.35 9: f -> g : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 1+C 4.97/2.35 4.97/2.35 2: g -> h : [ 0>=C ], cost: 1 4.97/2.35 4.97/2.35 10: g -> h : A'=0, B'=2^A*B, [ 0>=C && A>=1 ], cost: 1+A 4.97/2.35 4.97/2.35 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Eliminated locations (on tree-shaped paths): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 12: f -> h : A'=1, B'=1, [ 0>=C ], cost: 2 4.97/2.35 4.97/2.35 13: f -> h : A'=0, B'=2, [ 0>=C ], cost: 3 4.97/2.35 4.97/2.35 14: f -> h : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 2+C 4.97/2.35 4.97/2.35 15: f -> h : A'=0, B'=2^(2^C), C'=0, [ C>=1 && 2^C>=1 ], cost: 2+C+2^C 4.97/2.35 4.97/2.35 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Eliminated locations (on tree-shaped paths): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 16: f -> i : A'=0, B'=0, [ 0>=C ], cost: 6 4.97/2.35 4.97/2.35 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 4.97/2.35 4.97/2.35 18: f -> i : A'=0, B'=0, C'=0, [ C>=1 && 2^C>=1 && 2^(2^C)>=1 ], cost: 3+2^(2^C)+C+2^C 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Applied pruning (of leafs and parallel rules): 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 4.97/2.35 4.97/2.35 18: f -> i : A'=0, B'=0, C'=0, [ C>=1 && 2^C>=1 && 2^(2^C)>=1 ], cost: 3+2^(2^C)+C+2^C 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 ### Computing asymptotic complexity ### 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Fully simplified ITS problem 4.97/2.35 4.97/2.35 Start location: f 4.97/2.35 4.97/2.35 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 4.97/2.35 4.97/2.35 18: f -> i : A'=0, B'=0, C'=0, [ C>=1 && 2^C>=1 && 2^(2^C)>=1 ], cost: 3+2^(2^C)+C+2^C 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Computing asymptotic complexity for rule 17 4.97/2.35 4.97/2.35 Could not solve the limit problem. 4.97/2.35 4.97/2.35 Resulting cost 0 has complexity: Unknown 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Computing asymptotic complexity for rule 18 4.97/2.35 4.97/2.35 Solved the limit problem by the following transformations: 4.97/2.35 4.97/2.35 Created initial limit problem: 4.97/2.35 4.97/2.35 3+2^(2^C)+C+2^C (+), 2^(2^C) (+/+!), C (+/+!), 2^C (+/+!) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (E), replacing 2^C (+/+!) by 1 (+/+!) and C (+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 1 (+/+!), 3+2^(2^C)+C+2^C (+), 2^(2^C) (+/+!), C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 1 (+/+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 3+2^(2^C)+C+2^C (+), 2^(2^C) (+/+!), C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 reducing general power, replacing 3+2^(2^C)+C+2^C (+) by 1 (+/+!) and 3+C+2*2^C (+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 1 (+/+!), 2^(2^C) (+/+!), C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 1 (+/+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 2^(2^C) (+/+!), C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 reducing general power, replacing 2^(2^C) (+/+!) by 1 (+/+!) and 2^C (+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 1 (+/+!), C (+), 2^C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 1 (+/+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 2^C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (E), replacing 2^C (+) by 1 (+/+!) and C (+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 1 (+/+!), C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 1 (+/+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 3+C+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (A), replacing 3+C+2*2^C (+) by C (+) and 3+2*2^C (+) using + limit vector (+,+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 3+2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (A), replacing 3+2*2^C (+) by 2*2^C (+) and 3 (+!) using + limit vector (+,+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 3 (+!), C (+), 2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 3 (+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 2*2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (A), replacing 2*2^C (+) by 2^C (+) and 2 (+!) using + limit vector (+,+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 2 (+!), 2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 2 (+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+), 2^C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (E), replacing 2^C (+) by 1 (+/+!) and C (+) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 1 (+/+!), C (+) [not solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 applying transformation rule (B), deleting 1 (+/+!) 4.97/2.35 4.97/2.35 resulting limit problem: 4.97/2.35 4.97/2.35 C (+) [solved] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Solution: 4.97/2.35 4.97/2.35 C / n 4.97/2.35 4.97/2.35 Resulting cost 3+2^n+2^(2^n)+n has complexity: Exp 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Found new complexity Exp. 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 Obtained the following overall complexity (w.r.t. the length of the input n): 4.97/2.35 4.97/2.35 Complexity: Exp 4.97/2.35 4.97/2.35 Cpx degree: Exp 4.97/2.35 4.97/2.35 Solved cost: 3+2^n+2^(2^n)+n 4.97/2.35 4.97/2.35 Rule cost: 3+2^(2^C)+C+2^C 4.97/2.35 4.97/2.35 Rule guard: [ C>=1 && 2^C>=1 && 2^(2^C)>=1 ] 4.97/2.35 4.97/2.35 4.97/2.35 4.97/2.35 WORST_CASE(EXP,?) 4.97/2.35 4.97/2.35 4.97/2.35 ---------------------------------------- 4.97/2.35 4.97/2.35 (4) 4.97/2.35 BOUNDS(EXP, INF) 4.97/2.37 EOF