/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(EXP, EXP). (0) CpxIntTrs (1) Koat Proof [FINISHED, 165 ms] (2) BOUNDS(1, EXP) (3) Loat Proof [FINISHED, 755 ms] (4) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f(A, B, C) -> Com_1(g(A, 1, 1)) :|: TRUE g(A, B, C) -> Com_1(g(A - 1, 2 * B, C)) :|: A > 0 g(A, B, C) -> Com_1(h(A, B, C)) :|: A <= 0 h(A, B, C) -> Com_1(h(A, B - 1, 2 * C)) :|: B > 0 h(A, B, C) -> Com_1(i(A, B, C)) :|: B <= 0 i(A, B, C) -> Com_1(i(A, B, C - 1)) :|: C > 0 The start-symbols are:[f_3] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, pow(2, Ar_2) * 2 + pow(2, pow(2, Ar_2) * 2) * 2 + Ar_2 + 5) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g(2*Ar_0, Ar_1, Ar_2 - 1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] (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(1, 1, Ar_2)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g(2*Ar_0, Ar_1, Ar_2 - 1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] (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 Pol(g) = 2 Pol(h) = 1 Pol(i) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2)) (Comp: ?, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(g(2*Ar_0, Ar_1, Ar_2 - 1)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] (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) = V_3 Pol(g) = V_3 Pol(h) = V_3 Pol(i) = V_3 Pol(koat_start) = V_3 orients all transitions weakly and the transition g(Ar_0, Ar_1, Ar_2) -> Com_1(g(2*Ar_0, Ar_1, Ar_2 - 1)) [ Ar_2 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2)) (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 ] (Comp: 2, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] (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_1 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("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) S("i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ]", 0-1) = ? 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 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) S("h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-1) = ? S("h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 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) S("h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ]", 0-1) = ? 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 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) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ]", 0-1) = 1 S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ]", 0-2) = Ar_2 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) 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 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 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-0) = 1 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-1) = 1 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-2) = Ar_2 orients the transitions h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] weakly and the transition h(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0 - 1, 2*Ar_1, Ar_2)) [ Ar_0 >= 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2)) (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 ] (Comp: 2, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (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 ] (Comp: 2, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] (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(i) = 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("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) 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) 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 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) 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) S("h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 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) 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) 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 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) S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ]", 0-1) = 1 S("g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ]", 0-2) = Ar_2 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) 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 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 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-0) = 1 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-1) = 1 S("f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2))", 0-2) = Ar_2 orients the transitions i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] weakly and the transition i(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1 - 1, Ar_2)) [ Ar_1 >= 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(g(1, 1, Ar_2)) (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 ] (Comp: 2, Cost: 1) g(Ar_0, Ar_1, Ar_2) -> Com_1(h(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (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 ] (Comp: 2, Cost: 1) h(Ar_0, Ar_1, Ar_2) -> Com_1(i(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 ] (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 ] (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, Ar_2) * 2 + pow(2, pow(2, Ar_2) * 2) * 2 + Ar_2 + 5 Time: 0.139 sec (SMT: 0.120 sec) ---------------------------------------- (2) BOUNDS(1, EXP) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f 0: f -> g : A'=1, B'=1, [], cost: 1 1: g -> g : A'=2*A, C'=-1+C, [ C>=1 ], cost: 1 2: g -> h : [ 0>=C ], cost: 1 3: h -> h : A'=-1+A, B'=2*B, [ A>=1 ], cost: 1 4: h -> i : [ 0>=A ], cost: 1 5: i -> i : B'=-1+B, [ B>=1 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: f -> g : A'=1, B'=1, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 1: g -> g : A'=2*A, C'=-1+C, [ C>=1 ], cost: 1 Accelerated rule 1 with metering function C, yielding the new rule 6. Removing the simple loops: 1. Accelerating simple loops of location 2. Accelerating the following rules: 3: h -> h : A'=-1+A, B'=2*B, [ A>=1 ], cost: 1 Accelerated rule 3 with metering function A, yielding the new rule 7. Removing the simple loops: 3. Accelerating simple loops of location 3. Accelerating the following rules: 5: i -> i : B'=-1+B, [ B>=1 ], cost: 1 Accelerated rule 5 with metering function B, yielding the new rule 8. Removing the simple loops: 5. Accelerated all simple loops using metering functions (where possible): Start location: f 0: f -> g : A'=1, B'=1, [], cost: 1 2: g -> h : [ 0>=C ], cost: 1 6: g -> g : A'=A*2^C, C'=0, [ C>=1 ], cost: C 4: h -> i : [ 0>=A ], cost: 1 7: h -> h : A'=0, B'=2^A*B, [ A>=1 ], cost: A 8: i -> i : B'=0, [ B>=1 ], cost: B Chained accelerated rules (with incoming rules): Start location: f 0: f -> g : A'=1, B'=1, [], cost: 1 9: f -> g : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 1+C 2: g -> h : [ 0>=C ], cost: 1 10: g -> h : A'=0, B'=2^A*B, [ 0>=C && A>=1 ], cost: 1+A 4: h -> i : [ 0>=A ], cost: 1 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B Removed unreachable locations (and leaf rules with constant cost): Start location: f 0: f -> g : A'=1, B'=1, [], cost: 1 9: f -> g : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 1+C 2: g -> h : [ 0>=C ], cost: 1 10: g -> h : A'=0, B'=2^A*B, [ 0>=C && A>=1 ], cost: 1+A 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B Eliminated locations (on tree-shaped paths): Start location: f 12: f -> h : A'=1, B'=1, [ 0>=C ], cost: 2 13: f -> h : A'=0, B'=2, [ 0>=C ], cost: 3 14: f -> h : A'=2^C, B'=1, C'=0, [ C>=1 ], cost: 2+C 15: f -> h : A'=0, B'=2^(2^C), C'=0, [ C>=1 && 2^C>=1 ], cost: 2+C+2^C 11: h -> i : B'=0, [ 0>=A && B>=1 ], cost: 1+B Eliminated locations (on tree-shaped paths): Start location: f 16: f -> i : A'=0, B'=0, [ 0>=C ], cost: 6 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 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 Applied pruning (of leafs and parallel rules): Start location: f 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 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 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f 17: f -> i : A'=2^C, B'=0, C'=0, [ C>=1 && 0>=2^C ], cost: 4+C 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 Computing asymptotic complexity for rule 17 Could not solve the limit problem. Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 18 Solved the limit problem by the following transformations: Created initial limit problem: 3+2^(2^C)+C+2^C (+), 2^(2^C) (+/+!), C (+/+!), 2^C (+/+!) [not solved] applying transformation rule (E), replacing 2^C (+/+!) by 1 (+/+!) and C (+) resulting limit problem: 3+2^(2^C)+C+2^C (+), 1 (+/+!), 2^(2^C) (+/+!), C (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 3+2^(2^C)+C+2^C (+), 2^(2^C) (+/+!), C (+) [not solved] reducing general power, replacing 3+2^(2^C)+C+2^C (+) by 1 (+/+!) and 3+C+2*2^C (+) resulting limit problem: 1 (+/+!), 2^(2^C) (+/+!), C (+), 3+C+2*2^C (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: 2^(2^C) (+/+!), C (+), 3+C+2*2^C (+) [not solved] reducing general power, replacing 2^(2^C) (+/+!) by 1 (+/+!) and 2^C (+) resulting limit problem: 1 (+/+!), C (+), 2^C (+), 3+C+2*2^C (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: C (+), 2^C (+), 3+C+2*2^C (+) [not solved] applying transformation rule (E), replacing 2^C (+) by 1 (+/+!) and C (+) resulting limit problem: 1 (+/+!), C (+), 3+C+2*2^C (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: C (+), 3+C+2*2^C (+) [not solved] applying transformation rule (A), replacing 3+C+2*2^C (+) by C (+) and 3+2*2^C (+) using + limit vector (+,+) resulting limit problem: C (+), 3+2*2^C (+) [not solved] applying transformation rule (A), replacing 3+2*2^C (+) by 2*2^C (+) and 3 (+!) using + limit vector (+,+!) resulting limit problem: C (+), 3 (+!), 2*2^C (+) [not solved] applying transformation rule (B), deleting 3 (+!) resulting limit problem: C (+), 2*2^C (+) [not solved] applying transformation rule (A), replacing 2*2^C (+) by 2^C (+) and 2 (+!) using + limit vector (+,+!) resulting limit problem: C (+), 2 (+!), 2^C (+) [not solved] applying transformation rule (B), deleting 2 (+!) resulting limit problem: C (+), 2^C (+) [not solved] applying transformation rule (E), replacing 2^C (+) by 1 (+/+!) and C (+) resulting limit problem: 1 (+/+!), C (+) [not solved] applying transformation rule (B), deleting 1 (+/+!) resulting limit problem: C (+) [solved] Solution: C / n Resulting cost 3+2^(2^n)+n+2^n has complexity: Exp Found new complexity Exp. Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Exp Cpx degree: Exp Solved cost: 3+2^(2^n)+n+2^n Rule cost: 3+2^(2^C)+C+2^C Rule guard: [ C>=1 && 2^C>=1 && 2^(2^C)>=1 ] WORST_CASE(EXP,?) ---------------------------------------- (4) BOUNDS(EXP, INF)