/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(1, 1 + Arg_1) + nat(Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 150 ms]
(2) BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))
(3) Loat Proof [FINISHED, 567 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval(A, B) -> Com_1(eval(A - B, B)) :|: A >= B + 1 && A >= 1 && B >= 1
eval(A, B) -> Com_1(eval(A - B, B)) :|: B >= A + 1 && A >= 1 && B >= 1 && A >= B + 1
eval(A, B) -> Com_1(eval(A, B - A)) :|: A >= B + 1 && A >= 1 && B >= 1 && B >= A
eval(A, B) -> Com_1(eval(A, B - A)) :|: B >= A + 1 && A >= 1 && B >= 1 && B >= A
start(A, B) -> Com_1(eval(A, B)) :|: TRUE

The start-symbols are:[start_2]


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(1) Koat2 Proof (FINISHED)
YES( ?, 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)})



Initial Complexity Problem:

  Start:  start  

  Program_Vars:  Arg_0, Arg_1  

  Temp_Vars:    

  Locations:  eval, start  

  Transitions:

  eval(Arg_0,Arg_1) -> eval(Arg_0-Arg_1,Arg_1):|:Arg_1+1 <= Arg_0 && 1 <= Arg_0 && 1 <= Arg_1

  eval(Arg_0,Arg_1) -> eval(Arg_0,Arg_1-Arg_0):|:Arg_0+1 <= Arg_1 && 1 <= Arg_0 && 1 <= Arg_1 && Arg_0 <= Arg_1

  start(Arg_0,Arg_1) -> eval(Arg_0,Arg_1):|:  



Timebounds: 

  Overall timebound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}

  0: eval->eval: max([0, Arg_0]) {O(n)}

  3: eval->eval: max([0, Arg_1]) {O(n)}

  4: start->eval: 1 {O(1)}



Costbounds:

  Overall costbound: 1+max([0, Arg_1])+max([0, Arg_0]) {O(n)}

  0: eval->eval: max([0, Arg_0]) {O(n)}

  3: eval->eval: max([0, Arg_1]) {O(n)}

  4: start->eval: 1 {O(1)}



Sizebounds:

`Lower:

  0: eval->eval, Arg_0: 1 {O(1)}

  0: eval->eval, Arg_1: 1 {O(1)}

  3: eval->eval, Arg_0: 1 {O(1)}

  3: eval->eval, Arg_1: 1 {O(1)}

  4: start->eval, Arg_0: Arg_0 {O(n)}

  4: start->eval, Arg_1: Arg_1 {O(n)}

`Upper:

  0: eval->eval, Arg_0: Arg_0 {O(n)}

  0: eval->eval, Arg_1: Arg_1 {O(n)}

  3: eval->eval, Arg_0: Arg_0 {O(n)}

  3: eval->eval, Arg_1: Arg_1 {O(n)}

  4: start->eval, Arg_0: Arg_0 {O(n)}

  4: start->eval, Arg_1: Arg_1 {O(n)}


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(2)
BOUNDS(1, max(1, 1 + Arg_1) + nat(Arg_0))

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(3) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: start

      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1

      1: eval -> eval : A'=A-B, [ B>=1+A && A>=1 && B>=1 && A>=1+B ], cost: 1

      2: eval -> eval : B'=-A+B, [ A>=1+B && A>=1 && B>=1 && B>=A ], cost: 1

      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 && B>=1 && B>=A ], cost: 1

      4: start -> eval : [], cost: 1



Removed rules with unsatisfiable guard:

   Start location: start

      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1

      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 && B>=1 && B>=A ], cost: 1

      4: start -> eval : [], cost: 1



Simplified all rules, resulting in:

   Start location: start

      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1

      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1

      4: start -> eval : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 0.

   Accelerating the following rules:

      0: eval -> eval : A'=A-B, [ A>=1+B && A>=1 && B>=1 ], cost: 1

      3: eval -> eval : B'=-A+B, [ B>=1+A && A>=1 ], cost: 1



   Accelerated rule 0 with backward acceleration, yielding the new rule 5.

   Accelerated rule 3 with backward acceleration, yielding the new rule 6.

   Removing the simple loops: 0 3.



Accelerated all simple loops using metering functions (where possible):

   Start location: start

      5: eval -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: k

      6: eval -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: k_1

      4: start -> eval : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start

      4: start -> eval : [], cost: 1

      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k

      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1



Removed unreachable locations (and leaf rules with constant cost):

   Start location: start

      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k

      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start

      7: start -> eval : A'=-k*B+A, [ A>=1+B && A>=1 && B>=1 && k>0 && -(-1+k)*B+A>=1+B && -(-1+k)*B+A>=1 ], cost: 1+k

      8: start -> eval : B'=B-A*k_1, [ B>=1+A && A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1



Computing asymptotic complexity for rule 7

   Simplified the guard:

      7: start -> eval : A'=-k*B+A, [ B>=1 && k>0 && -(-1+k)*B+A>=1+B ], cost: 1+k

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -(-1+k)*B+A-B (+/+!), k (+/+!), 1+k (+), B (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {k==n,A==2*n,B==1}

      resulting limit problem:

      [solved]



   Solved the limit problem by the following transformations:

      Created initial limit problem:

      -(-1+k)*B+A-B (+/+!), k (+/+!), 1+k (+), B (+/+!) [not solved]



      applying transformation rule (C) using substitution {B==1}

      resulting limit problem:

      1 (+/+!), k (+/+!), 1+k (+), -k+A (+/+!) [not solved]



      applying transformation rule (B), deleting 1 (+/+!)

      resulting limit problem:

      k (+/+!), 1+k (+), -k+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {k==n,A==2*n}

      resulting limit problem:

      [solved]



   Solution:

      k / n

      A / 2*n

      B / 1

   Resulting cost 1+n has complexity: Poly(n^1)



Found new complexity Poly(n^1).



Computing asymptotic complexity for rule 8

   Simplified the guard:

      8: start -> eval : B'=B-A*k_1, [ A>=1 && k_1>0 && -A*(-1+k_1)+B>=1+A ], cost: 1+k_1

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+k_1 (+), -A*(-1+k_1)-A+B (+/+!), A (+/+!), k_1 (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==1,k_1==n,B==2*n}

      resulting limit problem:

      [solved]



   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+k_1 (+), -A*(-1+k_1)-A+B (+/+!), A (+/+!), k_1 (+/+!) [not solved]



      applying transformation rule (C) using substitution {A==1}

      resulting limit problem:

      1 (+/+!), 1+k_1 (+), -k_1+B (+/+!), k_1 (+/+!) [not solved]



      applying transformation rule (B), deleting 1 (+/+!)

      resulting limit problem:

      1+k_1 (+), -k_1+B (+/+!), k_1 (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {k_1==n,B==2*n}

      resulting limit problem:

      [solved]



   Solution:

      A / 1

      k_1 / n

      B / 2*n

   Resulting cost 1+n has 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: 1+n

   Rule cost:   1+k

   Rule guard:  [ B>=1 && k>0 && -(-1+k)*B+A>=1+B ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)