/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 21 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 113 ms]
(4) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
l0(A, B) -> Com_1(l1(0, B)) :|: TRUE
l1(A, B) -> Com_1(l1(A + 1, B - 1)) :|: B >= 1
l1(A, B) -> Com_1(l2(A, B)) :|: 0 >= B

The start-symbols are:[l0_2]


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(1) Koat Proof (FINISHED)
YES(?, Ar_1 + 2)



Initial complexity problem:

1:	T:

		(Comp: ?, Cost: 1)    l0(Ar_0, Ar_1) -> Com_1(l1(0, Ar_1))

		(Comp: ?, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ]

		(Comp: ?, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(l0(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)    l0(Ar_0, Ar_1) -> Com_1(l1(0, Ar_1))

		(Comp: ?, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ]

		(Comp: ?, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(l0) = 1

	Pol(l1) = 1

	Pol(l2) = 0

	Pol(koat_start) = 1

orients all transitions weakly and the transition

	l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_1 ]

strictly and produces the following problem:

3:	T:

		(Comp: 1, Cost: 1)    l0(Ar_0, Ar_1) -> Com_1(l1(0, Ar_1))

		(Comp: ?, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ]

		(Comp: 1, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_1 ]

		(Comp: 1, Cost: 0)    koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



A polynomial rank function with

	Pol(l0) = V_2

	Pol(l1) = V_2

	Pol(l2) = V_2

	Pol(koat_start) = V_2

orients all transitions weakly and the transition

	l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ]

strictly and produces the following problem:

4:	T:

		(Comp: 1, Cost: 1)       l0(Ar_0, Ar_1) -> Com_1(l1(0, Ar_1))

		(Comp: Ar_1, Cost: 1)    l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ]

		(Comp: 1, Cost: 1)       l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_1 ]

		(Comp: 1, Cost: 0)       koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ]

	start location:	koat_start

	leaf cost:	0



Complexity upper bound Ar_1 + 2



Time: 0.032 sec (SMT: 0.029 sec)


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(2)
BOUNDS(1, n^1)

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(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1

      2: l1 -> l2 : [ 0>=B ], cost: 1



Checking for constant complexity:

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

      0: l0 -> l1 : A'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: l1 -> l1 : A'=1+A, B'=-1+B, [ B>=1 ], cost: 1



   Accelerated rule 1 with metering function B, yielding the new rule 3.

   Removing the simple loops: 1.



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

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      3: l1 -> l1 : A'=A+B, B'=0, [ B>=1 ], cost: B



Chained accelerated rules (with incoming rules):

   Start location: l0

      0: l0 -> l1 : A'=0, [], cost: 1

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



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

   Start location: l0

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: l0

      4: l0 -> l1 : A'=B, B'=0, [ B>=1 ], cost: 1+B



Computing asymptotic complexity for rule 4

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      1+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 1+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: 1+n

   Rule cost:   1+B

   Rule guard:  [ B>=1 ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)