/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(X) -> a(X)
            f(a(g(X))) -> b(X)
            f(f(X)) -> f(a(b(f(X))))
        - Signature:
            {b/1,f/1} / {a/1,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(a) = [1] x1 + [13]
            p(b) = [1] x1 + [1] 
            p(f) = [1] x1 + [1] 
            p(g) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          f(a(g(X))) = [1] X + [14]
                     > [1] X + [1] 
                     = b(X)        
          
          
          Following rules are (at-least) weakly oriented:
             b(X) =  [1] X + [1]  
                  >= [1] X + [13] 
                  =  a(X)         
          
          f(f(X)) =  [1] X + [2]  
                  >= [1] X + [16] 
                  =  f(a(b(f(X))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(X) -> a(X)
            f(f(X)) -> f(a(b(f(X))))
        - Weak TRS:
            f(a(g(X))) -> b(X)
        - Signature:
            {b/1,f/1} / {a/1,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(a) = [1] x1 + [0]
            p(b) = [1] x1 + [9]
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [9]
          
          Following rules are strictly oriented:
          b(X) = [1] X + [9]
               > [1] X + [0]
               = a(X)       
          
          
          Following rules are (at-least) weakly oriented:
          f(a(g(X))) =  [1] X + [9]  
                     >= [1] X + [9]  
                     =  b(X)         
          
             f(f(X)) =  [1] X + [0]  
                     >= [1] X + [9]  
                     =  f(a(b(f(X))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(X)) -> f(a(b(f(X))))
        - Weak TRS:
            b(X) -> a(X)
            f(a(g(X))) -> b(X)
        - Signature:
            {b/1,f/1} / {a/1,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 0] x1 + [1]
                 [0 0]      [0]
          p(b) = [1 0] x1 + [7]
                 [0 0]      [4]
          p(f) = [1 2] x1 + [1]
                 [0 0]      [5]
          p(g) = [1 2] x1 + [6]
                 [0 0]      [0]
        
        Following rules are strictly oriented:
        f(f(X)) = [1 2] X + [12]
                  [0 0]     [5] 
                > [1 2] X + [10]
                  [0 0]     [5] 
                = f(a(b(f(X)))) 
        
        
        Following rules are (at-least) weakly oriented:
              b(X) =  [1 0] X + [7]
                      [0 0]     [4]
                   >= [1 0] X + [1]
                      [0 0]     [0]
                   =  a(X)         
        
        f(a(g(X))) =  [1 2] X + [8]
                      [0 0]     [5]
                   >= [1 0] X + [7]
                      [0 0]     [4]
                   =  b(X)         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            b(X) -> a(X)
            f(a(g(X))) -> b(X)
            f(f(X)) -> f(a(b(f(X))))
        - Signature:
            {b/1,f/1} / {a/1,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))