/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^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> d(x1)
            a(f(x1)) -> a(x1)
            b(a(x1)) -> a(b(x1))
            d(c(x1)) -> f(a(b(b(c(x1)))))
            d(f(x1)) -> f(a(b(x1)))
        - Signature:
            {a/1,b/1,d/1} / {c/1,f/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + 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 + [0]
            p(c) = [1] x1 + [0]
            p(d) = [1] x1 + [7]
            p(f) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          d(c(x1)) = [1] x1 + [7]     
                   > [1] x1 + [0]     
                   = f(a(b(b(c(x1)))))
          
          d(f(x1)) = [1] x1 + [7]     
                   > [1] x1 + [0]     
                   = f(a(b(x1)))      
          
          
          Following rules are (at-least) weakly oriented:
          a(b(x1)) =  [1] x1 + [0]
                   >= [1] x1 + [7]
                   =  d(x1)       
          
          a(f(x1)) =  [1] x1 + [0]
                   >= [1] x1 + [0]
                   =  a(x1)       
          
          b(a(x1)) =  [1] x1 + [0]
                   >= [1] x1 + [0]
                   =  a(b(x1))    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> d(x1)
            a(f(x1)) -> a(x1)
            b(a(x1)) -> a(b(x1))
        - Weak TRS:
            d(c(x1)) -> f(a(b(b(c(x1)))))
            d(f(x1)) -> f(a(b(x1)))
        - Signature:
            {a/1,b/1,d/1} / {c/1,f/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + 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 + [0]
            p(c) = [1] x1 + [2]
            p(d) = [1] x1 + [1]
            p(f) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          a(f(x1)) = [1] x1 + [1]
                   > [1] x1 + [0]
                   = a(x1)       
          
          
          Following rules are (at-least) weakly oriented:
          a(b(x1)) =  [1] x1 + [0]     
                   >= [1] x1 + [1]     
                   =  d(x1)            
          
          b(a(x1)) =  [1] x1 + [0]     
                   >= [1] x1 + [0]     
                   =  a(b(x1))         
          
          d(c(x1)) =  [1] x1 + [3]     
                   >= [1] x1 + [3]     
                   =  f(a(b(b(c(x1)))))
          
          d(f(x1)) =  [1] x1 + [2]     
                   >= [1] x1 + [1]     
                   =  f(a(b(x1)))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> d(x1)
            b(a(x1)) -> a(b(x1))
        - Weak TRS:
            a(f(x1)) -> a(x1)
            d(c(x1)) -> f(a(b(b(c(x1)))))
            d(f(x1)) -> f(a(b(x1)))
        - Signature:
            {a/1,b/1,d/1} / {c/1,f/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, 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 2 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 0]      [0]
                 [0 1 0] x1 + [1]
                 [0 0 0]      [0]
          p(b) = [1 1 0]      [0]
                 [0 1 0] x1 + [0]
                 [0 0 0]      [1]
          p(c) = [1 0 0]      [1]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [0]
          p(d) = [1 1 0]      [0]
                 [0 1 0] x1 + [1]
                 [0 0 0]      [0]
          p(f) = [1 0 0]      [0]
                 [0 1 1] x1 + [0]
                 [0 0 0]      [0]
        
        Following rules are strictly oriented:
        b(a(x1)) = [1 1 0]      [1]
                   [0 1 0] x1 + [1]
                   [0 0 0]      [1]
                 > [1 1 0]      [0]
                   [0 1 0] x1 + [1]
                   [0 0 0]      [0]
                 = a(b(x1))        
        
        
        Following rules are (at-least) weakly oriented:
        a(b(x1)) =  [1 1 0]      [0] 
                    [0 1 0] x1 + [1] 
                    [0 0 0]      [0] 
                 >= [1 1 0]      [0] 
                    [0 1 0] x1 + [1] 
                    [0 0 0]      [0] 
                 =  d(x1)            
        
        a(f(x1)) =  [1 0 0]      [0] 
                    [0 1 1] x1 + [1] 
                    [0 0 0]      [0] 
                 >= [1 0 0]      [0] 
                    [0 1 0] x1 + [1] 
                    [0 0 0]      [0] 
                 =  a(x1)            
        
        d(c(x1)) =  [1 0 0]      [1] 
                    [0 0 0] x1 + [1] 
                    [0 0 0]      [0] 
                 >= [1 0 0]      [1] 
                    [0 0 0] x1 + [1] 
                    [0 0 0]      [0] 
                 =  f(a(b(b(c(x1)))))
        
        d(f(x1)) =  [1 1 1]      [0] 
                    [0 1 1] x1 + [1] 
                    [0 0 0]      [0] 
                 >= [1 1 0]      [0] 
                    [0 1 0] x1 + [1] 
                    [0 0 0]      [0] 
                 =  f(a(b(x1)))      
        
* Step 4: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> d(x1)
        - Weak TRS:
            a(f(x1)) -> a(x1)
            b(a(x1)) -> a(b(x1))
            d(c(x1)) -> f(a(b(b(c(x1)))))
            d(f(x1)) -> f(a(b(x1)))
        - Signature:
            {a/1,b/1,d/1} / {c/1,f/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + Applied Processor:
        NaturalMI {miDimension = 4, miDegree = 2, 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 2 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 1 0]      [1]
                 [0 1 1 1] x1 + [0]
                 [0 0 0 0]      [0]
                 [0 0 0 0]      [0]
          p(b) = [1 1 0 0]      [0]
                 [0 1 0 0] x1 + [0]
                 [0 0 0 1]      [0]
                 [0 0 0 0]      [0]
          p(c) = [1 1 1 0]      [1]
                 [0 0 0 0] x1 + [0]
                 [0 0 0 0]      [0]
                 [0 0 0 0]      [1]
          p(d) = [1 1 0 1]      [0]
                 [0 1 0 1] x1 + [0]
                 [0 0 0 0]      [0]
                 [0 0 0 0]      [0]
          p(f) = [1 0 1 0]      [0]
                 [0 1 1 1] x1 + [1]
                 [0 0 0 1]      [0]
                 [0 0 0 0]      [0]
        
        Following rules are strictly oriented:
        a(b(x1)) = [1 1 0 1]      [1]
                   [0 1 0 1] x1 + [0]
                   [0 0 0 0]      [0]
                   [0 0 0 0]      [0]
                 > [1 1 0 1]      [0]
                   [0 1 0 1] x1 + [0]
                   [0 0 0 0]      [0]
                   [0 0 0 0]      [0]
                 = d(x1)             
        
        
        Following rules are (at-least) weakly oriented:
        a(f(x1)) =  [1 0 1 1]      [1]
                    [0 1 1 2] x1 + [1]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 >= [1 0 1 0]      [1]
                    [0 1 1 1] x1 + [0]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 =  a(x1)             
        
        b(a(x1)) =  [1 1 2 1]      [1]
                    [0 1 1 1] x1 + [0]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 >= [1 1 0 1]      [1]
                    [0 1 0 1] x1 + [0]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 =  a(b(x1))          
        
        d(c(x1)) =  [1 1 1 0]      [2]
                    [0 0 0 0] x1 + [1]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 >= [1 1 1 0]      [2]
                    [0 0 0 0] x1 + [1]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 =  f(a(b(b(c(x1))))) 
        
        d(f(x1)) =  [1 1 2 1]      [1]
                    [0 1 1 1] x1 + [1]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 >= [1 1 0 1]      [1]
                    [0 1 0 1] x1 + [1]
                    [0 0 0 0]      [0]
                    [0 0 0 0]      [0]
                 =  f(a(b(x1)))       
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(x1)) -> d(x1)
            a(f(x1)) -> a(x1)
            b(a(x1)) -> a(b(x1))
            d(c(x1)) -> f(a(b(b(c(x1)))))
            d(f(x1)) -> f(a(b(x1)))
        - Signature:
            {a/1,b/1,d/1} / {c/1,f/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))