MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { plus(x, s(s(y))) -> s(plus(s(x), y))
  , plus(s(s(x)), y) -> s(plus(x, s(y)))
  , plus(s(0()), y) -> s(y)
  , plus(0(), y) -> y
  , ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
  , ack(s(x), 0()) -> ack(x, s(0()))
  , ack(0(), y) -> s(y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { plus^#(x, s(s(y))) -> c_1(plus^#(s(x), y))
  , plus^#(s(s(x)), y) -> c_2(plus^#(x, s(y)))
  , plus^#(s(0()), y) -> c_3()
  , plus^#(0(), y) -> c_4()
  , ack^#(s(x), s(y)) ->
    c_5(ack^#(x, plus(y, ack(s(x), y))),
        plus^#(y, ack(s(x), y)),
        ack^#(s(x), y))
  , ack^#(s(x), 0()) -> c_6(ack^#(x, s(0())))
  , ack^#(0(), y) -> c_7() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { plus^#(x, s(s(y))) -> c_1(plus^#(s(x), y))
  , plus^#(s(s(x)), y) -> c_2(plus^#(x, s(y)))
  , plus^#(s(0()), y) -> c_3()
  , plus^#(0(), y) -> c_4()
  , ack^#(s(x), s(y)) ->
    c_5(ack^#(x, plus(y, ack(s(x), y))),
        plus^#(y, ack(s(x), y)),
        ack^#(s(x), y))
  , ack^#(s(x), 0()) -> c_6(ack^#(x, s(0())))
  , ack^#(0(), y) -> c_7() }
Weak Trs:
  { plus(x, s(s(y))) -> s(plus(s(x), y))
  , plus(s(s(x)), y) -> s(plus(x, s(y)))
  , plus(s(0()), y) -> s(y)
  , plus(0(), y) -> y
  , ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
  , ack(s(x), 0()) -> ack(x, s(0()))
  , ack(0(), y) -> s(y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {3,4,7} by applications of
Pre({3,4,7}) = {1,2,5,6}. Here rules are labeled as follows:

  DPs:
    { 1: plus^#(x, s(s(y))) -> c_1(plus^#(s(x), y))
    , 2: plus^#(s(s(x)), y) -> c_2(plus^#(x, s(y)))
    , 3: plus^#(s(0()), y) -> c_3()
    , 4: plus^#(0(), y) -> c_4()
    , 5: ack^#(s(x), s(y)) ->
         c_5(ack^#(x, plus(y, ack(s(x), y))),
             plus^#(y, ack(s(x), y)),
             ack^#(s(x), y))
    , 6: ack^#(s(x), 0()) -> c_6(ack^#(x, s(0())))
    , 7: ack^#(0(), y) -> c_7() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { plus^#(x, s(s(y))) -> c_1(plus^#(s(x), y))
  , plus^#(s(s(x)), y) -> c_2(plus^#(x, s(y)))
  , ack^#(s(x), s(y)) ->
    c_5(ack^#(x, plus(y, ack(s(x), y))),
        plus^#(y, ack(s(x), y)),
        ack^#(s(x), y))
  , ack^#(s(x), 0()) -> c_6(ack^#(x, s(0()))) }
Weak DPs:
  { plus^#(s(0()), y) -> c_3()
  , plus^#(0(), y) -> c_4()
  , ack^#(0(), y) -> c_7() }
Weak Trs:
  { plus(x, s(s(y))) -> s(plus(s(x), y))
  , plus(s(s(x)), y) -> s(plus(x, s(y)))
  , plus(s(0()), y) -> s(y)
  , plus(0(), y) -> y
  , ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
  , ack(s(x), 0()) -> ack(x, s(0()))
  , ack(0(), y) -> s(y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ plus^#(s(0()), y) -> c_3()
, plus^#(0(), y) -> c_4()
, ack^#(0(), y) -> c_7() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { plus^#(x, s(s(y))) -> c_1(plus^#(s(x), y))
  , plus^#(s(s(x)), y) -> c_2(plus^#(x, s(y)))
  , ack^#(s(x), s(y)) ->
    c_5(ack^#(x, plus(y, ack(s(x), y))),
        plus^#(y, ack(s(x), y)),
        ack^#(s(x), y))
  , ack^#(s(x), 0()) -> c_6(ack^#(x, s(0()))) }
Weak Trs:
  { plus(x, s(s(y))) -> s(plus(s(x), y))
  , plus(s(s(x)), y) -> s(plus(x, s(y)))
  , plus(s(0()), y) -> s(y)
  , plus(0(), y) -> y
  , ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
  , ack(s(x), 0()) -> ack(x, s(0()))
  , ack(0(), y) -> s(y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      Following exception was raised:
        stack overflow
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..