/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 6 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 187 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalSimpleSinglestart(A, B) -> Com_1(evalSimpleSingleentryin(A, B)) :|: TRUE evalSimpleSingleentryin(A, B) -> Com_1(evalSimpleSinglebb3in(0, B)) :|: TRUE evalSimpleSinglebb3in(A, B) -> Com_1(evalSimpleSinglebbin(A, B)) :|: B >= A + 1 evalSimpleSinglebb3in(A, B) -> Com_1(evalSimpleSinglereturnin(A, B)) :|: A >= B evalSimpleSinglebbin(A, B) -> Com_1(evalSimpleSinglebb3in(A + 1, B)) :|: TRUE evalSimpleSinglereturnin(A, B) -> Com_1(evalSimpleSinglestop(A, B)) :|: TRUE The start-symbols are:[evalSimpleSinglestart_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_1 + 6) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalSimpleSinglestart(Ar_0, Ar_1) -> Com_1(evalSimpleSingleentryin(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSingleentryin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(0, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalSimpleSinglebbin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestart(Ar_0, Ar_1)) [ 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) evalSimpleSinglestart(Ar_0, Ar_1) -> Com_1(evalSimpleSingleentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalSimpleSingleentryin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(0, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalSimpleSinglebbin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalSimpleSinglestart) = 2 Pol(evalSimpleSingleentryin) = 2 Pol(evalSimpleSinglebb3in) = 2 Pol(evalSimpleSinglebbin) = 2 Pol(evalSimpleSinglereturnin) = 1 Pol(evalSimpleSinglestop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalSimpleSinglestart(Ar_0, Ar_1) -> Com_1(evalSimpleSingleentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalSimpleSingleentryin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(0, Ar_1)) (Comp: ?, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalSimpleSinglebbin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalSimpleSinglestart) = V_2 Pol(evalSimpleSingleentryin) = V_2 Pol(evalSimpleSinglebb3in) = -V_1 + V_2 Pol(evalSimpleSinglebbin) = -V_1 + V_2 - 1 Pol(evalSimpleSinglereturnin) = -V_1 + V_2 Pol(evalSimpleSinglestop) = -V_1 + V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalSimpleSinglestart(Ar_0, Ar_1) -> Com_1(evalSimpleSingleentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalSimpleSingleentryin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(0, Ar_1)) (Comp: Ar_1, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalSimpleSinglebbin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestart(Ar_0, Ar_1)) [ 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) evalSimpleSinglestart(Ar_0, Ar_1) -> Com_1(evalSimpleSingleentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalSimpleSingleentryin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(0, Ar_1)) (Comp: Ar_1, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebbin(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalSimpleSinglebb3in(Ar_0, Ar_1) -> Com_1(evalSimpleSinglereturnin(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) evalSimpleSinglebbin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglebb3in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalSimpleSinglereturnin(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalSimpleSinglestart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 6 Time: 0.036 sec (SMT: 0.029 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalSimpleSinglestart 0: evalSimpleSinglestart -> evalSimpleSingleentryin : [], cost: 1 1: evalSimpleSingleentryin -> evalSimpleSinglebb3in : A'=0, [], cost: 1 2: evalSimpleSinglebb3in -> evalSimpleSinglebbin : [ B>=1+A ], cost: 1 3: evalSimpleSinglebb3in -> evalSimpleSinglereturnin : [ A>=B ], cost: 1 4: evalSimpleSinglebbin -> evalSimpleSinglebb3in : A'=1+A, [], cost: 1 5: evalSimpleSinglereturnin -> evalSimpleSinglestop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalSimpleSinglestart -> evalSimpleSingleentryin : [], cost: 1 Removed unreachable and leaf rules: Start location: evalSimpleSinglestart 0: evalSimpleSinglestart -> evalSimpleSingleentryin : [], cost: 1 1: evalSimpleSingleentryin -> evalSimpleSinglebb3in : A'=0, [], cost: 1 2: evalSimpleSinglebb3in -> evalSimpleSinglebbin : [ B>=1+A ], cost: 1 4: evalSimpleSinglebbin -> evalSimpleSinglebb3in : A'=1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalSimpleSinglestart 6: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=0, [], cost: 2 7: evalSimpleSinglebb3in -> evalSimpleSinglebb3in : A'=1+A, [ B>=1+A ], cost: 2 Accelerating simple loops of location 2. Accelerating the following rules: 7: evalSimpleSinglebb3in -> evalSimpleSinglebb3in : A'=1+A, [ B>=1+A ], cost: 2 Accelerated rule 7 with metering function -A+B, yielding the new rule 8. Removing the simple loops: 7. Accelerated all simple loops using metering functions (where possible): Start location: evalSimpleSinglestart 6: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=0, [], cost: 2 8: evalSimpleSinglebb3in -> evalSimpleSinglebb3in : A'=B, [ B>=1+A ], cost: -2*A+2*B Chained accelerated rules (with incoming rules): Start location: evalSimpleSinglestart 6: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=0, [], cost: 2 9: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=B, [ B>=1 ], cost: 2+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalSimpleSinglestart 9: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=B, [ B>=1 ], cost: 2+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalSimpleSinglestart 9: evalSimpleSinglestart -> evalSimpleSinglebb3in : A'=B, [ B>=1 ], cost: 2+2*B Computing asymptotic complexity for rule 9 Solved the limit problem by the following transformations: Created initial limit problem: 2+2*B (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 2+2*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+2*n Rule cost: 2+2*B Rule guard: [ B>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)