Problem AG01 innermost 4.32

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.32

stdout:

MAYBE

Problem:
 f(x,c(y)) -> f(x,s(f(y,y)))
 f(s(x),s(y)) -> f(x,s(c(s(y))))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.32

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.32

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(x, c(y)) -> f(x, s(f(y, y)))
     , f(s(x), s(y)) -> f(x, s(c(s(y))))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: NA
             ------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {2}, Uargs(c) = {}, Uargs(s) = {1}, Uargs(f^#) = {2},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 2] x1 + [1 2 1] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                c(x1) = [1 1 0] x1 + [1]
                        [0 1 2]      [3]
                        [0 0 0]      [1]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [1]
                        [0 0 1]      [2]
                f^#(x1, x2) = [3 3 3] x1 + [1 2 0] x2 + [0]
                              [3 3 3]      [3 3 3]      [0]
                              [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {2}, Uargs(c) = {}, Uargs(s) = {1}, Uargs(f^#) = {2},
                 Uargs(c_0) = {1}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 1] x1 + [1 1 0] x2 + [1]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [1]
                c(x1) = [1 1 0] x1 + [0]
                        [0 1 2]      [3]
                        [0 0 0]      [0]
                s(x1) = [1 0 1] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                f^#(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [3 3 3]      [3 3 3]      [0]
                              [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}.
           
           * Path {1}->{2}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
                  , f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))
                  , f(x, c(y)) -> f(x, s(f(y, y)))
                  , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}.
           
           * Path {1}->{2}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
                  , f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))
                  , f(x, c(y)) -> f(x, s(f(y, y)))
                  , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.32

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.32

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(x, c(y)) -> f(x, s(f(y, y)))
     , f(s(x), s(y)) -> f(x, s(c(s(y))))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}.
           
           * Path {1}->{2}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
                  , f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))
                  , f(x, c(y)) -> f(x, s(f(y, y)))
                  , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
              , 2: f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}.
           
           * Path {1}->{2}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(x, c(y)) -> f(x, s(f(y, y)))
                , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(x, c(y)) -> c_0(f^#(x, s(f(y, y))))
                  , f^#(s(x), s(y)) -> c_1(f^#(x, s(c(s(y)))))
                  , f(x, c(y)) -> f(x, s(f(y, y)))
                  , f(s(x), s(y)) -> f(x, s(c(s(y))))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.