/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 16 ms]
(2) BOUNDS(1, n^1)
(3) Loat Proof [FINISHED, 322 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 + 1)) :|: A >= B + 1
eval(A, B) -> Com_1(eval(A, B + 1)) :|: B >= A + 1 && A >= B + 1
eval(A, B) -> Com_1(eval(A + 1, B)) :|: A >= B + 1 && B >= A
eval(A, B) -> Com_1(eval(A + 1, B)) :|: B >= A + 1 && B >= A
start(A, B) -> Com_1(eval(A, B)) :|: TRUE

The start-symbols are:[start_2]


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(1) Koat Proof (FINISHED)
YES(?, 2*Ar_0 + 2*Ar_1 + 2)



Initial complexity problem:

1:	T:

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

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

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

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

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



Testing for reachability in the complexity graph removes the following transitions from problem 1:

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

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

We thus obtain the following problem:

2:	T:

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

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

		(Comp: ?, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval(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 2 produces the following problem:

3:	T:

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

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

		(Comp: 1, Cost: 1)    start(Ar_0, Ar_1) -> Com_1(eval(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(eval) = V_1 - V_2

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(eval(Ar_0, Ar_1))", 0-0) = Ar_0

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

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

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

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

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

orients the transitions

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

weakly and the transition

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

strictly and produces the following problem:

4:	T:

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

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

		(Comp: 1, Cost: 1)              start(Ar_0, Ar_1) -> Com_1(eval(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(eval) = -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(eval(Ar_0, Ar_1))", 0-0) = Ar_0

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

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

	S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1

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

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

orients the transitions

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

weakly and the transition

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

strictly and produces the following problem:

5:	T:

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

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

		(Comp: 1, Cost: 1)                  start(Ar_0, Ar_1) -> Com_1(eval(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 2*Ar_0 + 2*Ar_1 + 2



Time: 0.047 sec (SMT: 0.039 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: start

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

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

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

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

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



Checking for constant complexity:

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

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



Removed rules with unsatisfiable guard:

   Start location: start

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

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

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



Simplified all rules, resulting in:

   Start location: start

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

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

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



   Accelerated rule 0 with metering function A-B, yielding the new rule 5.

   Accelerated rule 3 with metering function -A+B, 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 : B'=A, [ A>=1+B ], cost: A-B

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

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



Chained accelerated rules (with incoming rules):

   Start location: start

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

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

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



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

   Start location: start

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

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



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: start

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

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



Computing asymptotic complexity for rule 7

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==0,B==-n}

      resulting limit problem:

      [solved]



   Solution:

      A / 0

      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+A-B

   Rule guard:  [ A>=1+B ]



WORST_CASE(Omega(n^1),?)


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(4)
BOUNDS(n^1, INF)