/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).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, 36 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 425 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v_n, v_x_0) -> Com_1(eval_start_bb0_in(v_n, v_x_0)) :|: TRUE eval_start_bb0_in(v_n, v_x_0) -> Com_1(eval_start_0(v_n, v_x_0)) :|: TRUE eval_start_0(v_n, v_x_0) -> Com_1(eval_start_1(v_n, v_x_0)) :|: TRUE eval_start_1(v_n, v_x_0) -> Com_1(eval_start_2(v_n, v_x_0)) :|: TRUE eval_start_2(v_n, v_x_0) -> Com_1(eval_start_3(v_n, v_x_0)) :|: TRUE eval_start_3(v_n, v_x_0) -> Com_1(eval_start_4(v_n, v_x_0)) :|: TRUE eval_start_4(v_n, v_x_0) -> Com_1(eval_start_bb1_in(v_n, 0)) :|: TRUE eval_start_bb1_in(v_n, v_x_0) -> Com_1(eval_start_bb2_in(v_n, v_x_0)) :|: v_x_0 < v_n eval_start_bb1_in(v_n, v_x_0) -> Com_1(eval_start_bb3_in(v_n, v_x_0)) :|: v_x_0 >= v_n eval_start_bb2_in(v_n, v_x_0) -> Com_1(eval_start_5(v_n, v_x_0)) :|: TRUE eval_start_5(v_n, v_x_0) -> Com_1(eval_start_6(v_n, v_x_0)) :|: TRUE eval_start_6(v_n, v_x_0) -> Com_1(eval_start_bb1_in(v_n, v_x_0 + 1)) :|: TRUE eval_start_bb3_in(v_n, v_x_0) -> Com_1(eval_start_stop(v_n, v_x_0)) :|: TRUE The start-symbols are:[eval_start_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 4*Ar_1 + 15) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalstartstart(Ar_0, Ar_1) -> Com_1(evalstartbb0in(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstartbb0in(Ar_0, Ar_1) -> Com_1(evalstart0(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart0(Ar_0, Ar_1) -> Com_1(evalstart1(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart1(Ar_0, Ar_1) -> Com_1(evalstart2(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart2(Ar_0, Ar_1) -> Com_1(evalstart3(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1) -> Com_1(evalstart4(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1) -> Com_1(evalstartbb1in(0, Ar_1)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1) -> Com_1(evalstart5(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1) -> Com_1(evalstart6(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalstartstart(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) evalstartstart(Ar_0, Ar_1) -> Com_1(evalstartbb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1) -> Com_1(evalstart0(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1) -> Com_1(evalstart1(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1) -> Com_1(evalstart2(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1) -> Com_1(evalstart3(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1) -> Com_1(evalstart4(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1) -> Com_1(evalstartbb1in(0, Ar_1)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1) -> Com_1(evalstart5(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1) -> Com_1(evalstart6(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalstartstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = 2 Pol(evalstartbb0in) = 2 Pol(evalstart0) = 2 Pol(evalstart1) = 2 Pol(evalstart2) = 2 Pol(evalstart3) = 2 Pol(evalstart4) = 2 Pol(evalstartbb1in) = 2 Pol(evalstartbb2in) = 2 Pol(evalstartbb3in) = 1 Pol(evalstart5) = 2 Pol(evalstart6) = 2 Pol(evalstartstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1) -> Com_1(evalstartbb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1) -> Com_1(evalstart0(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1) -> Com_1(evalstart1(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1) -> Com_1(evalstart2(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1) -> Com_1(evalstart3(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1) -> Com_1(evalstart4(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1) -> Com_1(evalstartbb1in(0, Ar_1)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1) -> Com_1(evalstart5(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1) -> Com_1(evalstart6(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalstartstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = V_2 + 1 Pol(evalstartbb0in) = V_2 + 1 Pol(evalstart0) = V_2 + 1 Pol(evalstart1) = V_2 + 1 Pol(evalstart2) = V_2 + 1 Pol(evalstart3) = V_2 + 1 Pol(evalstart4) = V_2 + 1 Pol(evalstartbb1in) = -V_1 + V_2 + 1 Pol(evalstartbb2in) = -V_1 + V_2 Pol(evalstartbb3in) = -V_1 + V_2 Pol(evalstart5) = -V_1 + V_2 Pol(evalstart6) = -V_1 + V_2 Pol(evalstartstop) = -V_1 + V_2 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1) -> Com_1(evalstartbb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1) -> Com_1(evalstart0(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1) -> Com_1(evalstart1(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1) -> Com_1(evalstart2(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1) -> Com_1(evalstart3(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1) -> Com_1(evalstart4(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1) -> Com_1(evalstartbb1in(0, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1) -> Com_1(evalstart5(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1) -> Com_1(evalstart6(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalstartstart(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) evalstartstart(Ar_0, Ar_1) -> Com_1(evalstartbb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1) -> Com_1(evalstart0(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1) -> Com_1(evalstart1(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1) -> Com_1(evalstart2(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1) -> Com_1(evalstart3(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1) -> Com_1(evalstart4(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1) -> Com_1(evalstartbb1in(0, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb2in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1) -> Com_1(evalstartbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 ] (Comp: Ar_1 + 1, Cost: 1) evalstartbb2in(Ar_0, Ar_1) -> Com_1(evalstart5(Ar_0, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evalstart5(Ar_0, Ar_1) -> Com_1(evalstart6(Ar_0, Ar_1)) (Comp: Ar_1 + 1, Cost: 1) evalstart6(Ar_0, Ar_1) -> Com_1(evalstartbb1in(Ar_0 + 1, Ar_1)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1) -> Com_1(evalstartstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalstartstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 4*Ar_1 + 15 Time: 0.051 sec (SMT: 0.039 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstartbb1in : A'=0, [], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1 8: evalstartbb1in -> evalstartbb3in : [ A>=B ], cost: 1 9: evalstartbb2in -> evalstart5 : [], cost: 1 10: evalstart5 -> evalstart6 : [], cost: 1 11: evalstart6 -> evalstartbb1in : A'=1+A, [], cost: 1 12: evalstartbb3in -> evalstartstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalstartstart -> evalstartbb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstartbb1in : A'=0, [], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1 9: evalstartbb2in -> evalstart5 : [], cost: 1 10: evalstart5 -> evalstart6 : [], cost: 1 11: evalstart6 -> evalstartbb1in : A'=1+A, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 18: evalstartstart -> evalstartbb1in : A'=0, [], cost: 7 21: evalstartbb1in -> evalstartbb1in : A'=1+A, [ B>=1+A ], cost: 4 Accelerating simple loops of location 7. Accelerating the following rules: 21: evalstartbb1in -> evalstartbb1in : A'=1+A, [ B>=1+A ], cost: 4 Accelerated rule 21 with metering function -A+B, yielding the new rule 22. Removing the simple loops: 21. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 18: evalstartstart -> evalstartbb1in : A'=0, [], cost: 7 22: evalstartbb1in -> evalstartbb1in : A'=B, [ B>=1+A ], cost: -4*A+4*B Chained accelerated rules (with incoming rules): Start location: evalstartstart 18: evalstartstart -> evalstartbb1in : A'=0, [], cost: 7 23: evalstartstart -> evalstartbb1in : A'=B, [ B>=1 ], cost: 7+4*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 23: evalstartstart -> evalstartbb1in : A'=B, [ B>=1 ], cost: 7+4*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 23: evalstartstart -> evalstartbb1in : A'=B, [ B>=1 ], cost: 7+4*B Computing asymptotic complexity for rule 23 Solved the limit problem by the following transformations: Created initial limit problem: 7+4*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 7+4*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: 7+4*n Rule cost: 7+4*B Rule guard: [ B>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)