/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^2), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, n^2).

(0) CpxIntTrs
(1) Koat Proof [FINISHED, 30 ms]
(2) BOUNDS(1, n^2)
(3) Loat Proof [FINISHED, 219 ms]
(4) BOUNDS(n^2, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
eval1(A, B) -> Com_1(eval2(A + 1, 1)) :|: A >= 0
eval2(A, B) -> Com_1(eval2(A, B + 1)) :|: A >= 0 && B >= 1 && A >= B
eval2(A, B) -> Com_1(eval1(A - 2, B)) :|: A >= 0 && B >= 1 && B >= A + 1
start(A, B) -> Com_1(eval1(A, B)) :|: TRUE

The start-symbols are:[start_2]


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(1) Koat Proof (FINISHED)
YES(?, 128*Ar_0 + 6*Ar_0^2 + 233)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(start(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: ?, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

		(Comp: ?, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(eval1) = 2*V_1 + 4

	Pol(eval2) = 2*V_1 + 1

	Pol(start) = 2*V_1 + 4

	Pol(koat_start) = 2*V_1 + 4

orients all transitions weakly and the transitions

	eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ]

	eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]

strictly and produces the following problem:

3:	T:

		(Comp: 2*Ar_0 + 4, Cost: 1)    eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]

		(Comp: ?, Cost: 1)             eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

		(Comp: 2*Ar_0 + 4, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: 1, Cost: 1)             start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)             koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(eval2) = V_1 - V_2 + 1

and size complexities

	S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0

	S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1

	S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-0) = Ar_0

	S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-1) = Ar_1

	S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = 3*Ar_0 + 54

	S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = ?

	S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-0) = 3*Ar_0 + 54

	S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-1) = ?

	S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-0) = 3*Ar_0 + 54

	S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-1) = 1

orients the transitions

	eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

weakly and the transition

	eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 2*Ar_0 + 4, Cost: 1)                   eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]

		(Comp: 6*Ar_0^2 + 124*Ar_0 + 224, Cost: 1)    eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ]

		(Comp: 2*Ar_0 + 4, Cost: 1)                   eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ]

		(Comp: 1, Cost: 1)                            start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))

		(Comp: 1, Cost: 0)                            koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound 128*Ar_0 + 6*Ar_0^2 + 233



Time: 0.056 sec (SMT: 0.048 sec)


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(2)
BOUNDS(1, n^2)

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: start

      0: eval1 -> eval2 : A'=1+A, B'=1, [ A>=0 ], cost: 1

      1: eval2 -> eval2 : B'=1+B, [ A>=0 && B>=1 && A>=B ], cost: 1

      2: eval2 -> eval1 : A'=-2+A, [ A>=0 && B>=1 && B>=1+A ], cost: 1

      3: start -> eval1 : [], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      3: start -> eval1 : [], cost: 1



Simplified all rules, resulting in:

   Start location: start

      0: eval1 -> eval2 : A'=1+A, B'=1, [ A>=0 ], cost: 1

      1: eval2 -> eval2 : B'=1+B, [ B>=1 && A>=B ], cost: 1

      2: eval2 -> eval1 : A'=-2+A, [ A>=0 && B>=1 && B>=1+A ], cost: 1

      3: start -> eval1 : [], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: eval2 -> eval2 : B'=1+B, [ B>=1 && A>=B ], cost: 1



   Accelerated rule 1 with metering function 1+A-B, yielding the new rule 4.

   Removing the simple loops: 1.



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

   Start location: start

      0: eval1 -> eval2 : A'=1+A, B'=1, [ A>=0 ], cost: 1

      2: eval2 -> eval1 : A'=-2+A, [ A>=0 && B>=1 && B>=1+A ], cost: 1

      4: eval2 -> eval2 : B'=1+A, [ B>=1 && A>=B ], cost: 1+A-B

      3: start -> eval1 : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start

      0: eval1 -> eval2 : A'=1+A, B'=1, [ A>=0 ], cost: 1

      5: eval1 -> eval2 : A'=1+A, B'=2+A, [ A>=0 ], cost: 2+A

      2: eval2 -> eval1 : A'=-2+A, [ A>=0 && B>=1 && B>=1+A ], cost: 1

      3: start -> eval1 : [], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: start

      6: eval1 -> eval1 : A'=-1+A, B'=2+A, [ A>=0 ], cost: 3+A

      3: start -> eval1 : [], cost: 1



Accelerating simple loops of location 0.

   Accelerating the following rules:

      6: eval1 -> eval1 : A'=-1+A, B'=2+A, [ A>=0 ], cost: 3+A



   Accelerated rule 6 with metering function 1+A, yielding the new rule 7.

   Removing the simple loops: 6.



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

   Start location: start

      7: eval1 -> eval1 : A'=-1, B'=2, [ A>=0 ], cost: 7/2+A*(1+A)+7/2*A-1/2*(1+A)^2

      3: start -> eval1 : [], cost: 1



Chained accelerated rules (with incoming rules):

   Start location: start

      3: start -> eval1 : [], cost: 1

      8: start -> eval1 : A'=-1, B'=2, [ A>=0 ], cost: 9/2+A*(1+A)+7/2*A-1/2*(1+A)^2



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

   Start location: start

      8: start -> eval1 : A'=-1, B'=2, [ A>=0 ], cost: 9/2+A*(1+A)+7/2*A-1/2*(1+A)^2



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start

      8: start -> eval1 : A'=-1, B'=2, [ A>=0 ], cost: 9/2+A*(1+A)+7/2*A-1/2*(1+A)^2



Computing asymptotic complexity for rule 8

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      4+7/2*A+1/2*A^2 (+), 1+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

   Resulting cost 4+7/2*n+1/2*n^2 has complexity: Poly(n^2)



Found new complexity Poly(n^2).



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Poly(n^2)

   Cpx degree:  2

   Solved cost: 4+7/2*n+1/2*n^2

   Rule cost:   9/2+A*(1+A)+7/2*A-1/2*(1+A)^2

   Rule guard:  [ A>=0 ]



WORST_CASE(Omega(n^2),?)


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(4)
BOUNDS(n^2, INF)